Convex Optimization & Euclidean Distance Geometry

836
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY DATTORRO Mεβοο

Transcript of Convex Optimization & Euclidean Distance Geometry

Page 1: Convex Optimization & Euclidean Distance Geometry

CONVEX

OPTIMIZATION

&

EUCLIDEAN

DISTANCE

GEOMETRY

DATTORRO

Mεβοο

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Dattorro

CONVEX

OPTIMIZATION

&

EUCLIDEAN

DISTANCE

GEOMETRY

Meboo

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Convex Optimization

&

Euclidean Distance Geometry

Jon Dattorro

Mεβoo Publishing

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Meboo Publishing USA

PO Box 12

Palo Alto, California 94302

Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo, 2005, v2010.02.04.

ISBN 0976401304 (English) ISBN 9780615193687 (International II)

This is version 2010.02.04: available in print, as conceived, in color.

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This searchable electronic color pdfBook is click-navigable within the text bypage, section, subsection, chapter, theorem, example, definition, cross reference,citation, equation, figure, table, and hyperlink. A pdfBook has no electronic copyprotection and can be read and printed by most computers. The publisher herebygrants the right to reproduce this work in any format but limited to personal use.© 2001-2010 Meboo

All rights reserved.

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for Jennie Columba

Antonio♦

& Sze Wan

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Prelude

The constant demands of my department and university and theever increasing work needed to obtain funding have stolen much ofmy precious thinking time, and I sometimes yearn for the halcyondays of Bell Labs.

−Steven Chu, Nobel laureate [79]

Convex Analysis is the calculus of inequalities while Convex Optimization isits application. Analysis is inherently the domain of a mathematician whileOptimization belongs to the engineer. Practitioners in the art of ConvexOptimization engage themselves with discovery of which hard problems,perhaps previously believed nonconvex, can be transformed into convexequivalents; because once convex form of a problem is found, then a globallyoptimal solution is close at hand − the hard work is finished: Finding convexexpression of a problem is itself, in a very real sense, the solution.

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

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There is a great race under way to determine which important problemscan be posed in a convex setting. Yet, that skill acquired by understandingthe geometry and application of Convex Optimization will remain more anart for some time to come; the reason being, there is generally no uniquetransformation of a given problem to its convex equivalent. This means,two researchers pondering the same problem are likely to formulate a convexequivalent differently; hence, one solution is likely different from the otherfor the same problem, and any convex combination of those two solutionsremains optimal. Any presumption of only one right or correct solutionbecomes nebulous. Study of equivalence & sameness, uniqueness, and dualitytherefore pervade study of Optimization.

Tremendous benefit accrues when an optimization problem can betransformed to its convex equivalent, primarily because any locally optimalsolution is then guaranteed globally optimal. Solving a nonlinear system,for example, by instead solving an equivalent convex optimization problemis therefore highly preferable.0.1 Yet it can be difficult for the engineer toapply theory without an understanding of Analysis.

These pages comprise my journal over an eight year period bridginggaps between engineer and mathematician; they constitute a translation,unification, and cohering of about three hundred papers, books, and reportsfrom several different fields of mathematics and engineering. Beacons ofhistorical accomplishment are cited throughout. Much of what is written herewill not be found elsewhere. Care to detail, clarity, accuracy, consistency,and typography accompanies removal of ambiguity and verbosity out ofrespect for the reader. Consequently there is much cross-referencing andbackground material provided in the text, footnotes, and appendices so asto be self-contained and to provide understanding of fundamental concepts.

−Jon Dattorro

Stanford, California

2009

0.1That is what motivates a convex optimization known as geometric programming,invented in 1960s, [58] [77] which has driven great advances in the electronic circuit designindustry. [33, §4.7] [244] [380] [383] [98] [176] [186] [187] [188] [189] [190] [260] [261] [305]

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Convex Optimization&Euclidean Distance Geometry

1 Overview 21

2 Convex geometry 352.1 Convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Vectorized-matrix inner product . . . . . . . . . . . . . . . . . 502.3 Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4 Halfspace, Hyperplane . . . . . . . . . . . . . . . . . . . . . . 722.5 Subspace representations . . . . . . . . . . . . . . . . . . . . . 842.6 Extreme, Exposed . . . . . . . . . . . . . . . . . . . . . . . . . 902.7 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.8 Cone boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.9 Positive semidefinite (PSD) cone . . . . . . . . . . . . . . . . . 1102.10 Conic independence (c.i.) . . . . . . . . . . . . . . . . . . . . . 1332.11 When extreme means exposed . . . . . . . . . . . . . . . . . . 1392.12 Convex polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 1402.13 Dual cone & generalized inequality . . . . . . . . . . . . . . . 148

3 Geometry of convex functions 2113.1 Convex function . . . . . . . . . . . . . . . . . . . . . . . . . . 2123.2 Conic origins of Lagrangian . . . . . . . . . . . . . . . . . . . 2163.3 Practical norm functions, absolute value . . . . . . . . . . . . 2163.4 Inverted functions and roots . . . . . . . . . . . . . . . . . . . 2263.5 Affine function . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.6 Epigraph, Sublevel set . . . . . . . . . . . . . . . . . . . . . . 233

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3.7 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.8 Convex matrix-valued function . . . . . . . . . . . . . . . . . . 2533.9 Quasiconvex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2603.10 Salient properties . . . . . . . . . . . . . . . . . . . . . . . . . 262

4 Semidefinite programming 2654.1 Conic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2664.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2764.3 Rank reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2914.4 Rank-constrained semidefinite program . . . . . . . . . . . . . 3004.5 Constraining cardinality . . . . . . . . . . . . . . . . . . . . . 3244.6 Cardinality and rank constraint examples . . . . . . . . . . . . 3444.7 Convex Iteration rank-1 . . . . . . . . . . . . . . . . . . . . . 380

5 Euclidean Distance Matrix 3855.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3865.2 First metric properties . . . . . . . . . . . . . . . . . . . . . . 3875.3 ∃ fifth Euclidean metric property . . . . . . . . . . . . . . . . 3875.4 EDM definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3925.5 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4245.6 Injectivity of D & unique reconstruction . . . . . . . . . . . . 4295.7 Embedding in affine hull . . . . . . . . . . . . . . . . . . . . . 4355.8 Euclidean metric versus matrix criteria . . . . . . . . . . . . . 4405.9 Bridge: Convex polyhedra to EDMs . . . . . . . . . . . . . . . 4485.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 4555.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 4585.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 4665.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 4705.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 476

6 Cone of distance matrices 4876.1 Defining EDM cone . . . . . . . . . . . . . . . . . . . . . . . . 4896.2 Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 4916.3

√EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 493

6.4 A geometry of completion . . . . . . . . . . . . . . . . . . . . 4946.5 EDM definition in 11T . . . . . . . . . . . . . . . . . . . . . . 5006.6 Correspondence to PSD cone SN−1

+ . . . . . . . . . . . . . . . 5086.7 Vectorization & projection interpretation . . . . . . . . . . . . 514

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6.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 519

6.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 534

6.10 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

7 Proximity problems 537

7.1 First prevalent problem: . . . . . . . . . . . . . . . . . . . . . 545

7.2 Second prevalent problem: . . . . . . . . . . . . . . . . . . . . 556

7.3 Third prevalent problem: . . . . . . . . . . . . . . . . . . . . . 568

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

A Linear algebra 581

A.1 Main-diagonal δ operator, λ , trace, vec . . . . . . . . . . . . 581

A.2 Semidefiniteness: domain of test . . . . . . . . . . . . . . . . . 586

A.3 Proper statements . . . . . . . . . . . . . . . . . . . . . . . . . 589

A.4 Schur complement . . . . . . . . . . . . . . . . . . . . . . . . 601

A.5 Eigenvalue decomposition . . . . . . . . . . . . . . . . . . . . 606

A.6 Singular value decomposition, SVD . . . . . . . . . . . . . . . 610

A.7 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

B Simple matrices 623

B.1 Rank-one matrix (dyad) . . . . . . . . . . . . . . . . . . . . . 624

B.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 630

B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 632

B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 637

C Some analytical optimal results 641

C.1 Properties of infima . . . . . . . . . . . . . . . . . . . . . . . . 641

C.2 Trace, singular and eigen values . . . . . . . . . . . . . . . . . 642

C.3 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 649

C.4 Two-sided orthogonal Procrustes . . . . . . . . . . . . . . . . 651

C.5 Nonconvex quadratics . . . . . . . . . . . . . . . . . . . . . . 656

D Matrix calculus 657

D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 657

D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 678

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E Projection 687E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 691E.2 I −P , Projection on algebraic complement . . . . . . . . . . 696E.3 Symmetric idempotent matrices . . . . . . . . . . . . . . . . . 697E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 702E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 703E.6 Vectorization interpretation, . . . . . . . . . . . . . . . . . . . 709E.7 Projection on matrix subspaces . . . . . . . . . . . . . . . . . 717E.8 Range/Rowspace interpretation . . . . . . . . . . . . . . . . . 721E.9 Projection on convex set . . . . . . . . . . . . . . . . . . . . . 721E.10 Alternating projection . . . . . . . . . . . . . . . . . . . . . . 735

F Notation and a few definitions 755

Bibliography 773

Index 803

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List of Figures

1 Overview 211 Orion nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Application of trilateration is localization . . . . . . . . . . . . 233 Molecular conformation . . . . . . . . . . . . . . . . . . . . . . 244 Face recognition . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Swiss roll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 USA map reconstruction . . . . . . . . . . . . . . . . . . . . . 287 Honeycomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Robotic vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Reconstruction of David . . . . . . . . . . . . . . . . . . . . . 3410 David by distance geometry . . . . . . . . . . . . . . . . . . . 34

2 Convex geometry 3511 Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812 Open, closed, convex sets . . . . . . . . . . . . . . . . . . . . . 4013 Intersection of line with boundary . . . . . . . . . . . . . . . . 4114 Tangentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415 Inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816 Inverse image under linear map . . . . . . . . . . . . . . . . . 4917 Tesseract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218 Linear injective mapping of Euclidean body . . . . . . . . . . 5419 Linear noninjective mapping of Euclidean body . . . . . . . . 5520 Convex hull of a random list of points . . . . . . . . . . . . . . 6121 Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322 Two Fantopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623 Convex hull of rank-1 matrices . . . . . . . . . . . . . . . . . . 68

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14 LIST OF FIGURES

24 A simplicial cone . . . . . . . . . . . . . . . . . . . . . . . . . 7125 Hyperplane illustrated ∂H is a partially bounding line . . . . 7326 Hyperplanes in R2 . . . . . . . . . . . . . . . . . . . . . . . . 7527 Affine independence . . . . . . . . . . . . . . . . . . . . . . . . 7828 z∈ C | aTz = κi . . . . . . . . . . . . . . . . . . . . . . . . . 8029 Hyperplane supporting closed set . . . . . . . . . . . . . . . . 8130 Minimizing hyperplane over affine set in nonnegative orthant . 8831 Closed convex set illustrating exposed and extreme points . . 9232 Two-dimensional nonconvex cone . . . . . . . . . . . . . . . . 9433 Nonconvex cone made from lines . . . . . . . . . . . . . . . . 9534 Nonconvex cone is convex cone boundary . . . . . . . . . . . . 9535 Union of convex cones is nonconvex cone . . . . . . . . . . . . 9536 Truncated nonconvex cone X . . . . . . . . . . . . . . . . . . 9637 Cone exterior is convex cone . . . . . . . . . . . . . . . . . . . 9638 Not a cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9839 Minimum element. Minimal element . . . . . . . . . . . . . . 10140 K is a pointed polyhedral cone not full-dimensional . . . . . . 10541 Exposed and extreme directions . . . . . . . . . . . . . . . . . 10942 Positive semidefinite cone . . . . . . . . . . . . . . . . . . . . 11243 Convex Schur-form set . . . . . . . . . . . . . . . . . . . . . . 11444 Projection of truncated PSD cone . . . . . . . . . . . . . . . . 11745 Circular cone showing axis of revolution . . . . . . . . . . . . 12346 Circular section . . . . . . . . . . . . . . . . . . . . . . . . . . 12447 Polyhedral inscription . . . . . . . . . . . . . . . . . . . . . . 12648 Conically (in)dependent vectors . . . . . . . . . . . . . . . . . 13449 Pointed six-faceted polyhedral cone and its dual . . . . . . . . 13650 Minimal set of generators for halfspace about origin . . . . . . 13851 Venn diagram for cones and polyhedra . . . . . . . . . . . . . 14252 Unit simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14453 Two views of a simplicial cone and its dual . . . . . . . . . . . 14654 Two equivalent constructions of dual cone . . . . . . . . . . . 15055 Dual polyhedral cone construction by right angle . . . . . . . 15156 K is a halfspace about the origin . . . . . . . . . . . . . . . . 15257 Iconic primal and dual objective functions . . . . . . . . . . . 15358 Orthogonal cones . . . . . . . . . . . . . . . . . . . . . . . . . 15859 Blades K and K∗

. . . . . . . . . . . . . . . . . . . . . . . . . 16060 Membership relation w.r.t orthant . . . . . . . . . . . . . . . . 17061 Shrouded polyhedral cone . . . . . . . . . . . . . . . . . . . . 174

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LIST OF FIGURES 15

62 Simplicial cone K in R2 and its dual . . . . . . . . . . . . . . . 17963 Monotone nonnegative cone KM+ and its dual . . . . . . . . . 18964 Monotone cone KM and its dual . . . . . . . . . . . . . . . . . 19165 Two views of monotone cone KM and its dual . . . . . . . . . 19266 First-order optimality condition . . . . . . . . . . . . . . . . . 196

3 Geometry of convex functions 21167 Convex functions having unique minimizer . . . . . . . . . . . 21368 Minimum/Minimal element, dual cone characterization . . . . 21569 1-norm ball B1 from compressed sensing/compressive sampling 21970 Cardinality minimization, signed versus unsigned variable . . 22171 1-norm variants . . . . . . . . . . . . . . . . . . . . . . . . . . 22172 Affine function . . . . . . . . . . . . . . . . . . . . . . . . . . 23073 Supremum of affine functions . . . . . . . . . . . . . . . . . . 23274 Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23375 Gradient in R2 evaluated on grid . . . . . . . . . . . . . . . . 24176 Quadratic function convexity in terms of its gradient . . . . . 24977 Contour plot of convex real function at selected levels . . . . . 25078 Iconic quasiconvex function . . . . . . . . . . . . . . . . . . . 260

4 Semidefinite programming 26579 Venn diagram of convex program classes . . . . . . . . . . . . 26980 Visualizing positive semidefinite cone in high dimension . . . . 27281 Primal/Dual transformations . . . . . . . . . . . . . . . . . . 28282 Convex iteration versus projection . . . . . . . . . . . . . . . 30383 Trace heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 30484 Sensor-network localization . . . . . . . . . . . . . . . . . . . . 30885 2-lattice of sensors and anchors for localization example . . . . 31186 3-lattice of sensors and anchors for localization example . . . . 31287 4-lattice of sensors and anchors for localization example . . . . 31388 5-lattice of sensors and anchors for localization example . . . . 31489 Uncertainty ellipsoids orientation and eccentricity . . . . . . . 31590 2-lattice localization solution . . . . . . . . . . . . . . . . . . . 31891 3-lattice localization solution . . . . . . . . . . . . . . . . . . . 31892 4-lattice localization solution . . . . . . . . . . . . . . . . . . . 31993 5-lattice localization solution . . . . . . . . . . . . . . . . . . . 31994 10-lattice localization solution . . . . . . . . . . . . . . . . . . 320

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16 LIST OF FIGURES

95 100 randomized noiseless sensor localization . . . . . . . . . . 32096 100 randomized sensors localization . . . . . . . . . . . . . . . 32197 1-norm heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 32698 Sparse sampling theorem . . . . . . . . . . . . . . . . . . . . . 33099 Signal dropout . . . . . . . . . . . . . . . . . . . . . . . . . . 334100 Signal dropout reconstruction . . . . . . . . . . . . . . . . . . 335101 Simplex with intersecting line problem in compressed sensing . 337102 Geometric interpretations of sparse sampling . . . . . . . . . . 340103 Permutation matrix column-norm and column-sum constraint 348104 max cut problem . . . . . . . . . . . . . . . . . . . . . . . . 357105 Shepp-Logan phantom . . . . . . . . . . . . . . . . . . . . . . 362106 MRI radial sampling pattern in Fourier domain . . . . . . . . 367107 Aliased phantom . . . . . . . . . . . . . . . . . . . . . . . . . 369108 Neighboring-pixel stencil on Cartesian grid . . . . . . . . . . . 371109 Differentiable almost everywhere . . . . . . . . . . . . . . . . . 373110 MIT logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376111 One-pixel camera . . . . . . . . . . . . . . . . . . . . . . . . . 377112 One-pixel camera - compression estimates . . . . . . . . . . . 378

5 Euclidean Distance Matrix 385113 Convex hull of three points . . . . . . . . . . . . . . . . . . . . 386114 Complete dimensionless EDM graph . . . . . . . . . . . . . . 389115 Fifth Euclidean metric property . . . . . . . . . . . . . . . . . 391116 Fermat point . . . . . . . . . . . . . . . . . . . . . . . . . . . 400117 Arbitrary hexagon in R3 . . . . . . . . . . . . . . . . . . . . . 402118 Kissing number . . . . . . . . . . . . . . . . . . . . . . . . . . 403119 Trilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408120 This EDM graph provides unique isometric reconstruction . . 412121 Two sensors • and three anchors . . . . . . . . . . . . . . . 412122 Two discrete linear trajectories of sensors . . . . . . . . . . . . 413123 Coverage in cellular telephone network . . . . . . . . . . . . . 416124 Contours of equal signal power . . . . . . . . . . . . . . . . . . 416125 Depiction of molecular conformation . . . . . . . . . . . . . . 418126 Square diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 427127 Orthogonal complements in SN abstractly oriented . . . . . . . 430128 Elliptope E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449129 Elliptope E2 interior to S2

+ . . . . . . . . . . . . . . . . . . . . 450

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LIST OF FIGURES 17

130 Smallest eigenvalue of −V TNDVN . . . . . . . . . . . . . . . . 456

131 Some entrywise EDM compositions . . . . . . . . . . . . . . . 456132 Map of United States of America . . . . . . . . . . . . . . . . 469133 Largest ten eigenvalues of −V T

N OVN . . . . . . . . . . . . . . 472134 Relative-angle inequality tetrahedron . . . . . . . . . . . . . . 479135 Nonsimplicial pyramid in R3 . . . . . . . . . . . . . . . . . . . 483

6 Cone of distance matrices 487136 Relative boundary of cone of Euclidean distance matrices . . . 490137 Intersection of EDM cone with hyperplane . . . . . . . . . . . 492138 Neighborhood graph . . . . . . . . . . . . . . . . . . . . . . . 495139 Trefoil knot untied . . . . . . . . . . . . . . . . . . . . . . . . 497140 Trefoil ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 499141 Example of VX selection to make an EDM . . . . . . . . . . . 502142 Vector VX spirals . . . . . . . . . . . . . . . . . . . . . . . . . 505143 Three views of translated negated elliptope . . . . . . . . . . . 512144 Halfline T on PSD cone boundary . . . . . . . . . . . . . . . . 516145 Vectorization and projection interpretation example . . . . . . 517146 Orthogonal complement of geometric center subspace . . . . . 522147 EDM cone construction by flipping PSD cone . . . . . . . . . 523148 Decomposing a member of polar EDM cone . . . . . . . . . . 528149 Ordinary dual EDM cone projected on S3

h . . . . . . . . . . . 533

7 Proximity problems 537150 Pseudo-Venn diagram . . . . . . . . . . . . . . . . . . . . . . 540151 Elbow placed in path of projection . . . . . . . . . . . . . . . 541152 Convex envelope . . . . . . . . . . . . . . . . . . . . . . . . . 561

A Linear algebra 581153 Geometrical interpretation of full SVD . . . . . . . . . . . . . 613

B Simple matrices 623154 Four fundamental subspaces of any dyad . . . . . . . . . . . . 625155 Four fundamental subspaces of a doublet . . . . . . . . . . . . 629156 Four fundamental subspaces of elementary matrix . . . . . . . 630157 Gimbal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

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18 LIST OF FIGURES

D Matrix calculus 657158 Convex quadratic bowl in R2× R . . . . . . . . . . . . . . . . 668

E Projection 687159 Action of pseudoinverse . . . . . . . . . . . . . . . . . . . . . . 688160 Nonorthogonal projection of x∈R3 on R(U)= R2 . . . . . . . 694161 Biorthogonal expansion of point x∈ aff K . . . . . . . . . . . . 705162 Dual interpretation of projection on convex set . . . . . . . . . 724163 Projection product on convex set in subspace . . . . . . . . . 733164 von Neumann-style projection of point b . . . . . . . . . . . . 736165 Alternating projection on two halfspaces . . . . . . . . . . . . 737166 Distance, optimization, feasibility . . . . . . . . . . . . . . . . 739167 Alternating projection on nonnegative orthant and hyperplane 742168 Geometric convergence of iterates in norm . . . . . . . . . . . 742169 Distance between PSD cone and iterate in A . . . . . . . . . . 747170 Dykstra’s alternating projection algorithm . . . . . . . . . . . 748171 Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . 750172 Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . . 751

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List of Tables2 Convex geometry

Table 2.9.2.3.1, rank versus dimension of S3

+ faces 119

Table 2.10.0.0.1, maximum number of c.i. directions 135

Cone Table 1 186

Cone Table S 187

Cone Table A 188

Cone Table 1* 193

4 Semidefinite programmingFaces of S3

+ corresponding to faces of S3

+ 273

5 Euclidean Distance MatrixPrecis 5.7.2: affine dimension in terms of rank 438

B Simple matricesAuxiliary V -matrix Table B.4.4 636

D Matrix calculusTable D.2.1, algebraic gradients and derivatives 679

Table D.2.2, trace Kronecker gradients 680

Table D.2.3, trace gradients and derivatives 681

Table D.2.4, logarithmic determinant gradients, derivatives 683

Table D.2.5, determinant gradients and derivatives 684

Table D.2.6, logarithmic derivatives 684

Table D.2.7, exponential gradients and derivatives 685© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

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Chapter 1

Overview

Convex Optimization

Euclidean Distance Geometry

People are so afraid of convex analysis.

−Claude Lemarechal, 2003

In layman’s terms, the mathematical science of Optimization is the studyof how to make a good choice when confronted with conflicting requirements.The qualifier convex means: when an optimal solution is found, then it isguaranteed to be a best solution; there is no better choice.

Any convex optimization problem has geometric interpretation. If a givenoptimization problem can be transformed to a convex equivalent, then thisinterpretive benefit is acquired. That is a powerful attraction: the ability tovisualize geometry of an optimization problem. Conversely, recent advancesin geometry and in graph theory hold convex optimization within their proofs’core. [391] [315]

This book is about convex optimization, convex geometry (withparticular attention to distance geometry), and nonconvex, combinatorial,and geometrical problems that can be relaxed or transformed into convexproblems. A virtual flood of new applications follows by epiphany that manyproblems, presumed nonconvex, can be so transformed. [10] [11] [57] [91][145] [147] [273] [294] [302] [358] [359] [388] [391] [33, §4.3, p.316-322]© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

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22 CHAPTER 1. OVERVIEW

Figure 1: Orion nebula. (astrophotography by Massimo Robberto)

Euclidean distance geometry is, fundamentally, a determination of pointconformation (configuration, relative position or location) by inference frominterpoint distance information. By inference we mean: e.g., given onlydistance information, determine whether there corresponds a realizableconformation of points; a list of points in some dimension that attains thegiven interpoint distances. Each point may represent simply location or,abstractly, any entity expressible as a vector in finite-dimensional Euclideanspace; e.g., distance geometry of music [104].

It is a common misconception to presume that some desired pointconformation cannot be recovered in absence of complete interpoint distanceinformation. We might, for example, want to realize a constellation givenonly interstellar distance (or, equivalently, parsecs from our Sun and relativeangular measurement; the Sun as vertex to two distant stars); called stellarcartography, an application evoked by Figure 1. At first it may seem O(N 2)data is required, yet there are many circumstances where this can be reducedto O(N ).

If we agree that a set of points can have a shape (three points can form

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x2

x4

x3

Figure 2: Application of trilateration (§5.4.2.3.5) is localization (determiningposition) of a radio signal source in 2 dimensions; more commonly known byradio engineers as the process “triangulation”. In this scenario, anchorsx2 , x3 , x4 are illustrated as fixed antennae. [205] The radio signal source(a sensor • x1) anywhere in affine hull of three antenna bases can beuniquely localized by measuring distance to each (dashed white arrowed linesegments). Ambiguity of lone distance measurement to sensor is representedby circle about each antenna. Trilateration is expressible as a semidefiniteprogram; hence, a convex optimization problem. [316]

a triangle and its interior, for example, four points a tetrahedron), thenwe can ascribe shape of a set of points to their convex hull. It should beapparent: from distance, these shapes can be determined only to within arigid transformation (rotation, reflection, translation).

Absolute position information is generally lost, given only distanceinformation, but we can determine the smallest possible dimension in whichan unknown list of points can exist; that attribute is their affine dimension(a triangle in any ambient space has affine dimension 2, for example). Incircumstances where stationary reference points are also provided, it becomespossible to determine absolute position or location; e.g., Figure 2.

23

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24 CHAPTER 1. OVERVIEW

Figure 3: [185] [113] Distance data collected via nuclear magnetic resonance(NMR) helped render this 3-dimensional depiction of a protein molecule.At the beginning of the 1980s, Kurt Wuthrich [Nobel laureate] developedan idea about how NMR could be extended to cover biological moleculessuch as proteins. He invented a systematic method of pairing each NMRsignal with the right hydrogen nucleus (proton) in the macromolecule. Themethod is called sequential assignment and is today a cornerstone of all NMRstructural investigations. He also showed how it was subsequently possible todetermine pairwise distances between a large number of hydrogen nuclei anduse this information with a mathematical method based on distance-geometryto calculate a three-dimensional structure for the molecule. −[277] [375] [179]

Geometric problems involving distance between points can sometimes bereduced to convex optimization problems. Mathematics of this combinedstudy of geometry and optimization is rich and deep. Its applicationhas already proven invaluable discerning organic molecular conformationby measuring interatomic distance along covalent bonds; e.g., Figure 3.[86] [348] [136] [45] Many disciplines have already benefitted and simplifiedconsequent to this theory; e.g., distance-based pattern recognition (Figure 4),localization in wireless sensor networks by measurement of intersensordistance along channels of communication, [46] [386] [44] wireless locationof a radio-signal source such as a cell phone by multiple measurements ofsignal strength, the global positioning system (GPS), and multidimensionalscaling which is a numerical representation of qualitative data by finding alow-dimensional scale.

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25

Figure 4: This coarsely discretized triangulated algorithmically flattenedhuman face (made by Kimmel & the Bronsteins [222]) represents a stagein machine recognition of human identity; called face recognition. Distancegeometry is applied to determine discriminating-features.

Euclidean distance geometry together with convex optimization have alsofound application in artificial intelligence: to machine learning by discerning naturally occurring manifolds in:

– Euclidean bodies (Figure 5, §6.4.0.0.1),

– Fourier spectra of kindred utterances [208],

– and photographic image sequences [369], to robotics ; e.g., automatic control of vehicles maneuvering information. (Figure 8)

by chapter

We study pervasive convex Euclidean bodies; their many manifestationsand representations. In particular, we make convex polyhedra, cones, anddual cones visceral through illustration in chapter 2 Convex geometrywhere geometric relation of polyhedral cones to nonorthogonal bases

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26 CHAPTER 1. OVERVIEW

Figure 5: Swiss roll from Weinberger & Saul [369]. The problem of manifoldlearning, illustrated for N = 800 data points sampled from a “Swiss roll” 1O.A discretized manifold is revealed by connecting each data point and its k=6nearest neighbors 2O. An unsupervised learning algorithm unfolds the Swissroll while preserving the local geometry of nearby data points 3O. Finally, thedata points are projected onto the two dimensional subspace that maximizestheir variance, yielding a faithful embedding of the original manifold 4O.

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27

(biorthogonal expansion) is examined. It is shown that coordinates areunique in any conic system whose basis cardinality equals or exceedsspatial dimension. The conic analogue to linear independence, called conicindependence, is introduced as a tool for study, analysis, and manipulation ofcones; a natural extension and next logical step in progression: linear, affine,conic. We explain conversion between halfspace- and vertex-description of aconvex cone, we motivate the dual cone and provide formulae for finding it,and we show how first-order optimality conditions or alternative systemsof linear inequality or linear matrix inequality can be explained by dualgeneralized inequalities with respect to convex cones. Arcane theorems ofalternative generalized inequality are, in fact, simply derived from conemembership relations ; generalizations of algebraic Farkas’ lemma translatedto geometry of convex cones.

Any convex optimization problem can be visualized geometrically. Desireto visualize in high dimension [Sagan, Cosmos−The Edge of Forever, 22:55′]is deeply embedded in the mathematical psyche. [1] Chapter 2 providestools to make visualization easier, and we teach how to visualize in highdimension. The concepts of face, extreme point, and extreme direction of aconvex Euclidean body are explained here; crucial to understanding convexoptimization. How to find the smallest face of any pointed closed convexcone containing convex set C , for example, is divulged; later shown to havepractical application to presolving convex programs. The convex cone ofpositive semidefinite matrices, in particular, is studied in depth: We interpret, for example, inverse image of the positive semidefinite

cone under affine transformation. (Example 2.9.1.0.2) Subsets of the positive semidefinite cone, discriminated by rankexceeding some lower bound, are convex. In other words, high-ranksubsets of the positive semidefinite cone boundary united with itsinterior are convex. (Theorem 2.9.2.6.3) There is a closed form forprojection on those convex subsets. The positive semidefinite cone is a circular cone in low dimension,while Gersgorin discs specify inscription of a polyhedral cone into thatpositive semidefinite cone. (Figure 47)

Chapter 3 Geometry of convex functions observes Fenchel’s analogybetween convex sets and functions: We explain, for example, how the realaffine function relates to convex functions as the hyperplane relates to convex

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28 CHAPTER 1. OVERVIEW

original reconstruction

Figure 6: About five thousand points along borders constituting UnitedStates were used to create an exhaustive matrix of interpoint distance for eachand every pair of points in an ordered set (a list); called Euclidean distancematrix. From that noiseless distance information, it is easy to reconstructmap exactly via Schoenberg criterion (886). (§5.13.1.0.1, confer Figure 132)Map reconstruction is exact (to within a rigid transformation) given anynumber of interpoint distances; the greater the number of distances, thegreater the detail (as it is for all conventional map preparation).

sets. Partly a toolbox of practical useful convex functions and a cookbookfor optimization problems, methods are drawn from the appendices aboutmatrix calculus for determining convexity and discerning geometry.

Semidefinite programming is reviewed in chapter 4 with particularattention to optimality conditions for prototypical primal and dual problems,their interplay, and a perturbation method for rank reduction of optimalsolutions (extant but not well known). Positive definite Farkas’ lemmais derived, and we also show how to determine if a feasible set belongsexclusively to a positive semidefinite cone boundary. An arguably goodthree-dimensional polyhedral analogue to the positive semidefinite cone of3×3 symmetric matrices is introduced: a new tool for visualizing coexistenceof low- and high-rank optimal solutions in six isomorphic dimensions anda mnemonic aid for understanding semidefinite programs. We find a

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29

minimum-cardinality Boolean solution to an instance of Ax = b :

minimizex

‖x‖0

subject to Ax = b

xi ∈ 0, 1 , i=1 . . . n

(656)

The sensor-network localization problem is solved in any dimension in thischapter. We introduce a method of convex iteration for constraining rankin the form rank G≤ ρ and cardinality in the form cardx≤ k . Cardinalityminimization is applied to a discrete image-gradient of the Shepp-Loganphantom, from Magnetic Resonance Imaging (MRI) in the field of medicalimaging, for which we find a new lower bound of 1.9% cardinality. Weshow how to handle polynomial constraints, and how to transform arank-constrained problem to a rank-1 problem.

The EDM is studied in chapter 5 Euclidean Distance Matrix; itsproperties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical properties of Euclidean metric;thereby, observing existence of an infinity of properties of the Euclideanmetric beyond triangle inequality. We proceed by deriving the fifth Euclideanmetric property and then explain why furthering this endeavor is inefficientbecause the ensuing criteria (while describing polyhedra in angle or area,volume, content, and so on ad infinitum) grow linearly in complexity andnumber with problem size.

Reconstruction methods are explained and applied to a map of theUnited States; e.g., Figure 6. We also experimentally test a conjecture ofBorg & Groenen by reconstructing a distorted but recognizable isotonic mapof the USA using only ordinal (comparative) distance data: Figure 132e-f.We demonstrate an elegant method for including dihedral (or torsion)angle constraints into a molecular conformation problem. We explain whytrilateration (a.k.a, localization) is a convex optimization problem. Weshow how to recover relative position given incomplete interpoint distanceinformation, and how to pose EDM problems or transform geometricalproblems to convex optimizations; e.g., kissing number of packed spheresabout a central sphere (solved in R3 by Isaac Newton).

The set of all Euclidean distance matrices forms a pointed closed convexcone called the EDM cone: EDMN . We offer a new proof of Schoenberg’s

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30 CHAPTER 1. OVERVIEW

Figure 7: These bees construct a honeycomb by solving a convex optimizationproblem (§5.4.2.3.4). The most dense packing of identical spheres about acentral sphere in 2 dimensions is 6. Sphere centers describe a regular lattice.

seminal characterization of EDMs:

D ∈ EDMN ⇔

−V TNDVN º 0

D ∈ SNh

(886)

Our proof relies on fundamental geometry; assuming, any EDM mustcorrespond to a list of points contained in some polyhedron (possibly atits vertices) and vice versa. It is known, but not obvious, this Schoenbergcriterion implies nonnegativity of the EDM entries; proved herein.

We characterize the eigenvalue spectrum of an EDM, and then devisea polyhedral spectral cone for determining membership of a given matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices;id est, a matrix is an EDM if and only if its nonincreasingly ordered vectorof eigenvalues belongs to a polyhedral spectral cone for EDMN ;

D ∈ EDMN ⇔

λ

([

0 1T

1 −D

])

∈[

RN+

R−

]

∩ ∂H

D ∈ SNh

(1101)

We will see: spectral cones are not unique.In chapter 6 Cone of distance matrices we explain a geometric

relationship between the cone of Euclidean distance matrices, two positivesemidefinite cones, and the elliptope. We illustrate geometric requirements,in particular, for projection of a given matrix on a positive semidefinite conethat establish its membership to the EDM cone. The faces of the EDM coneare described, but still open is the question whether all its faces are exposedas they are for the positive semidefinite cone.

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31

Figure 8: Robotic vehicles in concert can move larger objects, guard aperimeter, perform surveillance, find land mines, or localize a plume of gas,liquid, or radio waves. [135]

The Schoenberg criterion (886) for identifying a Euclidean distancematrix is revealed to be a discretized membership relation (dual generalizedinequalities, a new Farkas’-like lemma) between the EDM cone and its

ordinary dual: EDMN∗. A matrix criterion for membership to the dual

EDM cone is derived that is simpler than the Schoenberg criterion:

D∗∈ EDMN∗ ⇔ δ(D∗1) − D∗ º 0 (1250)

There is a concise equality, relating the convex cone of Euclidean distancematrices to the positive semidefinite cone, apparently overlooked in theliterature; an equality between two large convex Euclidean bodies:

EDMN = SNh ∩

(

SN⊥c − SN

+

)

(1244)

In chapter 7 Proximity problems we explore methods of solutionto a few fundamental and prevalent Euclidean distance matrix proximityproblems; the problem of finding that distance matrix closest, in some sense,

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32 CHAPTER 1. OVERVIEW

to a given matrix H :

minimizeD

‖−V (D − H)V ‖2F

subject to rank V DV ≤ ρ

D ∈ EDMN

minimize√

D‖ √

D − H‖2F

subject to rank V DV ≤ ρ√

D ∈√

EDMN

minimizeD

‖D − H‖2F

subject to rank V DV ≤ ρ

D ∈ EDMN

minimize√

D‖−V (

√D − H)V ‖2

F

subject to rank V DV ≤ ρ√

D ∈√

EDMN

(1286)

We apply a convex iteration method for constraining rank. Known heuristicsfor rank minimization are also explained. We offer new geometrical proof, ofa famous discovery by Eckart & Young in 1936 [126], with particular regardto Euclidean projection of a point on that generally nonconvex subset ofthe positive semidefinite cone boundary comprising all semidefinite matriceshaving rank not exceeding a prescribed bound ρ . We explain how thisproblem is transformed to a convex optimization for any rank ρ .

appendices

Toolboxes are provided so as to be more self-contained: linear algebra (appendix A is primarily concerned with properstatements of semidefiniteness for square matrices), simple matrices (dyad, doublet, elementary, Householder, Schoenberg,orthogonal, etcetera, in appendix B), collection of known analytical solutions to some important optimizationproblems (appendix C), matrix calculus remains somewhat unsystematized when comparedto ordinary calculus (appendix D concerns matrix-valued functions,matrix differentiation and directional derivatives, Taylor series, andtables of first- and second-order gradients and matrix derivatives),

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33 an elaborate exposition offering insight into orthogonal andnonorthogonal projection on convex sets (the connection betweenprojection and positive semidefiniteness, for example, or betweenprojection and a linear objective function in appendix E), Matlab code on Wıκımization to discriminate EDMs, to determineconic independence, to reduce or constrain rank of an optimal solutionto a semidefinite program, compressed sensing (compressive sampling)for digital image and audio signal processing, and two distinct methodsof reconstructing a map of the United States: one given only distancedata, the other given only comparative distance data.

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34 CHAPTER 1. OVERVIEW

Figure 9: Three-dimensional reconstruction of David from distance data.

Figure 10: Digital Michelangelo Project , Stanford University. Measuringdistance to David by laser rangefinder. Spatial resolution is 0.29mm.

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Chapter 2

Convex geometry

Convexity has an immensely rich structure and numerousapplications. On the other hand, almost every “convex” idea canbe explained by a two-dimensional picture.

−Alexander Barvinok [26, p.vii]

As convex geometry and linear algebra are inextricably bonded by asymmetry(linear inequality), we provide much background material on linear algebra(especially in the appendices) although a reader is assumed comfortable with[328] [330] [199] or any other intermediate-level text. The essential referencesto convex analysis are [196] [303]. The reader is referred to [327] [26] [368][41] [59] [300] [355] for a comprehensive treatment of convexity. There isrelatively less published pertaining to convex matrix-valued functions. [211][200, §6.6] [291]

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

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36 CHAPTER 2. CONVEX GEOMETRY

2.1 Convex set

A set C is convex iff for all Y , Z∈ C and 0≤µ≤1

µY + (1 − µ)Z ∈ C (1)

Under that defining condition on µ , the linear sum in (1) is called a convexcombination of Y and Z . If Y and Z are points in real finite-dimensionalEuclidean vector space Rn or Rm×n (matrices), then (1) represents theclosed line segment joining them. All line segments are thereby convex sets.Apparent from the definition, a convex set is a connected set. [254, §3.4, §3.5][41, p.2] A convex set can, but does not necessarily, contain the origin 0.

An ellipsoid centered at x = a (Figure 13, p.41), given matrix C∈ Rm×n

x∈Rn | ‖C(x − a)‖2 = (x − a)TCTC(x − a) ≤ 1 (2)

is a good icon for a convex set.2.1

2.1.1 subspace

A nonempty subset R of real Euclidean vector space Rn is called a subspace(§2.5) if every vector2.2 of the form αx + βy , for α, β∈R , is in R whenevervectors x and y are. [246, §2.3] A subspace is a convex set containing theorigin, by definition. [303, p.4] Any subspace is therefore open in the sensethat it contains no boundary, but closed in the sense [254, §2]

R + R = R (3)

It is not difficult to showR = −R (4)

as is true for any subspace R , because x∈R ⇔ −x∈R . Given any x∈RR = x + R (5)

The intersection of an arbitrary collection of subspaces remains asubspace. Any subspace not constituting the entire ambient vector space Rn

is a proper subspace; e.g.,2.3 any line through the origin in two-dimensionalEuclidean space R2. The vector space Rn is itself a conventional subspace,inclusively, [223, §2.1] although not proper.

2.1This particular definition is slablike (Figure 11) in Rn when C has nontrivial nullspace.2.2A vector is assumed, throughout, to be a column vector.2.3We substitute abbreviation e.g. in place of the Latin exempli gratia.

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2.1. CONVEX SET 37

2.1.2 linear independence

Arbitrary given vectors in Euclidean space Γi∈ Rn, i=1 . . . N are linearlyindependent (l.i.) if and only if, for all ζ∈ RN

Γ1 ζ1 + · · · + ΓN−1 ζN−1 − ΓN ζN = 0 (6)

has only the trivial solution ζ = 0 ; in other words, iff no vector from thegiven set can be expressed as a linear combination of those remaining.Geometrically, two nontrivial vector subspaces are linearly independent iffthey intersect only at the origin.

2.1.2.1 preservation of linear independence

(confer §2.4.2.4, §2.10.1) Linear transformation preserves linear dependence.[223, p.86] Conversely, linear independence can be preserved under lineartransformation. Given Y = [ y1 y2 · · · yN ]∈ RN×N , consider the mapping

T (Γ) : Rn×N → Rn×N , Γ Y (7)

whose domain is the set of all matrices Γ∈Rn×N holding a linearlyindependent set columnar. Linear independence of Γyi∈ Rn, i=1 . . . Ndemands, by definition, there exist no nontrivial solution ζ∈ RN to

Γy1 ζi + · · · + ΓyN−1 ζN−1 − ΓyN ζN = 0 (8)

By factoring out Γ , we see that triviality is ensured by linear independenceof yi∈ RN.

2.1.3 Orthant:

name given to a closed convex set that is the higher-dimensionalgeneralization of quadrant from the classical Cartesian partition of R2 ; aCartesian cone. The most common is the nonnegative orthant Rn

+ or Rn×n+

(analogue to quadrant I) to which membership denotes nonnegative vector-or matrix-entries respectively; e.g.,

Rn+ , x∈Rn | xi≥ 0 ∀ i (9)

The nonpositive orthant Rn− or Rn×n

− (analogue to quadrant III) denotesnegative and 0 entries. Orthant convexity2.4 is easily verified bydefinition (1).

2.4All orthants are selfdual simplicial cones. (§2.13.5.1, §2.12.3.1.1)

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38 CHAPTER 2. CONVEX GEOMETRY

Figure 11: A slab is a convex Euclidean body infinite in extent but notaffine. Illustrated in R2, it may be constructed by intersecting two opposinghalfspaces whose bounding hyperplanes are parallel but not coincident.Because number of halfspaces used in its construction is finite, slab is apolyhedron (§2.12). (Cartesian axes + and vector inward-normal, to eachhalfspace-boundary, are drawn for reference.)

2.1.4 affine set

A nonempty affine set (from the word affinity) is any subset of Rn that is atranslation of some subspace. Any affine set is convex and open so containsno boundary: e.g., empty set ∅ , point, line, plane, hyperplane (§2.4.2),subspace, etcetera. For some parallel 2.5 subspace R and any point x ∈ A

A is affine ⇔ A = x + R= y | y − x∈R

(10)

The intersection of an arbitrary collection of affine sets remains affine. Theaffine hull of a set C⊆Rn (§2.3.1) is the smallest affine set containing it.

2.1.5 dimension

Dimension of an arbitrary set S is the dimension of its affine hull; [368, p.14]

dimS , dim aff S = dim aff(S − s) , s∈S (11)

the same as dimension of the subspace parallel to that affine set aff S whennonempty. Hence dimension (of a set) is synonymous with affine dimension.[196, A.2.1]

2.5Two affine sets are said to be parallel when one is a translation of the other. [303, p.4]

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2.1. CONVEX SET 39

2.1.6 empty set versus empty interior

Emptiness ∅ of a set is handled differently than interior in the classicalliterature. It is common for a nonempty convex set to have empty interior;e.g., paper in the real world: An ordinary flat sheet of paper is a nonempty convex set having empty

interior in R3 but nonempty interior relative to its affine hull.

2.1.6.1 relative interior

Although it is always possible to pass to a smaller ambient Euclidean spacewhere a nonempty set acquires an interior [26, §II.2.3], we prefer the qualifierrelative which is the conventional fix to this ambiguous terminology.2.6 Sowe distinguish interior from relative interior throughout: [327] [368] [355] Classical interior int C is defined as a union of points: x is an interior

point of C⊆ Rn if there exists an open ball of dimension n and nonzeroradius centered at x that is contained in C . Relative interior rel int C of a convex set C⊆Rn is interior relative toits affine hull.2.7

Thus defined, it is common (though confusing) for int C the interior of C tobe empty while its relative interior is not: this happens whenever dimensionof its affine hull is less than dimension of the ambient space (dim aff C< n ;e.g., were C paper) or in the exception when C is a single point; [254, §2.2.1]

rel intx , affx = x , intx = ∅ , x∈Rn (12)

In any case, closure of the relative interior of a convex set C always yieldsclosure of the set itself;

rel int C = C (13)

Closure is invariant to translation. If C is convex then rel int C and C areconvex. [196, p.24] If C has nonempty interior, then

rel int C= int C (14)

2.6Superfluous mingling of terms as in relatively nonempty set would be an unfortunateconsequence. From the opposite perspective, some authors use the term full orfull-dimensional to describe a set having nonempty interior.2.7Likewise for relative boundary (§2.1.7.2), although relative closure is superfluous.

[196, §A.2.1]

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40 CHAPTER 2. CONVEX GEOMETRY

(a)

(b)

(c)

R2

Figure 12: (a) Closed convex set. (b) Neither open, closed, or convex. YetPSD cone can remain convex in absence of certain boundary components(§2.9.2.6.3). Nonnegative orthant with origin excluded (§2.6) and positiveorthant with origin adjoined [303, p.49] are convex. (c) Open convex set.

Given the intersection of convex set C with affine set Arel int(C ∩ A) = rel int(C) ∩ A ⇐ rel int(C) ∩ A 6= ∅ (15)

Because an affine set A is open

rel intA = A (16)

2.1.7 classical boundary

(confer §2.1.7.2) Boundary of a set C is the closure of C less its interior;

∂ C = C \ int C (17)

[53, §1.1] which follows from the fact

int C = C (18)

and presumption of nonempty interior.2.8 Implications are:

2.8Otherwise, for x∈Rn as in (12), [254, §2.1-§2.3]

intx = ∅ = ∅

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2.1. CONVEX SET 41

(a)

(b)

(c)

R

R2

R3

Figure 13: (a) Ellipsoid in R is a line segment whose boundary comprises twopoints. Intersection of line with ellipsoid in R , (b) in R2, (c) in R3. Eachellipsoid illustrated has entire boundary constituted by zero-dimensionalfaces; in fact, by vertices (§2.6.1.0.1). Intersection of line with boundaryis a point at entry to interior. These same facts hold in higher dimension. int C = C \∂ C a bounded open set has boundary defined but not contained in the set interior of an open set is equivalent to the set itself;

from which an open set is defined: [254, p.109]

C is open ⇔ int C = C (19)

C is closed ⇔ int C = C (20)

The set illustrated in Figure 12b is not open because it is not equivalentto its interior, for example, it is not closed because it does not containits boundary, and it is not convex because it does not contain all convexcombinations of its boundary points.

the empty set is both open and closed.

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42 CHAPTER 2. CONVEX GEOMETRY

2.1.7.1 Line intersection with boundary

A line can intersect the boundary of a convex set in any dimension at apoint demarcating the line’s entry to the set interior. On one side of thatentry-point along the line is the exterior of the set, on the other side is theset interior. In other words, starting from any point of a convex set, a move toward the interior is

an immediate entry into the interior. [26, §II.2]

When a line intersects the interior of a convex body in any dimension, theboundary appears to the line to be as thin as a point. This is intuitivelyplausible because, for example, a line intersects the boundary of the ellipsoidsin Figure 13 at a point in R , R2, and R3. Such thinness is a remarkablefact when pondering visualization of convex polyhedra (§2.12, §5.14.3) infour dimensions, for example, having boundaries constructed from otherthree-dimensional convex polyhedra called faces.

We formally define face in (§2.6). For now, we observe the boundary of aconvex body to be entirely constituted by all its faces of dimension lower thanthe body itself. For example: The ellipsoids in Figure 13 have boundariescomposed only of zero-dimensional faces. The two-dimensional slab inFigure 11 is an unbounded polyhedron having one-dimensional faces makingits boundary. The three-dimensional bounded polyhedron in Figure 20 haszero-, one-, and two-dimensional polygonal faces constituting its boundary.

2.1.7.1.1 Example. Intersection of line with boundary in R6.The convex cone of positive semidefinite matrices S3

+ (§2.9) in the ambientsubspace of symmetric matrices S3 (§2.2.2.0.1) is a six-dimensional Euclideanbody in isometrically isomorphic R6 (§2.2.1). The boundary of thepositive semidefinite cone in this dimension comprises faces having only thedimensions 0, 1, and 3 ; id est, ρ(ρ+1)/2, ρ=0, 1, 2.

Unique minimum-distance projection PX (§E.9) of any point X∈ S3 onthat cone S3

+ is known in closed form (§7.1.2). Given, for example, λ∈ int R3

+

and diagonalization (§A.5.1) of exterior point

X = QΛQT∈ S3, Λ ,

λ1 0λ2

0 −λ3

(21)

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2.1. CONVEX SET 43

where Q∈R3×3 is an orthogonal matrix, then the projection on S3

+ in R6 is

PX = Q

λ1 0λ2

0 0

QT∈ S3

+ (22)

This positive semidefinite matrix PX nearest X thus has rank 2, found bydiscarding all negative eigenvalues. The line connecting these two points isX + (PX−X)t | t∈R where t=0 ⇔ X and t=1 ⇔ PX . Becausethis line intersects the boundary of the positive semidefinite cone S3

+ atpoint PX and passes through its interior (by assumption), then the matrixcorresponding to an infinitesimally positive perturbation of t there shouldreside interior to the cone (rank 3). Indeed, for ε an arbitrarily small positiveconstant,

X + (PX−X)t|t=1+ε = Q(Λ+(PΛ−Λ)(1+ε))QT = Q

λ1 0λ2

0 ελ3

QT∈ int S3

+

(23)2

2.1.7.1.2 Example. Tangential line intersection with boundary.A higher-dimensional boundary ∂C of a convex Euclidean body C is simplya dimensionally larger set through which a line can pass when it does notintersect the body’s interior. Still, for example, a line existing in five ormore dimensions may pass tangentially (intersecting no point interior to C[215, §15.3]) through a single point relatively interior to a three-dimensionalface on ∂C . Let’s understand why by inductive reasoning.

Figure 14a shows a vertical line-segment whose boundary comprisesits two endpoints. For a line to pass through the boundary tangentially(intersecting no point relatively interior to the line-segment), it must exist inan ambient space of at least two dimensions. Otherwise, the line is confinedto the same one-dimensional space as the line-segment and must pass alongthe segment to reach the end points.

Figure 14b illustrates a two-dimensional ellipsoid whose boundary isconstituted entirely by zero-dimensional faces. Again, a line must exist inat least two dimensions to tangentially pass through any single arbitrarilychosen point on the boundary (without intersecting the ellipsoid interior).

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44 CHAPTER 2. CONVEX GEOMETRY

R2

(a) (b)

(c) (d)

R3

Figure 14: Line tangential: (a) (b) to relative interior of a zero-dimensionalface in R2, (c) (d) to relative interior of a one-dimensional face in R3.

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2.1. CONVEX SET 45

Now let’s move to an ambient space of three dimensions. Figure 14cshows a polygon rotated into three dimensions. For a line to pass through itszero-dimensional boundary (one of its vertices) tangentially, it must existin at least the two dimensions of the polygon. But for a line to passtangentially through a single arbitrarily chosen point in the relative interiorof a one-dimensional face on the boundary as illustrated, it must exist in atleast three dimensions.

Figure 14d illustrates a solid circular cone (drawn truncated) whoseone-dimensional faces are halflines emanating from its pointed end (vertex ).This cone’s boundary is constituted solely by those one-dimensional halflines.A line may pass through the boundary tangentially, striking only onearbitrarily chosen point relatively interior to a one-dimensional face, if itexists in at least the three-dimensional ambient space of the cone.

From these few examples, way deduce a general rule (without proof): A line may pass tangentially through a single arbitrarily chosen pointrelatively interior to a k-dimensional face on the boundary of a convexEuclidean body if the line exists in dimension at least equal to k+2.

Now the interesting part, with regard to Figure 20 showing a boundedpolyhedron in R3 ; call it P : A line existing in at least four dimensionsis required in order to pass tangentially (without hitting intP) through asingle arbitrary point in the relative interior of any two-dimensional polygonalface on the boundary of polyhedron P . Now imagine that polyhedron P isitself a three-dimensional face of some other polyhedron in R4. To pass aline tangentially through polyhedron P itself, striking only one point fromits relative interior rel intP as claimed, requires a line existing in at leastfive dimensions.2.9

It is not too difficult to deduce: A line may pass through a single arbitrarily chosen point interior to ak-dimensional convex Euclidean body (hitting no other interior point)if that line exists in dimension at least equal to k+1.

In layman’s terms, this means: a being capable of navigating four spatialdimensions (one dimension beyond our physical reality) could see insidethree-dimensional objects. 2

2.9This rule can help determine whether there exists unique solution to a convexoptimization problem whose feasible set is an intersecting line; e.g., the trilaterationproblem (§5.4.2.3.5).

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46 CHAPTER 2. CONVEX GEOMETRY

2.1.7.2 Relative boundary

The classical definition of boundary of a set C presumes nonempty interior:

∂ C = C \ int C (17)

More suitable to study of convex sets is the relative boundary ; defined[196, §A.2.1.2]

rel ∂ C , C \ rel int C (24)

boundary relative to affine hull of C .In the exception when C is a single point x , (12)

rel ∂x = x\x = ∅ , x∈Rn (25)

A bounded convex polyhedron (§2.3.2, §2.12.0.0.1) in subspace R , forexample, has boundary constructed from two points, in R2 from atleast three line segments, in R3 from convex polygons, while a convexpolychoron (a bounded polyhedron in R4 [370]) has boundary constructedfrom three-dimensional convex polyhedra. A halfspace is partially boundedby a hyperplane; its interior therefore excludes that hyperplane. An affineset has no relative boundary.

2.1.8 intersection, sum, difference, product

2.1.8.0.1 Theorem. Intersection. [303, §2, thm.6.5]Intersection of an arbitrary collection of convex sets Ci is convex. For afinite collection of N sets, a necessarily nonempty intersection of relativeinterior

⋂Ni=1rel int Ci = rel int

⋂Ni=1Ci equals relative interior of intersection.

And for a possibly infinite collection,⋂ Ci =

⋂ Ci . ⋄

In converse this theorem is implicitly false in so far as a convex set canbe formed by the intersection of sets that are not. Unions of convex sets aregenerally not convex. [196, p.22]

Vector sum of two convex sets C1 and C2 is convex [196, p.24]

C1+ C2 = x + y | x ∈ C1 , y ∈ C2 (26)

but not necessarily closed unless at least one set is closed and bounded.

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2.1. CONVEX SET 47

By additive inverse, we can similarly define vector difference of two convexsets

C1− C2 = x − y | x ∈ C1 , y ∈ C2 (27)

which is convex. Applying this definition to nonempty convex set C1 , itsself-difference C1− C1 is generally nonempty, nontrivial, and convex; e.g.,for any convex cone K , (§2.7.2) the set K − K constitutes its affine hull.[303, p.15]

Cartesian product of convex sets

C1× C2 =

[

xy

]

| x ∈ C1 , y ∈ C2

=

[

C1

C2

]

(28)

remains convex. The converse also holds; id est, a Cartesian product isconvex iff each set is. [196, p.23]

Convex results are also obtained for scaling κ C of a convex set C ,rotation/reflection Q C , or translation C+ α ; each similarly defined.

Given any operator T and convex set C , we are prone to write T (C)meaning

T (C) , T (x) | x∈ C (29)

Given linear operator T , it therefore follows from (26),

T (C1 + C2) = T (x + y) | x∈ C1 , y∈ C2= T (x) + T (y) | x∈ C1 , y∈ C2= T (C1) + T (C2)

(30)

2.1.9 inverse image

While epigraph (§3.6) of a convex function must be convex, it generallyholds that inverse image (Figure 15) of a convex function is not. The mostprominent examples to the contrary are affine functions (§3.5):

2.1.9.0.1 Theorem. Inverse image. [303, §3]Let f be a mapping from Rp×k to Rm×n. The image of a convex set C under any affine function

f(C) = f(X) | X∈ C ⊆ Rm×n (31)

is convex.

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48 CHAPTER 2. CONVEX GEOMETRY

f

f

f−1(F ) F

C

f(C)

(b)

(a)

Figure 15: (a) Image of convex set in domain of any convex function f isconvex, but there is no converse. (b) Inverse image under convex function f .

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2.1. CONVEX SET 49

xp

Rn A†Ax = xp

Axp = b

Ax = b

R(AT) R(A)

N (A) N (AT)

Aη = 0η

x=xp+ η

b

Rm

b

xp =A†b

00x

Figure 16:(confer Figure 159) Action of linear map represented by A∈ Rm×n :Component of vector x in nullspace N (A) maps to origin while componentin rowspace R(AT) maps to range R(A). For any A∈Rm×n, AA†Ax = b(§E) and inverse image of b ∈R(A) is a nonempty affine set: xp+ N (A). Inverse image of a convex set F ,

f−1(F ) = X | f(X)∈F ⊆ Rp×k (32)

a single- or many-valued mapping, under any affine function f isconvex. ⋄

In particular, any affine transformation of an affine set remains affine.[303, p.8] Ellipsoids are invariant to any [sic] affine transformation.

Although not precluded, this inverse image theorem does not require auniquely invertible mapping f . Figure 16, for example, mechanizes inverseimage under a general linear map. Example 2.9.1.0.2 and Example 3.6.0.0.2offer further applications.

Each converse of this two-part theorem is generally false; id est, givenf affine, a convex image f(C) does not imply that set C is convex, andneither does a convex inverse image f−1(F ) imply set F is convex. Acounter-example, invalidating a converse, is easy to visualize when the affinefunction is an orthogonal projector [328] [246]:

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50 CHAPTER 2. CONVEX GEOMETRY

2.1.9.0.2 Corollary. Projection on subspace.2.10 (1896) [303, §3]Orthogonal projection of a convex set on a subspace or nonempty affine setis another convex set. ⋄

Again, the converse is false. Shadows, for example, are umbral projectionsthat can be convex when the body providing the shade is not.

2.2 Vectorized-matrix inner product

Euclidean space Rn comes equipped with a linear vector inner-product

〈y , z〉 , yTz (33)

We prefer those angle brackets to connote a geometric rather than algebraicperspective; e.g., vector y might represent a hyperplane normal (§2.4.2).Two vectors are orthogonal (perpendicular) to one another if and only iftheir inner product vanishes;

y ⊥ z ⇔ 〈y , z〉 = 0 (34)

When orthogonal vectors each have unit norm, then they are orthonormal.A vector inner-product defines Euclidean norm (vector 2-norm)

‖y‖2 = ‖y‖ ,√

yTy , ‖y‖ = 0 ⇔ y = 0 (35)

For linear operation A on a vector, represented by a real matrix, the adjointoperation AT is transposition and defined for matrix A by [223, §3.10]

〈y ,ATz〉 , 〈Ay, z〉 (36)

The vector inner-product for matrices is calculated just as it is for vectors;by first transforming a matrix in Rp×k to a vector in Rpk by concatenatingits columns in the natural order. For lack of a better term, we shall callthat linear bijective (one-to-one and onto [223, App.A1.2]) transformation

2.10For hyperplane representations see §2.4.2. For projection of convex sets on hyperplanessee [368, §6.6]. A nonempty affine set is called an affine subset (§2.3.1.0.1). Orthogonalprojection of points on affine subsets is reviewed in §E.4.

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2.2. VECTORIZED-MATRIX INNER PRODUCT 51

vectorization. For example, the vectorization of Y = [ y1 y2 · · · yk ]∈ Rp×k

[162] [324] is

vec Y ,

y1

y2...yk

∈ Rpk (37)

Then the vectorized-matrix inner-product is trace of matrix inner-product;for Z ∈ Rp×k, [59, §2.6.1] [196, §0.3.1] [379, §8] [362, §2.2]

〈Y , Z〉 , tr(Y TZ) = vec(Y )Tvec Z (38)

where (§A.1.1)

tr(Y TZ) = tr(ZY T) = tr(YZT) = tr(ZTY ) = 1T(Y Z)1 (39)

and where denotes the Hadamard product 2.11 of matrices [155, §1.1.4]. Theadjoint operation AT on a matrix can therefore be defined in like manner:

〈Y , ATZ〉 , 〈AY , Z〉 (40)

Take any element C1 from a matrix-valued set in Rp×k, for example, andconsider any particular dimensionally compatible real vectors v and w .Then vector inner-product of C1 with vwT is

〈vwT, C1〉 = 〈v , C1w〉 = vTC1w = tr(wvTC1) = 1T(

(vwT) C1

)

1 (41)

Further, linear bijective vectorization is distributive with respect toHadamard product; id est,

vec(Y Z) = vec(Y ) vec(Z) (42)

2.2.0.0.1 Example. Application of inverse image theorem.Suppose set C ⊆ Rp×k were convex. Then for any particular vectors v∈Rp

and w∈Rk, the set of vector inner-products

Y , vTCw = 〈vwT, C〉 ⊆ R (43)

2.11Hadamard product is a simple entrywise product of corresponding entries from twomatrices of like size; id est, not necessarily square. A commutative operation, theHadamard product can be extracted from within a Kronecker product. [199, p.475]

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52 CHAPTER 2. CONVEX GEOMETRY

(a) (b)

R2 R3

Figure 17: (a) Cube in R3 projected on paper-plane R2. Subspace projectionoperator is not an isomorphism because new adjacencies are introduced.(b) Tesseract is a projection of hypercube in R4 on R3.

is convex. It is easy to show directly that convex combination of elementsfrom Y remains an element of Y .2.12 Instead given convex set Y , C mustbe convex consequent to inverse image theorem 2.1.9.0.1.

More generally, vwT in (43) may be replaced with any particular matrixZ∈ Rp×k while convexity of set 〈Z , C〉⊆ R persists. Further, by replacingv and w with any particular respective matrices U and W of dimensioncompatible with all elements of convex set C , then set UTCW is convex bythe inverse image theorem because it is a linear mapping of C . 2

2.2.1 Frobenius’

2.2.1.0.1 Definition. Isomorphic.An isomorphism of a vector space is a transformation equivalent to a linearbijective mapping. Image and inverse image under the transformationoperator are then called isomorphic vector spaces.

2.12To verify that, take any two elements C1 and C2 from the convex matrix-valued set C ,and then form the vector inner-products (43) that are two elements of Y by definition.Now make a convex combination of those inner products; videlicet, for 0≤µ≤1

µ 〈vwT, C1〉 + (1 − µ) 〈vwT, C2〉 = 〈vwT, µ C1 + (1 − µ)C2〉

The two sides are equivalent by linearity of inner product. The right-hand side remainsa vector inner-product of vwT with an element µ C1 + (1 − µ)C2 from the convex set C ;hence, it belongs to Y . Since that holds true for any two elements from Y , then it mustbe a convex set. ¨

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2.2. VECTORIZED-MATRIX INNER PRODUCT 53

Isomorphic vector spaces are characterized by preservation of adjacency ;id est, if v and w are points connected by a line segment in one vectorspace, then their images will be connected by a line segment in the other.Two Euclidean bodies may be considered isomorphic if there exists anisomorphism, of their vector spaces, under which the bodies correspond.[364, §I.1] Projection (§E) is not an isomorphism, Figure 17 for example;hence, perfect reconstruction (inverse projection) is generally impossiblewithout additional information.

When Z =Y ∈ Rp×k in (38), Frobenius’ norm is resultant from vectorinner-product; (confer (1649))

‖Y ‖2F = ‖ vec Y ‖2

2 = 〈Y , Y 〉 = tr(Y TY )

=∑

i, j

Y 2ij =

i

λ(Y TY )i =∑

i

σ(Y )2i

(44)

where λ(Y TY )i is the ith eigenvalue of Y TY , and σ(Y )i the ith singularvalue of Y . Were Y a normal matrix (§A.5.1), then σ(Y )= |λ(Y )|[390, §8.1] thus

‖Y ‖2F =

i

λ(Y )2i = ‖λ(Y )‖2

2 = 〈λ(Y ) , λ(Y )〉 = 〈Y , Y 〉 (45)

The converse (45) ⇒ normal matrix Y also holds. [199, §2.5.4]Frobenius’ norm is the Euclidean norm of vectorized matrices. Because

the metrics are equivalent, for X∈ Rp×k

‖ vec X−vec Y ‖2 = ‖X−Y ‖F (46)

and because vectorization (37) is a linear bijective map, then vector spaceRp×k is isometrically isomorphic with vector space Rpk in the Euclidean senseand vec is an isometric isomorphism2.13 of Rp×k. Because of this Euclideanstructure, all the known results from convex analysis in Euclidean space Rn

carry over directly to the space of real matrices Rp×k.

2.2.1.1 Injective linear operators

Injective mapping (transformation) means one-to-one mapping; synonymouswith uniquely invertible linear mapping on Euclidean space. Linear injectivemappings are fully characterized by lack of a nontrivial nullspace.

2.13Given matrix A , its range R(A) (§2.5) is isometrically isomorphic with its vectorizedrange vecR(A) but not with R(vec A).

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54 CHAPTER 2. CONVEX GEOMETRY

R2

R3

T

dim dom T = dimR(T )

Figure 18: Linear injective mapping Tx=Ax : R2→ R3 of Euclidean bodyremains two-dimensional under mapping represented by skinny full-rankmatrix A∈ R3×2 ; two bodies are isomorphic by Definition 2.2.1.0.1.

2.2.1.1.1 Definition. Isometric isomorphism.An isometric isomorphism of a vector space having a metric defined on it is alinear bijective mapping T that preserves distance; id est, for all x, y∈dom T

‖Tx − Ty‖ = ‖x − y‖ (47)

Then the isometric isomorphism T is a bijective isometry.

Unitary linear operator Q : Rk → Rk, represented by orthogonal matrixQ∈Rk×k (§B.5), is an isometric isomorphism; e.g., discrete Fourier transformvia (812). Suppose T (X)= UXQ , for example. Then we say Frobenius’norm is orthogonally invariant ; meaning, for X,Y ∈ Rp×k and dimensionallycompatible orthonormal matrix 2.14 U and orthogonal matrix Q

‖U(X−Y )Q‖F = ‖X−Y ‖F (48)

Yet isometric operator T : R2 → R3, represented by A =

1 00 10 0

on R2,

is injective but not a surjective map to R3. [223, §1.6, §2.6] This operator Tcan therefore be a bijective isometry only with respect to its range.

2.14 any matrix U whose columns are orthonormal with respect to each other (UTU = I );these include the orthogonal matrices.

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2.2. VECTORIZED-MATRIX INNER PRODUCT 55

R3R3

PT

B

PT (B)

x PTx

Figure 19: Linear noninjective mapping PTx=A†Ax : R3→ R3 ofthree-dimensional Euclidean body B has affine dimension 2 under projectionon rowspace of fat full-rank matrix A∈ R2×3. Set of coefficients of orthogonalprojection TB= Ax |x∈B is isomorphic with projection P (TB) [sic].

Any linear injective transformation on Euclidean space is uniquelyinvertible on its range. In fact, any linear injective transformation has arange whose dimension equals that of its domain. In other words, for anyinvertible linear transformation T [ibidem]

dim dom(T ) = dimR(T ) (49)

e.g., T represented by skinny-or-square full-rank matrices. (Figure 18) Animportant consequence of this fact is: Affine dimension, of any n-dimensional Euclidean body in domain of

operator T , is invariant to linear injective transformation.

2.2.1.2 Noninjective linear operators

Mappings in Euclidean space created by noninjective linear operators can becharacterized in terms of an orthogonal projector (§E). Consider noninjectivelinear operator Tx =Ax : Rn→ Rm represented by fat matrix A∈Rm×n

(m< n). What can be said about the nature of this m-dimensional mapping?

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56 CHAPTER 2. CONVEX GEOMETRY

Concurrently, consider injective linear operator Py=A†y : Rm→ Rn

where R(A†)=R(AT). P (Ax)= PTx achieves projection of vector x onthe row space R(AT). (§E.3.1) This means vector Ax can be succinctlyinterpreted as coefficients of orthogonal projection.

Pseudoinverse matrix A† is skinny and full-rank, so operator Py is a linearbijection with respect to its range R(A†). By Definition 2.2.1.0.1, imageP (TB) of projection PT (B) on R(AT) in Rn must therefore be isomorphicwith the set of projection coefficients TB= Ax |x∈B in Rm and have thesame affine dimension by (49). To illustrate, we present a three-dimensionalEuclidean body B in Figure 19 where any point x in the nullspace N (A)maps to the origin.

2.2.2 Symmetric matrices

2.2.2.0.1 Definition. Symmetric matrix subspace.Define a subspace of RM×M : the convex set of all symmetric M×M matrices;

SM ,

A∈RM×M | A=AT

⊆ RM×M (50)

This subspace comprising symmetric matrices SM is isomorphic with thevector space RM(M+1)/2 whose dimension is the number of free variables in asymmetric M×M matrix. The orthogonal complement [328] [246] of SM is

SM⊥ ,

A∈RM×M | A=−AT

⊂ RM×M (51)

the subspace of antisymmetric matrices in RM×M ; id est,

SM⊕ SM⊥ = RM×M (52)

where unique vector sum ⊕ is defined on page 758.

All antisymmetric matrices are hollow by definition (have 0 maindiagonal). Any square matrix A∈RM×M can be written as a sum of itssymmetric and antisymmetric parts: respectively,

A =1

2(A +AT) +

1

2(A −AT) (53)

The symmetric part is orthogonal in RM2

to the antisymmetric part; videlicet,

tr(

(A +AT)(A −AT))

= 0 (54)

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2.2. VECTORIZED-MATRIX INNER PRODUCT 57

In the ambient space of real matrices, the antisymmetric matrix subspacecan be described

SM⊥ =

1

2(A −AT) | A∈RM×M

⊂ RM×M (55)

because any matrix in SM is orthogonal to any matrix in SM⊥. Furtherconfined to the ambient subspace of symmetric matrices, because ofantisymmetry, SM⊥ would become trivial.

2.2.2.1 Isomorphism of symmetric matrix subspace

When a matrix is symmetric in SM , we may still employ the vectorizationtransformation (37) to RM2

; vec , an isometric isomorphism. We mightinstead choose to realize in the lower-dimensional subspace RM(M+1)/2 byignoring redundant entries (below the main diagonal) during transformation.Such a realization would remain isomorphic but not isometric. Lack ofisometry is a spatial distortion due now to disparity in metric between RM 2

and RM(M+1)/2. To realize isometrically in RM(M+1)/2, we must make acorrection: For Y = [Yij]∈ SM we take symmetric vectorization [211, §2.2.1]

svec Y ,

Y11√2Y12

Y22√2Y13√2Y23

Y33...YMM

∈ RM(M+1)/2 (56)

where all entries off the main diagonal have been scaled. Now for Z ∈ SM

〈Y , Z〉 , tr(Y TZ) = vec(Y )Tvec Z = svec(Y )Tsvec Z (57)

Then because the metrics become equivalent, for X∈ SM

‖ svec X − svec Y ‖2 = ‖X − Y ‖F (58)

and because symmetric vectorization (56) is a linear bijective mapping, thensvec is an isometric isomorphism of the symmetric matrix subspace. In other

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58 CHAPTER 2. CONVEX GEOMETRY

words, SM is isometrically isomorphic with RM(M+1)/2 in the Euclidean senseunder transformation svec .

The set of all symmetric matrices SM forms a proper subspace in RM×M ,so for it there exists a standard orthonormal basis in isometrically isomorphicRM(M+1)/2

Eij ∈ SM =

eieTi , i = j = 1 . . . M

1√2

(

eieTj + eje

Ti

)

, 1 ≤ i < j ≤ M

(59)

where M(M + 1)/2 standard basis matrices Eij are formed from thestandard basis vectors

ei =

[

1 , i = j0 , i 6= j

, j = 1 . . . M

]

∈ RM (60)

Thus we have a basic orthogonal expansion for Y ∈ SM

Y =M

j=1

j∑

i=1

〈Eij , Y 〉Eij (61)

whose coefficients

〈Eij , Y 〉 =

Yii , i = 1 . . . M

Yij

√2 , 1 ≤ i < j ≤ M

(62)

correspond to entries of the symmetric vectorization (56).

2.2.3 Symmetric hollow subspace

2.2.3.0.1 Definition. Hollow subspaces. [349]Define a subspace of RM×M : the convex set of all (real) symmetric M×Mmatrices having 0 main diagonal;

RM×Mh ,

A∈RM×M | A=AT, δ(A) = 0

⊂ RM×M (63)

where the main diagonal of A∈RM×M is denoted (§A.1)

δ(A) ∈ RM (1391)

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2.2. VECTORIZED-MATRIX INNER PRODUCT 59

Operating on a vector, linear operator δ naturally returns a diagonal matrix;δ(δ(A)) is a diagonal matrix. Operating recursively on a vector Λ∈RN ordiagonal matrix Λ∈ SN , operator δ(δ(Λ)) returns Λ itself;

δ2(Λ) ≡ δ(δ(Λ)) = Λ (1393)

The subspace RM×Mh (63) comprising (real) symmetric hollow matrices is

isomorphic with subspace RM(M−1)/2 ; its orthogonal complement is

RM×M⊥h ,

A∈RM×M | A=−AT + 2δ2(A)

⊆ RM×M (64)

the subspace of antisymmetric antihollow matrices in RM×M ; id est,

RM×Mh ⊕ RM×M⊥

h = RM×M (65)

Yet defined instead as a proper subspace of ambient SM

SMh ,

A∈ SM | δ(A) = 0

≡ RM×Mh ⊂ SM (66)

the orthogonal complement SM⊥h of symmetric hollow subspace SM

h ,

SM⊥h ,

A∈ SM | A=δ2(A)

⊆ SM (67)

called symmetric antihollow subspace, is simply the subspace of diagonalmatrices; id est,

SMh ⊕ SM⊥

h = SM (68)

Any matrix A∈RM×M can be written as a sum of its symmetric hollowand antisymmetric antihollow parts: respectively,

A =

(

1

2(A +AT) − δ2(A)

)

+

(

1

2(A −AT) + δ2(A)

)

(69)

The symmetric hollow part is orthogonal to the antisymmetric antihollowpart in RM2

; videlicet,

tr

((

1

2(A +AT) − δ2(A)

)(

1

2(A −AT) + δ2(A)

))

= 0 (70)

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60 CHAPTER 2. CONVEX GEOMETRY

because any matrix in subspace RM×Mh is orthogonal to any matrix in the

antisymmetric antihollow subspace

RM×M⊥h =

1

2(A −AT) + δ2(A) | A∈RM×M

⊆ RM×M (71)

of the ambient space of real matrices; which reduces to the diagonal matricesin the ambient space of symmetric matrices

SM⊥h =

δ2(A) | A∈SM

=

δ(u) | u∈RM

⊆ SM (72)

In anticipation of their utility with Euclidean distance matrices (EDMs)in §5, for symmetric hollow matrices we introduce the linear bijectivevectorization dvec that is the natural analogue to symmetric matrixvectorization svec (56): for Y = [Yij]∈ SM

h

dvec Y ,√

2

Y12

Y13

Y23

Y14

Y24

Y34...YM−1,M

∈ RM(M−1)/2 (73)

Like svec , dvec is an isometric isomorphism on the symmetric hollowsubspace. For X∈ SM

h

‖ dvec X − dvec Y ‖2 = ‖X − Y ‖F (74)

The set of all symmetric hollow matrices SMh forms a proper subspace

in RM×M , so for it there must be a standard orthonormal basis inisometrically isomorphic RM(M−1)/2

Eij ∈ SMh =

1√2

(

eieTj + eje

Ti

)

, 1 ≤ i < j ≤ M

(75)

where M(M−1)/2 standard basis matrices Eij are formed from the standardbasis vectors ei∈ RM .

The symmetric hollow majorization corollary A.1.2.0.2 characterizeseigenvalues of symmetric hollow matrices.

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2.3. HULLS 61

Figure 20: Convex hull of a random list of points in R3. Some pointsfrom that generating list reside interior to this convex polyhedron (§2.12).[370, Convex Polyhedron] (Avis-Fukuda-Mizukoshi)

2.3 Hulls

We focus on the affine, convex, and conic hulls: convex sets that may beregarded as kinds of Euclidean container or vessel united with its interior.

2.3.1 Affine hull, affine dimension

Affine dimension of any set in Rn is the dimension of the smallest affine set(empty set, point, line, plane, hyperplane (§2.4.2), translated subspace, Rn)that contains it. For nonempty sets, affine dimension is the same as dimensionof the subspace parallel to that affine set. [303, §1] [196, §A.2.1]

Ascribe the points in a list xℓ ∈ Rn, ℓ=1 . . . N to the columns ofmatrix X :

X = [ x1 · · · xN ] ∈ Rn×N (76)

In particular, we define affine dimension r of the N -point list X asdimension of the smallest affine set in Euclidean space Rn that contains X ;

r , dim aff X (77)

Affine dimension r is a lower bound sometimes called embedding dimension.[349] [181] That affine set A in which those points are embedded is uniqueand called the affine hull [327, §2.1];

A , aff xℓ∈ Rn, ℓ=1 . . . N = aff X= x1 + Rxℓ − x1 , ℓ=2 . . . N = Xa | aT1 = 1 ⊆ Rn (78)

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62 CHAPTER 2. CONVEX GEOMETRY

for which we call list X a set of generators. Hull A is parallel to subspace

Rxℓ − x1 , ℓ=2 . . . N = R(X − x11T) ⊆ Rn (79)

where R(A) = Ax | ∀x (140)

Given some arbitrary set C and any x∈ Caff C = x + aff(C − x) (80)

where aff(C−x) is a subspace.

2.3.1.0.1 Definition. Affine subset.We analogize affine subset to subspace,2.15 defining it to be any nonemptyaffine set (§2.1.4).

aff ∅ , ∅ (81)

The affine hull of a point x is that point itself;

affx = x (82)

The affine hull of two distinct points is the unique line through them.(Figure 21) The affine hull of three noncollinear points in any dimensionis that unique plane containing the points, and so on. The subspace ofsymmetric matrices Sm is the affine hull of the cone of positive semidefinitematrices; (§2.9)

aff Sm+ = Sm (83)

2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [214]The set of all m×m rank-1 correlation matrices is defined by all the binaryvectors y in Rm (confer §5.9.1.0.1)

yyT∈ Sm+ | δ(yyT)=1 (84)

Affine hull of the rank-1 correlation matrices is equal to the set of normalizedsymmetric matrices; id est,

affyyT∈ Sm+ | δ(yyT)=1 = A∈ Sm | δ(A)=1 (85)

2

2.15The popular term affine subspace is an oxymoron.

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2.3. HULLS 63

affine hull (drawn truncated)

convex hull

conic hull (truncated)

range or span is a plane (truncated)

A

C

K

R

Figure 21: Given two points in Euclidean vector space of any dimension,their various hulls are illustrated. Each hull is a subset of range; generally,A , C , K ⊆ R ∋ 0. (Cartesian axes drawn for reference.)

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64 CHAPTER 2. CONVEX GEOMETRY

2.3.1.0.3 Exercise. Affine hull of correlation matrices.Prove (85) via definition of affine hull. Find the convex hull instead. H

2.3.1.1 Partial order induced by RN+ and SM

+

Notation aº 0 means vector a belongs to the nonnegative orthant RN+ ,

while a≻ 0 means vector a belongs to the nonnegative orthant’sinterior int RN

+ , whereas aº b denotes comparison of vector a to vector bon RN with respect to the nonnegative orthant; id est, aº b means a− bbelongs to the nonnegative orthant, but neither a or b necessarily belongsto that orthant. With particular respect to the nonnegative orthant,aº b ⇔ ai ≥ bi ∀ i (348). More generally, aºK b or a≻K b denotescomparison with respect to pointed closed convex cone K , but equivalencewith entrywise comparison does not hold. (§2.7.2.2)

The symbol ≥ is reserved for scalar comparison on the real line R withrespect to the nonnegative real line R+ as in aTy ≥ b . Comparison ofmatrices with respect to the positive semidefinite cone SM

+ , like I ºAº 0in Example 2.3.2.0.1, is explained in §2.9.0.1.

2.3.2 Convex hull

The convex hull [196, §A.1.4] [303] of any bounded2.16 list or set of N pointsX∈ Rn×N forms a unique bounded convex polyhedron (confer §2.12.0.0.1)whose vertices constitute some subset of that list;

P , conv xℓ , ℓ=1 . . . N = conv X = Xa | aT1 = 1, a º 0 ⊆ Rn

(86)

Union of relative interior and relative boundary (§2.1.7.2) of the polyhedroncomprise its convex hull P , the smallest closed convex set that contains thelist X ; e.g., Figure 20. Given P , the generating list xℓ is not unique.But because every bounded polyhedron is the convex hull of its vertices,[327, §2.12.2] the vertices of P comprise a minimal set of generators.

Given some arbitrary set C⊆Rn, its convex hull conv C is equivalent tothe smallest convex set containing it. (confer §2.4.1.1.1) The convex hull is a

2.16An arbitrary set C in Rn is bounded iff it can be contained in a Euclidean ball havingfinite radius. [110, §2.2] (confer §5.7.3.0.1) The smallest ball containing C has radiusinfx

supy∈C

‖x−y‖ and center x⋆ whose determination is a convex problem because supy∈C

‖x−y‖is a convex function of x ; but the supremum may be difficult to ascertain.

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2.3. HULLS 65

subset of the affine hull;

conv C ⊆ aff C = aff C = aff C = aff conv C (87)

Any closed bounded convex set C is equal to the convex hull of its boundary;

C = conv ∂C (88)

conv ∅ , ∅ (89)

2.3.2.0.1 Example. Hull of rank-k projection matrices. [139] [284][11, §4.1] [289, §3] [235, §2.4] [236] Convex hull of the set comprising outerproduct of orthonormal matrices has equivalent expression: for 1 ≤ k ≤ N(§2.9.0.1)

conv

UUT | U ∈ RN×k, UTU = I

=

A∈ SN | I º A º 0 , 〈I , A〉=k

⊂ SN+

(90)

This important convex body we call Fantope (after mathematician Ky Fan).In case k = 1, there is slight simplification: ((1584), Example 2.9.2.4.1)

conv

UUT | U ∈ RN , UTU = 1

=

A∈ SN | A º 0 , 〈I , A〉=1

(91)

In case k = N , the Fantope is identity matrix I . More generally, the set

UUT | U ∈ RN×k, UTU = I

(92)

comprises the extreme points (§2.6.0.0.1) of its convex hull. By (1439), eachand every extreme point UUT has only k nonzero eigenvalues λ and theyall equal 1 ; id est, λ(UUT)1:k = λ(UTU) = 1. So Frobenius’ norm of eachand every extreme point equals the same constant

‖UUT‖2F = k (93)

Each extreme point simultaneously lies on the boundary of the positivesemidefinite cone (when k < N , §2.9) and on the boundary of a hypersphere

of dimension k(N− k2+ 1

2) and radius

k(1− kN

) centered at kN

I along

the ray (base 0) through the identity matrix I in isomorphic vector space

RN(N+1)/2 (§2.2.2.1).Figure 22 illustrates extreme points (92) comprising the boundary of a

Fantope, the boundary of a disc corresponding to k = 1, N = 2 ; but thatcircumscription does not hold in higher dimension. (§2.9.2.5) 2

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66 CHAPTER 2. CONVEX GEOMETRY

α√

γ

I

svec ∂ S2

+

[

α ββ γ

]

Figure 22: Two Fantopes. Circle (radius 1/√

2), shown here on boundary ofpositive semidefinite cone S2

+ in isometrically isomorphic R3 from Figure 42,comprises boundary of a Fantope (90) in this dimension (k = 1, N = 2). Lonepoint illustrated is identity matrix I , interior to PSD cone, and is thatFantope corresponding to k = 2, N = 2. (View is from inside PSD conelooking toward origin.)

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2.3. HULLS 67

2.3.2.0.2 Example. Nuclear norm ball : convex hull of rank-1 matrices.From (91), in Example 2.3.2.0.1, we learn that the convex hull of normalizedsymmetric rank-1 matrices is a slice of the positive semidefinite cone.In §2.9.2.4 we find the convex hull of all symmetric rank-1 matrices to bethe entire positive semidefinite cone.

In the present example we abandon symmetry; instead posing, what isthe convex hull of bounded nonsymmetric rank-1 matrices:

convuvT | ‖uvT‖ ≤ 1 , u∈ Rm, v∈ Rn = X∈ Rm×n |∑

i

σ(X)i ≤ 1 (94)

where σ(X) is a vector of singular values. (Since ‖uvT‖= ‖u‖‖v‖ (1576),norm of each vector constituting a dyad uvT (§B.1) in the hull is effectivelybounded above by 1.)

Proof. (⇐) Suppose∑

σ(X)i ≤ 1. As in §A.6, define a singularvalue decomposition: X = UΣV T where U = [u1 . . . uminm,n]∈Rm×minm,n,

V = [v1 . . . vminm,n]∈Rn×minm,n, and whose sum of singular values is∑

σ(X)i = tr Σ = κ≤ 1. Then we may write X =∑ σi

κ

√κui

√κvT

i which is aconvex combination of dyads each of whose norm does not exceed 1. (Srebro)

(⇒) Now suppose we are given a convex combination of dyadsX =

αi uivTi such that

αi =1, αi≥ 0 ∀ i , and ‖uivTi ‖≤ 1 ∀ i .

Then by triangle inequality for singular values [200, cor.3.4.3]∑

σ(X)i ≤∑

σ(αi uivTi )=

αi‖uivTi ‖≤

αi . ¨

Given any particular dyad uvTp in the convex hull, because its polar −uvT

p

and every convex combination of the two belong to that hull, then the uniqueline containing those two points ±uvT

p (their affine combination (78)) mustintersect the hull’s boundary at the normalized dyads ±uvT | ‖uvT‖=1.Any point formed by convex combination of dyads in the hull must thereforebe expressible as a convex combination of dyads on the boundary: Figure 23.

convuvT | ‖uvT‖ ≤ 1 , u∈ Rm, v∈ Rn ≡ convuvT | ‖uvT‖ = 1 , u∈ Rm, v∈ Rn(95)

id est, dyads may be normalized and the hull’s boundary contains them;

∂X∈ Rm×n |∑

i

σ(X)i ≤ 1 ⊇ uvT | ‖uvT‖ = 1 , u∈ Rm, v∈ Rn (96)

Normalized dyads constitute the set of extreme points (§2.6.0.0.1) of thisnuclear norm ball which is, therefore, their convex hull. 2

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68 CHAPTER 2. CONVEX GEOMETRY

xyTp

−xyTp

uvTp

−uvTp

‖xyT‖=1

‖−xyT‖=1

‖uvT‖=1

‖−uvT‖=1

X∈ Rm×n | ∑

i

σ(X)i ≤ 1

‖uvT‖≤1

0

Figure 23: uvTp is a convex combination of normalized dyads ‖±uvT‖=1 ;

similarly for xyTp . Any point in line segment joining xyT

p to uvTp is expressible

as a convex combination of two to four points indicated on boundary. Affinedimension of vectorized nuclear norm ball in X∈R2×2 is 2 because (1576)σ1 = ‖u‖‖v‖=1 by (95).

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2.3. HULLS 69

2.3.2.0.3 Exercise. Convex hull of outer product.Describe the interior of a Fantope.Find the convex hull of nonorthogonal projection matrices (§E.1.1):

UV T | U ∈ RN×k, V ∈ RN×k, V TU = I (97)

Find the convex hull of nonsymmetric matrices bounded under some norm:

UV T | U ∈ Rm×k, V ∈ Rn×k, ‖UV T‖ ≤ 1 (98)

H

2.3.2.0.4 Example. Permutation polyhedron. [198] [313] [257]A permutation matrix Ξ is formed by interchanging rows and columns ofidentity matrix I . Since Ξ is square and ΞTΞ = I , the set of all permutationmatrices is a proper subset of the nonconvex manifold of orthogonal matrices(§B.5). In fact, the only orthogonal matrices having all nonnegative entriesare permutations of the identity.

Regarding the permutation matrices as a set of points in Euclidean space,its convex hull is a bounded polyhedron (§2.12) described (Birkhoff, 1946)

convΞ = Πi(I∈ Sn)∈ Rn×n, i=1 . . . n! = X∈ Rn×n | XT1=1, X1=1, X≥ 0(99)

where Πi is a linear operator here representing the ith permutation. Thispolyhedral hull, whose n! vertices are the permutation matrices, is alsoknown as the set of doubly stochastic matrices. The permutation matricesare the minimum cardinality (fewest nonzero entries) doubly stochasticmatrices. The only orthogonal matrices belonging to this polyhedron arethe permutation matrices.

It is remarkable that n! permutation matrices can be describedas the extreme points (§2.6.0.0.1) of a bounded polyhedron, of affinedimension (n−1)2, that is itself described by only 2n equalities (2n−1linearly independent equality constraints in nonnegative variables). ByCaratheodory’s theorem, conversely, any doubly stochastic matrix can bedescribed as a convex combination of at most (n−1)2+1 permutationmatrices. [199, §8.7] This polyhedron, then, can be a device for relaxing aninteger, combinatorial, or Boolean optimization problem.2.17 [64] [279, §3.1]

2

2.17Relaxation replaces an objective function with its convex envelope or expands a feasibleset to one that is convex. Dantzig first showed in 1951 that, by this device, the so-calledassignment problem can be formulated as a linear program. [312] [26, §II.5]

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70 CHAPTER 2. CONVEX GEOMETRY

2.3.2.0.5 Example. Convex hull of orthonormal matrices. [27, §1.2]Consider rank-k matrices U ∈ Rn×k such that UTU = I . These are theorthonormal matrices; a closed bounded submanifold, of all orthogonalmatrices, having dimension nk − 1

2k(k + 1) [50]. Their convex hull is

expressed, for 1 ≤ k ≤ n

convU ∈ Rn×k | UTU = I = X∈ Rn×k | ‖X‖2 ≤ 1= X∈ Rn×k | ‖XTa‖ ≤ ‖a‖ ∀ a∈Rn (100)

When k=n , matrices U are orthogonal and the convex hull is called thespectral norm ball which is the set of all contractions. [200, p.158] [326, p.313]The orthogonal matrices then constitute the extreme points (§2.6.0.0.1) ofthis hull.

By Schur complement (§A.4), the spectral norm ‖X‖2 constraining largestsingular value σ(X)1 can be expressed as a semidefinite constraint

‖X‖2 ≤ 1 ⇔[

I XXT I

]

º 0 (101)

because of equivalence XTX¹ I ⇔ σ(X) ¹ 1 with singular values. (1532)(1426) (1427) 2

2.3.3 Conic hull

In terms of a finite-length point list (or set) arranged columnar in X∈ Rn×N

(76), its conic hull is expressed

K , cone xℓ , ℓ=1 . . . N = cone X = Xa | a º 0 ⊆ Rn (102)

id est, every nonnegative combination of points from the list. Conic hull ofany finite-length list forms a polyhedral cone [196, §A.4.3] (§2.12.1.0.1; e.g.,Figure 49a); the smallest closed convex cone (§2.7.2) that contains the list.

By convention, the aberration [327, §2.1]

cone ∅ , 0 (103)

Given some arbitrary set C , it is apparent

conv C ⊆ cone C (104)

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2.3. HULLS 71

Figure 24: A simplicial cone (§2.12.3.1.1) in R3 whose boundary is drawntruncated; constructed using A∈ R3×3 and C = 0 in (266). By the mostfundamental definition of a cone (§2.7.1), entire boundary can be constructedfrom an aggregate of rays emanating exclusively from the origin. Eachof three extreme directions corresponds to an edge (§2.6.0.0.3); they areconically, affinely, and linearly independent for this cone. Because thisset is polyhedral, exposed directions are in one-to-one correspondence withextreme directions; there are only three. Its extreme directions give rise towhat is called a vertex-description of this polyhedral cone; simply, the conichull of extreme directions. Obviously this cone can also be constructed byintersection of three halfspaces; hence the equivalent halfspace-description.

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72 CHAPTER 2. CONVEX GEOMETRY

2.3.4 Vertex-description

The conditions in (78), (86), and (102) respectively define an affinecombination, convex combination, and conic combination of elements fromthe set or list. Whenever a Euclidean body can be described as somehull or span of a set of points, then that representation is loosely calleda vertex-description and those points are called generators.

2.4 Halfspace, Hyperplane

A two-dimensional affine subset is called a plane. An (n−1)-dimensionalaffine subset in Rn is called a hyperplane. [303] [196] Every hyperplanepartially bounds a halfspace (which is convex but not affine).

2.4.1 Halfspaces H+ and H−

Euclidean space Rn is partitioned in two by any hyperplane ∂H ; id est,H− + H+ = Rn. The resulting (closed convex) halfspaces, both partiallybounded by ∂H , may be described as an asymmetry

H− = y | aTy ≤ b = y | aT(y − yp) ≤ 0 ⊂ Rn (105)

H+ = y | aTy ≥ b = y | aT(y − yp) ≥ 0 ⊂ Rn (106)

where nonzero vector a∈Rn is an outward-normal to the hyperplane partiallybounding H− while an inward-normal with respect to H+ . For any vectory− yp that makes an obtuse angle with normal a , vector y will lie in thehalfspace H− on one side (shaded in Figure 25) of the hyperplane while acuteangles denote y in H+ on the other side.

An equivalent more intuitive representation of a halfspace comes aboutwhen we consider all the points in Rn closer to point d than to point c orequidistant, in the Euclidean sense; from Figure 25,

H− = y | ‖y − d‖ ≤ ‖y − c‖ (107)

This representation, in terms of proximity, is resolved with the moreconventional representation of a halfspace (105) by squaring both sides ofthe inequality in (107);

H− =

y | (c − d)Ty ≤ ‖c‖2 − ‖d‖2

2

=

y | (c − d)T(

y − c + d

2

)

≤ 0

(108)

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2.4. HALFSPACE, HYPERPLANE 73

∂H = y | aT(y − yp)=0 = N (aT) + yp

N (aT)=y | aTy=0

c

dy

yp

a

H+

H−

Figure 25: Hyperplane illustrated ∂H is a line partially bounding halfspacesH−= y | aT(y − yp)≤ 0 and H+ = y | aT(y − yp)≥ 0 in R2. Shaded isa rectangular piece of semiinfinite H− with respect to which vector a isoutward-normal to bounding hyperplane; vector a is inward-normal withrespect to H+ . Halfspace H− contains nullspace N (aT) (dashed linethrough origin) because aTyp > 0. Hyperplane, halfspace, and nullspace areeach drawn truncated. Points c and d are equidistant from hyperplane, andvector c− d is normal to it. ∆ is distance from origin to hyperplane.

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74 CHAPTER 2. CONVEX GEOMETRY

2.4.1.1 PRINCIPLE 1: Halfspace-description of convex sets

The most fundamental principle in convex geometry follows from thegeometric Hahn-Banach theorem [246, §5.12] [18, §1] [129, §I.1.2] whichguarantees any closed convex set to be an intersection of halfspaces.

2.4.1.1.1 Theorem. Halfspaces. [196, §A.4.2(b)] [41, §2.4]A closed convex set in Rn is equivalent to the intersection of all halfspacesthat contain it. ⋄

Intersection of multiple halfspaces in Rn may be represented using amatrix constant A

i

Hi− = y | ATy ¹ b = y | AT(y − yp) ¹ 0 (109)

i

Hi+ = y | ATy º b = y | AT(y − yp) º 0 (110)

where b is now a vector, and the ith column of A is normal to a hyperplane∂Hi partially bounding Hi . By the halfspaces theorem, intersections likethis can describe interesting convex Euclidean bodies such as polyhedra andcones, giving rise to the term halfspace-description.

2.4.2 Hyperplane ∂H representations

Every hyperplane ∂H is an affine set parallel to an (n−1)-dimensionalsubspace of Rn ; it is itself a subspace if and only if it contains the origin.

dim ∂H = n − 1 (111)

so a hyperplane is a point in R , a line in R2, a plane in R3, and so on. Everyhyperplane can be described as the intersection of complementary halfspaces;[303, §19]

∂H = H− ∩ H+ = y | aTy ≤ b , aTy ≥ b = y | aTy = b (112)

a halfspace-description. Assuming normal a∈Rn to be nonzero, then anyhyperplane in Rn can be described as the solution set to vector equationaTy = b (illustrated in Figure 25 and Figure 26 for R2 );

∂H , y | aTy = b = y | aT(y−yp) = 0 = Zξ+yp | ξ∈Rn−1 ⊂ Rn (113)

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2.4. HALFSPACE, HYPERPLANE 75

1

1

1

1

1

1

1

1

1

−1

−1

−1

−1

−1

−1

−1

−1

−1

a =

[

11

]

b =

[

−1−1

]

c =

[

−11

]

d =

[

1−1

]

e =

[

10

]

y | aTy=1

y | aTy=−1

y | bTy=−1

y | bTy=1

y | cTy=1

y | cTy=−1

y | dTy=−1

y | dTy=1

y | eTy=−1 y | eTy=1

(a) (b)

(c) (d)

(e)

Figure 26: (a)-(d) Hyperplanes in R2 (truncated). Movement in normaldirection increases vector inner-product. This visual concept is exploitedto attain analytical solution of linear programs; e.g., Example 2.4.2.6.2,Exercise 2.5.1.2.2, Example 3.5.0.0.2, [59, exer.4.8-exer.4.20]. Each graph isalso interpretable as a contour plot of a real affine function of two variablesas in Figure 72. (e) Ratio |β|/‖α‖ from x | αTx = β represents radius ofhypersphere about 0 supported by hyperplane whose normal is α .

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76 CHAPTER 2. CONVEX GEOMETRY

All solutions y constituting the hyperplane are offset from the nullspace ofaT by the same constant vector yp∈Rn that is any particular solution toaTy=b ; id est,

y = Zξ + yp (114)

where the columns of Z∈ Rn×n−1 constitute a basis for the nullspaceN (aT) = x∈Rn | aTx=0 .2.18

Conversely, given any point yp in Rn, the unique hyperplane containingit having normal a is the affine set ∂H (113) where b equals aTyp andwhere a basis for N (aT) is arranged in Z columnar. Hyperplane dimensionis apparent from the dimensions of Z ; that hyperplane is parallel to thespan of its columns.

2.4.2.0.1 Exercise. Hyperplane scaling.Given normal y , draw a hyperplane x∈R2 | xTy =1. Suppose z = 1

2y .

On the same plot, draw the hyperplane x∈R2 | xTz =1. Now supposez = 2y , then draw the last hyperplane again with this new z . What is theapparent effect of scaling normal y ? H

2.4.2.0.2 Example. Distance from origin to hyperplane.Given the (shortest) distance ∆∈R+ from the origin to a hyperplanehaving normal vector a , we can find its representation ∂H by droppinga perpendicular. The point thus found is the orthogonal projection of theorigin on ∂H (§E.5.0.0.5), equal to a∆/‖a‖ if the origin is known a priorito belong to halfspace H− (Figure 25), or equal to −a∆/‖a‖ if the originbelongs to halfspace H+ ; id est, when H−∋0

∂H =

y | aT(y − a∆/‖a‖) = 0

=

y | aTy = ‖a‖∆

(115)

or when H+∋0

∂H =

y | aT(y + a∆/‖a‖) = 0

=

y | aTy = −‖a‖∆

(116)

Knowledge of only distance ∆ and normal a thus introduces ambiguity intothe hyperplane representation. 2

2.18We will later find this expression for y in terms of nullspace of aT (more generally, ofmatrix AT (142)) to be a useful device for eliminating affine equality constraints, much aswe did here.

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2.4. HALFSPACE, HYPERPLANE 77

2.4.2.1 Matrix variable

Any halfspace in Rmn may be represented using a matrix variable. Forvariable Y ∈ Rm×n, given constants A∈Rm×n and b = 〈A , Yp〉 ∈ R

H− = Y ∈ Rmn | 〈A , Y 〉 ≤ b = Y ∈ Rmn | 〈A , Y −Yp〉 ≤ 0 (117)

H+ = Y ∈ Rmn | 〈A , Y 〉 ≥ b = Y ∈ Rmn | 〈A , Y −Yp〉 ≥ 0 (118)

Recall vector inner-product from §2.2: 〈A , Y 〉= tr(ATY )= vec(A)Tvec(Y ).Hyperplanes in Rmn may, of course, also be represented using matrix

variables.

∂H = Y | 〈A , Y 〉 = b = Y | 〈A , Y −Yp〉 = 0 ⊂ Rmn (119)

Vector a from Figure 25 is normal to the hyperplane illustrated. Likewise,nonzero vectorized matrix A is normal to hyperplane ∂H ;

A ⊥ ∂H in Rmn (120)

2.4.2.2 Vertex-description of hyperplane

Any hyperplane in Rn may be described as affine hull of a minimal set ofpoints xℓ ∈ Rn, ℓ = 1 . . . n arranged columnar in a matrix X∈ Rn×n : (78)

∂H = affxℓ ∈ Rn, ℓ = 1 . . . n , dim affxℓ ∀ ℓ=n−1

= aff X , dim aff X = n−1

= x1 + Rxℓ − x1 , ℓ=2 . . . n , dimRxℓ − x1 , ℓ=2 . . . n=n−1

= x1 + R(X − x11T) , dimR(X − x11

T) = n−1

(121)

whereR(A) = Ax | ∀x (140)

2.4.2.3 Affine independence, minimal set

For any particular affine set, a minimal set of points constituting itsvertex-description is an affinely independent generating set and vice versa.

Arbitrary given points xi∈ Rn, i=1 . . . N are affinely independent(a.i.) if and only if, over all ζ∈ RN Ä ζT1=1, ζk = 0 (confer §2.1.2)

xi ζi + · · · + xj ζj − xk = 0 , i 6= · · · 6=j 6=k = 1 . . . N (122)

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78 CHAPTER 2. CONVEX GEOMETRY

A1

A2A3

0

Figure 27: Any one particular point of three points illustrated does not belongto affine hull Ai (i∈ 1, 2 , 3, each drawn truncated) of points remaining.Three corresponding vectors in R2 are, therefore, affinely independent (butneither linearly or conically independent).

has no solution ζ ; in words, iff no point from the given set can be expressedas an affine combination of those remaining. We deduce

l.i. ⇒ a.i. (123)

Consequently, xi , i=1 . . . N is an affinely independent set if and only ifxi−x1 , i=2 . . . N is a linearly independent (l.i.) set. [202, §3] (Figure 27)

This is equivalent to the property that the columns of

[

X1T

]

(for X∈ Rn×N

as in (76)) form a linearly independent set. [196, §A.1.3]

2.4.2.4 Preservation of affine independence

Independence in the linear (§2.1.2.1), affine, and conic (§2.10.1) senses canbe preserved under linear transformation. Suppose a matrix X∈ Rn×N (76)holds an affinely independent set in its columns. Consider a transformationon the domain of such matrices

T (X) : Rn×N → Rn×N , XY (124)

where fixed matrix Y , [ y1 y2 · · · yN ]∈ RN×N represents linear operator T .Affine independence of Xyi∈ Rn, i=1 . . . N demands (by definition (122))

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2.4. HALFSPACE, HYPERPLANE 79

there exist no solution ζ∈ RN Ä ζT1=1, ζk = 0, to

Xyi ζi + · · · + Xyj ζj − Xyk = 0 , i 6= · · · 6=j 6=k = 1 . . . N (125)

By factoring out X , we see that is ensured by affine independence ofyi∈ RN and by R(Y )∩N (X) = 0 where

N (A) = x | Ax=0 (141)

2.4.2.5 Affine maps

Affine transformations preserve affine hulls. Given any affine mapping T ofvector spaces and some arbitrary set C [303, p.8]

aff(T C) = T (aff C) (126)

2.4.2.6 PRINCIPLE 2: Supporting hyperplane

The second most fundamental principle of convex geometry also follows fromthe geometric Hahn-Banach theorem [246, §5.12] [18, §1] that guaranteesexistence of at least one hyperplane in Rn supporting a full-dimensionalconvex set2.19 at each point on its boundary.

The partial boundary ∂H of a halfspace that contains arbitrary set Y iscalled a supporting hyperplane ∂H to Y when the hyperplane contains atleast one point of Y . [303, §11]

2.4.2.6.1 Definition. Supporting hyperplane ∂H .Assuming set Y and some normal a 6=0 reside in opposite halfspaces2.20

(Figure 29a), then a hyperplane supporting Y at yp∈ ∂Y is

∂H− =

y | aT(y − yp) = 0 , yp∈Y , aT(z − yp)≤ 0 ∀ z∈Y

(127)

Given normal a , instead, the supporting hyperplane is

∂H− =

y | aTy = supaTz | z∈Y

(128)

2.19It is customary to speak of a hyperplane supporting set C but not containing C ;called nontrivial support. [303, p.100] Hyperplanes in support of lower-dimensional bodiesare admitted.2.20Normal a belongs to H+ by definition.

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80 CHAPTER 2. CONVEX GEOMETRY

C

a

z∈ R2 | aTz = κ1z∈ R2 | aTz = κ2

z∈ R2 | aTz = κ3

H−

H+

0 > κ3 > κ2 > κ1

Figure 28: (confer Figure 72) Each linear contour, of equal inner product invector z with normal a , represents ith hyperplane in R2 parametrized byscalar κi . Inner product κi increases in direction of normal a . ith linesegment z∈ C | aTz = κi in convex set C⊂ R2 represents intersection withhyperplane. (Cartesian axes drawn for reference.)

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2.4. HALFSPACE, HYPERPLANE 81

yp

yp

Y

Y

∂H−

∂H+

(a)

(b)

a

a

tradition

nontraditional

H+

H−

H+

H−

Figure 29: (a) Hyperplane ∂H− (127) supporting closed set Y∈R2.Vector a is inward-normal to hyperplane with respect to halfspace H+ ,but outward-normal with respect to set Y . A supporting hyperplane canbe considered the limit of an increasing sequence in the normal-direction likethat in Figure 28. (b) Hyperplane ∂H+ nontraditionally supporting Y .Vector a is inward-normal to hyperplane now with respect to bothhalfspace H+ and set Y . Tradition [196] [303] recognizes only positivenormal polarity in support function σY as in (128); id est, normal a ,figure (a). But both interpretations of supporting hyperplane are useful.

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82 CHAPTER 2. CONVEX GEOMETRY

where real function

σY(a) = supaTz | z∈Y (531)

is called the support function for Y .An equivalent but nontraditional representation2.21 for a supporting

hyperplane is obtained by reversing polarity of normal a ; (1641)

∂H+ =

y | aT(y − yp) = 0 , yp∈Y , aT(z − yp)≥ 0 ∀ z∈Y

=

y | aTy = − infaTz | z∈Y = sup−aTz | z∈Y (129)

where normal a and set Y both now reside in H+ . (Figure 29b)When a supporting hyperplane contains only a single point of Y , that

hyperplane is termed strictly supporting.2.22

A full-dimensional set that has a supporting hyperplane at every pointon its boundary, conversely, is convex. A convex set C⊂ Rn, for example,can be expressed as the intersection of all halfspaces partially bounded byhyperplanes supporting it; videlicet, [246, p.135]

C =⋂

a∈Rn

y | aTy ≤ σC(a)

(130)

by the halfspaces theorem (§2.4.1.1.1).There is no geometric difference between supporting hyperplane ∂H+ or

∂H− or ∂H and2.23 an ordinary hyperplane ∂H coincident with them.

2.4.2.6.2 Example. Minimization over hypercube.Consider minimization of a linear function over a hypercube, given vector c

minimizex

cTx

subject to −1 ¹ x ¹ 1(131)

2.21 useful for constructing the dual cone; e.g., Figure 54b. Tradition would instead haveus construct the polar cone; which is, the negative dual cone.2.22Rockafellar terms a strictly supporting hyperplane tangent to Y if it is unique there;[303, §18, p.169] a definition we do not adopt because our only criterion for tangency isintersection exclusively with a relative boundary. Hiriart-Urruty & Lemarechal [196, p.44](confer [303, p.100]) do not demand any tangency of a supporting hyperplane.2.23If vector-normal polarity is unimportant, we may instead signify a supportinghyperplane by ∂H .

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2.4. HALFSPACE, HYPERPLANE 83

This convex optimization problem is called a linear program 2.24 because theobjective of minimization is linear and the constraints describe a polyhedron(intersection of a finite number of halfspaces and hyperplanes). Applyinggraphical concepts from Figure 26, Figure 28, and Figure 29, an optimalsolution can be shown to be x⋆ =− sgn(c) but is not necessarily unique.Because an optimal solution always exists at a hypercube vertex (§2.6.1.0.1)regardless of the value of nonzero vector c [92], mathematicians see thisgeometry as a means to relax a discrete problem (whose desired solution isinteger or combinatorial, confer Example 4.2.3.1.1). [235, §3.1] [236] 2

2.4.2.6.3 Exercise. Unbounded below.Suppose instead we minimize over the unit hypersphere in Example 2.4.2.6.2;‖x‖ ≤ 1. What is an expression for optimal solution now? Is that programstill linear?

Now suppose minimization of absolute value in (131). Are the followingprograms equivalent for some arbitrary real convex set C ? (confer (495))

minimizex∈R

|x|subject to −1 ≤ x ≤ 1

x ∈ C≡

minimizeα , β

α + β

subject to 1 ≥ β ≥ 0

1 ≥ α ≥ 0

α − β ∈ C

(132)

Many optimization problems of interest and some methods of solutionrequire nonnegative variables. The method illustrated below splits a variableinto parts; x = α − β (extensible to vectors). Under what conditions onvector a and scalar b is an optimal solution x⋆ negative infinity?

minimizeα∈R , β∈R

α − β

subject to β ≥ 0

α ≥ 0

aT

[

αβ

]

= b

(133)

Minimization of the objective function2.25 entails maximization of β . H

2.24The term program has its roots in economics. It was originally meant with regard toa plan or to efficient organization of some industrial process. [92, §2]2.25The objective is the function that is argument to minimization or maximization.

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84 CHAPTER 2. CONVEX GEOMETRY

2.4.2.7 PRINCIPLE 3: Separating hyperplane

The third most fundamental principle of convex geometry again follows fromthe geometric Hahn-Banach theorem [246, §5.12] [18, §1] [129, §I.1.2] thatguarantees existence of a hyperplane separating two nonempty convex setsin Rn whose relative interiors are nonintersecting. Separation intuitivelymeans each set belongs to a halfspace on an opposing side of the hyperplane.There are two cases of interest:

1) If the two sets intersect only at their relative boundaries (§2.1.7.2), thenthere exists a separating hyperplane ∂H containing the intersection butcontaining no points relatively interior to either set. If at least one ofthe two sets is open, conversely, then the existence of a separatinghyperplane implies the two sets are nonintersecting. [59, §2.5.1]

2) A strictly separating hyperplane ∂H intersects the closure of neither set;its existence is guaranteed when intersection of the closures is emptyand at least one set is bounded. [196, §A.4.1]

2.4.3 Angle between hyperspaces

Given halfspace-descriptions, dihedral angle between hyperplanes orhalfspaces is defined as the angle between their defining normals. Givennormals a and b respectively describing ∂Ha and ∂Hb , for example

Á(∂Ha , ∂Hb) , arccos

( 〈a , b〉‖a‖ ‖b‖

)

radians (134)

2.5 Subspace representations

There are two common forms of expression for Euclidean subspaces, bothcoming from elementary linear algebra: range form R and nullspace form N ;a.k.a, vertex-description and halfspace-description respectively.

The fundamental vector subspaces associated with a matrix A∈Rm×n

[328, §3.1] are ordinarily related by orthogonal complement

R(AT) ⊥ N (A) , N (AT) ⊥ R(A) (135)

R(AT) ⊕ N (A) = Rn , N (AT) ⊕ R(A) = Rm (136)

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2.5. SUBSPACE REPRESENTATIONS 85

and of dimension:

dimR(AT) = dimR(A) = rank A ≤ minm,n (137)

with complementarity

dimN (A) = n − rank A , dimN (AT) = m − rank A (138)

These equations (135)-(138) comprise the fundamental theorem of linearalgebra. [328, p.95, p.138]

From these four fundamental subspaces, the rowspace and range identifyone form of subspace description (range form or vertex-description (§2.3.4))

R(AT) , span AT = ATy | y∈Rm = x∈Rn | ATy=x , y∈R(A) (139)

R(A) , span A = Ax | x∈Rn = y∈Rm | Ax=y , x∈R(AT) (140)

while the nullspaces identify the second common form (nullspace form orhalfspace-description (112))

N (A) , x∈Rn | Ax=0 (141)

N (AT) , y∈Rm | ATy=0 (142)

Range forms (139) (140) are realized as the respective span of the columnvectors in matrices AT and A , whereas nullspace form (141) or (142) is thesolution set to a linear equation similar to hyperplane definition (113). Yetbecause matrix A generally has multiple rows, halfspace-description N (A) isactually the intersection of as many hyperplanes through the origin; for (141),each row of A is normal to a hyperplane while each row of AT is a normalfor (142).

2.5.0.0.1 Exercise. Subspace algebra.Given

R(A) + N (AT) = R(B) + N (BT) (143)

prove

R(A) ⊇ N (BT) ⇔ N (AT) ⊆ R(B) (144)

R(A) ⊇ R(B) ⇔ N (AT) ⊆ N (BT) (145)

e.g., Theorem A.3.1.0.6. H

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86 CHAPTER 2. CONVEX GEOMETRY

2.5.1 Subspace or affine subset. . .

Any particular vector subspace Rp can be described as N (A) the nullspaceof some matrix A or as R(B) the range of some matrix B .

More generally, we have the choice of expressing an n−m-dimensionalaffine subset in Rn as the intersection of m hyperplanes, or as the offset spanof n−m vectors:

2.5.1.1 . . . as hyperplane intersection

Any affine subset A of dimension n−m can be described as an intersectionof m hyperplanes in Rn ; given fat (m≤n) full-rank (rank = minm , n)matrix

A ,

aT1...

aTm

∈ Rm×n (146)

and vector b∈Rm,

A , x∈Rn | Ax= b =m⋂

i=1

x | aTi x= bi

(147)

a halfspace-description. (112)

For example: The intersection of any two independent2.26 hyperplanesin R3 is a line, whereas three independent hyperplanes intersect at apoint. In R4, the intersection of two independent hyperplanes is a plane(Example 2.5.1.2.1), whereas three hyperplanes intersect at a line, four at apoint, and so on. A describes a subspace whenever b = 0 in (147).

For n>k

A ∩ Rk = x∈Rn | Ax= b ∩ Rk =m⋂

i=1

x∈Rk | ai(1 :k)Tx= bi

(148)

The result in §2.4.2.2 is extensible; id est, any affine subset A also has avertex-description:

2.26Hyperplanes are said to be independent iff the normals defining them are linearlyindependent.

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2.5. SUBSPACE REPRESENTATIONS 87

2.5.1.2 . . . as span of nullspace basis

Alternatively, we may compute a basis for nullspace of matrix A in (146) andthen equivalently express affine subset A as its span plus an offset: Define

Z , basisN (A)∈ Rn×n−m (149)

so AZ = 0. Then we have the vertex-description in Z ,

A = x∈Rn | Ax = b =

Zξ + xp | ξ∈ Rn−m

⊆ Rn (150)

the offset span of n−m column vectors, where xp is any particular solutionto Ax = b . For example, A describes a subspace whenever xp = 0.

2.5.1.2.1 Example. Intersecting planes in 4-space.Two planes can intersect at a point in four-dimensional Euclidean vectorspace. It is easy to visualize intersection of two planes in three dimensions;a line can be formed. In four dimensions it is harder to visualize. So let’sresort to the tools acquired.

Suppose an intersection of two hyperplanes in four dimensions is specifiedby a fat full-rank matrix A1∈ R2×4 (m = 2, n = 4) as in (147):

A1 ,

x∈R4

[

a11 a12 a13 a14

a21 a22 a23 a24

]

x = b1

(151)

The nullspace of A1 is two dimensional (from Z in (150)), so A1 representsa plane in four dimensions. Similarly define a second plane in terms ofA2∈ R2×4 :

A2 ,

x∈R4

[

a31 a32 a33 a34

a41 a42 a43 a44

]

x = b2

(152)

If the two planes are independent (meaning any line in one is linearlyindependent of any line from the other), they will intersect at a point because

then

[

A1

A2

]

is invertible;

A1 ∩ A2 =

x∈R4

[

A1

A2

]

x =

[

b1

b2

]

(153)

2

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88 CHAPTER 2. CONVEX GEOMETRY

(a)

(b)

A = x |Ax = b

A = x |Ax = b

x | cTx = α

x | cTx = α

c

c

Figure 30: Minimizing hyperplane over affine set A in nonnegativeorthant R2

+ . (a) Optimal solution is • . (b) Optimal objective α⋆ =−∞.

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2.5. SUBSPACE REPRESENTATIONS 89

2.5.1.2.2 Exercise. Linear program.Minimize a hyperplane over affine set A in the nonnegative orthant

minimizex

cTx

subject to Ax = bx º 0

(154)

where A = x | Ax = b. Two cases of interest are drawn in Figure 30.Graphically illustrate and explain optimal solutions indicated in the caption.Why is α⋆ negative in both cases? Is there solution on the vertical axis? H

2.5.2 Intersection of subspaces

The intersection of nullspaces associated with two matrices A∈ Rm×n andB∈ Rk×n can be expressed most simply as

N (A) ∩ N (B) = N([

AB

])

, x∈Rn |[

AB

]

x = 0 (155)

the nullspace of their rowwise concatenation.

Suppose the columns of a matrix Z constitute a basis for N (A) while thecolumns of a matrix W constitute a basis for N (BZ). Then [155, §12.4.2]

N (A) ∩ N (B) = R(ZW ) (156)

If each basis is orthonormal, then the columns of ZW constitute anorthonormal basis for the intersection.

In the particular circumstance A and B are each positive semidefinite[21, §6], or in the circumstance A and B are two linearly independent dyads(§B.1.1), then

N (A) ∩ N (B) = N (A + B) ,

A,B∈ SM+

orA + B = u1v

T1 + u2v

T2 (l.i.)

(157)

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90 CHAPTER 2. CONVEX GEOMETRY

2.6 Extreme, Exposed

2.6.0.0.1 Definition. Extreme point.An extreme point xε of a convex set C is a point, belonging to its closure C[41, §3.3], that is not expressible as a convex combination of points in Cdistinct from xε ; id est, for xε∈ C and all x1 , x2∈ C \xε

µx1 + (1 − µ)x2 6= xε , µ ∈ [0, 1] (158)

In other words, xε is an extreme point of C if and only if xε is not apoint relatively interior to any line segment in C . [355, §2.10]

Borwein & Lewis offer: [53, §4.1.6] An extreme point of a convex set C isa point xε in C whose relative complement C \xε is convex.

The set consisting of a single point C=xε is itself an extreme point.

2.6.0.0.2 Theorem. Extreme existence. [303, §18.5.3] [26, §II.3.5]A nonempty closed convex set containing no lines has at least one extremepoint. ⋄

2.6.0.0.3 Definition. Face, edge. [196, §A.2.3] A face F of convex set C is a convex subset F⊆ C such that everyclosed line segment x1x2 in C , having a relatively interior point(x∈ rel int x1x2) in F , has both endpoints in F . The zero-dimensionalfaces of C constitute its extreme points. The empty set ∅ and C itselfare conventional faces of C . [303, §18] All faces F are extreme sets by definition; id est, for F⊆ C and allx1 , x2∈ C\F

µx1 + (1 − µ)x2 /∈ F , µ ∈ [0, 1] (159) A one-dimensional face of a convex set is called an edge.

Dimension of a face is the penultimate number of affinely independentpoints (§2.4.2.3) belonging to it;

dimF = supρ

dimx2− x1 , x3− x1 , . . . , xρ− x1 | xi∈F , i=1 . . . ρ (160)

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2.6. EXTREME, EXPOSED 91

The point of intersection in C with a strictly supporting hyperplaneidentifies an extreme point, but not vice versa. The nonempty intersection ofany supporting hyperplane with C identifies a face, in general, but not viceversa. To acquire a converse, the concept exposed face requires introduction:

2.6.1 Exposure

2.6.1.0.1 Definition. Exposed face, exposed point, vertex, facet.[196, §A.2.3, A.2.4] F is an exposed face of an n-dimensional convex set C iff there is a

supporting hyperplane ∂H to C such that

F = C ∩ ∂H (161)

Only faces of dimension −1 through n−1 can be exposed by ahyperplane. An exposed point, the definition of vertex, is equivalent to azero-dimensional exposed face; the point of intersection with a strictlysupporting hyperplane. A facet is an (n−1)-dimensional exposed face of an n-dimensionalconvex set C ; facets exist in one-to-one correspondence with the(n−1)-dimensional faces.2.27 exposed points = extreme pointsexposed faces ⊆ faces

2.6.1.1 Density of exposed points

For any closed convex set C , its exposed points constitute a dense subset ofits extreme points; [303, §18] [332] [327, §3.6, p.115] dense in the sense [370]that closure of that subset yields the set of extreme points.

For the convex set illustrated in Figure 31, point B cannot be exposedbecause it relatively bounds both the facet AB and the closed quarter circle,each bounding the set. Since B is not relatively interior to any line segmentin the set, then B is an extreme point by definition. Point B may be regardedas the limit of some sequence of exposed points beginning at vertex C .

2.27This coincidence occurs simply because the facet’s dimension is the same as thedimension of the supporting hyperplane exposing it.

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92 CHAPTER 2. CONVEX GEOMETRY

BA

D

C

Figure 31: Closed convex set in R2. Point A is exposed hence extreme;a classical vertex. Point B is extreme but not an exposed point. Point Cis exposed and extreme; zero-dimensional exposure makes it a vertex.Point D is neither an exposed or extreme point although it belongs to aone-dimensional exposed face. [196, §A.2.4] [327, §3.6] Closed face AB isexposed; a facet. The arc is not a conventional face, yet it is composedentirely of extreme points. Union of all rotations of this entire set aboutits vertical edge produces another convex set in three dimensions havingno edges; but that convex set produced by rotation about horizontal edgecontaining D has edges.

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2.6. EXTREME, EXPOSED 93

2.6.1.2 Face transitivity and algebra

Faces of a convex set enjoy transitive relation. If F1 is a face (an extreme set)of F2 which in turn is a face of F3 , then it is always true that F1 is aface of F3 . (The parallel statement for exposed faces is false. [303, §18])For example, any extreme point of F2 is an extreme point of F3 ; inthis example, F2 could be a face exposed by a hyperplane supportingpolyhedron F3 . [219, def.115/6, p.358] Yet it is erroneous to presume thata face, of dimension 1 or more, consists entirely of extreme points. Nor is aface of dimension 2 or more entirely composed of edges, and so on.

For the polyhedron in R3 from Figure 20, for example, the nonemptyfaces exposed by a hyperplane are the vertices, edges, and facets; thereare no more. The zero-, one-, and two-dimensional faces are in one-to-onecorrespondence with the exposed faces in that example.

Define the smallest face F , that contains some element G , of a convexset C :

F(C ∋G) (162)

videlicet, C ⊇ F(C ∋G) ∋ G . An affine set has no faces except itself and theempty set. The smallest face, that contains G , of the intersection of convexset C with an affine set A [235, §2.4] [236]

F((C∩A)∋G) = F(C ∋G) ∩ A (163)

equals the intersection of A with the smallest face, that contains G , of set C .

2.6.1.3 Conventional boundary

(confer §2.1.7.2) Relative boundary

rel ∂ C = C \ rel int C (24)

is equivalent to:

2.6.1.3.1 Definition. Conventional boundary of convex set. [196, §C.3.1]The relative boundary ∂ C of a nonempty convex set C is the union of allexposed faces of C .

Equivalence to (24) comes about because it is conventionally presumedthat any supporting hyperplane, central to the definition of exposure, doesnot contain C . [303, p.100] Any face F of convex set C (that is not C itself)belongs to rel ∂ C . (§2.8.2.1)

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94 CHAPTER 2. CONVEX GEOMETRY

0

X

X

(a)0

(b)

Figure 32: (a) Two-dimensional nonconvex cone drawn truncated. Boundaryof this cone is itself a cone. Each polar half is itself a convex cone. (b) Thisconvex cone (drawn truncated) is a line through the origin in any dimension.It has no relative boundary, while its relative interior comprises entire line.

2.7 Cones

In optimization, convex cones achieve prominence because they generalizesubspaces. Most compelling is the projection analogy: Projection on asubspace can be ascertained from projection on its orthogonal complement(§E), whereas projection on a closed convex cone can be determined fromprojection instead on its algebraic complement (§2.13, §E.9.2.1); called thepolar cone.

2.7.0.0.1 Definition. Ray.The one-dimensional set

ζΓ + B | ζ ≥ 0 , Γ 6= 0 ⊂ Rn (164)

defines a halfline called a ray in nonzero direction Γ∈Rn having baseB∈Rn. When B=0, a ray is the conic hull of direction Γ ; hence aconvex cone.

Relative boundary of a single ray, base 0 in any dimension, is the originbecause that is the union of all exposed faces not containing the entire set.Its relative interior is the ray itself excluding the origin.

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2.7. CONES 95

0

Figure 33: This nonconvex cone in R2 is a pair of lines through the origin.[246, §2.4]

0

Figure 34: Boundary of a convex cone in R2 is a nonconvex cone; a pair ofrays emanating from the origin.

X

X

Figure 35: Union of two pointed closed convex cones is nonconvex cone X .

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96 CHAPTER 2. CONVEX GEOMETRY

X

X

Figure 36: Truncated nonconvex cone X = x ∈ R2 | x1 ≥ x2 , x1 x2 ≥ 0.Boundary is also a cone. [246, §2.4] Cartesian axes drawn for reference. Eachhalf (about the origin) is itself a convex cone.

0

X

Figure 37: Nonconvex cone X drawn truncated in R2. Boundary is also acone. [246, §2.4] Cone exterior is convex cone.

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2.7. CONES 97

2.7.1 Cone defined

A set X is called, simply, cone if and only if

Γ ∈ X ⇒ ζΓ ∈ X for all ζ ≥ 0 (165)

where X denotes closure of cone X . An example of such a cone is the unionof two opposing quadrants; e.g., X =x∈R2 | x1 x2≥ 0 which is not convex.[368, §2.5] Similar examples are shown in Figure 32 and Figure 36.

All cones can be defined by an aggregate of rays emanating exclusivelyfrom the origin (but not all cones are convex). Hence all closed cones containthe origin 0 and are unbounded, excepting the simplest cone 0. Theempty set ∅ is not a cone, but its conic hull is;

cone ∅ = 0 (103)

2.7.2 Convex cone

We call the set K⊆ RM a convex cone iff

Γ1 , Γ2 ∈ K ⇒ ζΓ1 + ξΓ2 ∈ K for all ζ , ξ ≥ 0 (166)

id est, if and only if any conic combination of elements from K belongs to itsclosure. Apparent from this definition, ζΓ1 ∈ K and ξΓ2 ∈ K ∀ ζ , ξ≥ 0 ;meaning, K is a cone. Set K is convex since, for any particular ζ , ξ≥ 0

µ ζΓ1 + (1 − µ) ξΓ2 ∈ K ∀µ ∈ [0, 1] (167)

because µ ζ , (1 − µ) ξ ≥ 0. Obviously,

X ⊃ K (168)

the set of all convex cones is a proper subset of all cones. The set ofconvex cones is a narrower but more familiar class of cone, any memberof which can be equivalently described as the intersection of a possibly(but not necessarily) infinite number of hyperplanes (through the origin)and halfspaces whose bounding hyperplanes pass through the origin; ahalfspace-description (§2.4). Convex cones need not be full-dimensional.

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98 CHAPTER 2. CONVEX GEOMETRY

Figure 38: Not a cone; ironically, the three-dimensional flared horn (withor without its interior) resembling mathematical symbol ≻ denoting strictcone membership and partial order.

More familiar convex cones are Lorentz cone (confer Figure 45)2.28

Kℓ =

[

xt

]

∈ Rn× R | ‖x‖ℓ ≤ t

, ℓ=2 (169)

and polyhedral cone (§2.12.1.0.1); e.g., any orthant generated by Cartesianhalf-axes (§2.1.3). Esoteric examples of convex cones include the point at theorigin, any line through the origin, any ray having the origin as base suchas the nonnegative real line R+ in subspace R , any halfspace partiallybounded by a hyperplane through the origin, the positive semidefinitecone SM

+ (182), the cone of Euclidean distance matrices EDMN (869)(§6), completely positive semidefinite matrices CCT |C≥ 0 [39, p.71], anysubspace, and Euclidean vector space Rn.

2.28 a.k.a: second-order cone, quadratic cone, circular cone (§2.9.2.5.1), unboundedice-cream cone united with its interior.

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2.7. CONES 99

2.7.2.1 cone invariance

More Euclidean bodies are cones, it seems, than are not.2.29 This class ofconvex body, the convex cone, is invariant to scaling, linear and single- ormany-valued inverse linear transformation, vector summation, and Cartesianproduct, but is not invariant to translation. [303, p.22]

2.7.2.1.1 Theorem. Cone intersection (nonempty). Intersection of an arbitrary collection of convex cones is a convex cone.[303, §2, §19] Intersection of an arbitrary collection of closed convex cones is a closedconvex cone. [254, §2.3] Intersection of a finite number of polyhedral cones (Figure 49 p.136,§2.12.1.0.1) remains a polyhedral cone. ⋄

The property pointedness is associated with a convex cone; but, pointed cone < convex cone (Figure 34, Figure 35)

2.7.2.1.2 Definition. Pointed convex cone. (confer §2.12.2.2)A convex cone K is pointed iff it contains no line. Equivalently, K is notpointed iff there exists any nonzero direction Γ∈ K such that −Γ ∈ K . Ifthe origin is an extreme point of K or, equivalently, if

K ∩ −K = 0 (170)

then K is pointed, and vice versa. [327, §2.10] A convex cone is pointed iffthe origin is the smallest nonempty face of its closure.

Then a pointed closed convex cone, by principle of separating hyperplane(§2.4.2.7), has a strictly supporting hyperplane at the origin. The simplestand only bounded [368, p.75] convex cone K= 0 ⊆ Rn is pointed, byconvention, but not full-dimensional. Its relative boundary is the empty set∅ (25) while its relative interior is the point 0 itself (12). The pointed convex

2.29confer Figures: 24 32 33 34 37 36 38 40 42 49 53 56 58 59 61 62 63 64 65 136149 172

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100 CHAPTER 2. CONVEX GEOMETRY

cone that is a halfline, emanating from the origin in Rn, has relative boundary0 while its relative interior is the halfline itself excluding 0. Pointed areany Lorentz cone, cone of Euclidean distance matrices EDMN in symmetrichollow subspace SN

h , and positive semidefinite cone SM+ in ambient SM .

2.7.2.1.3 Theorem. Pointed cones. [53, §3.3.15, exer.20]A closed convex cone K⊂ Rn is pointed if and only if there exists a normal αsuch that the set

C , x∈K | 〈x , α〉 = 1 (171)

is closed, bounded, and K= cone C . Equivalently, K is pointed if and onlyif there exists a vector β normal to a hyperplane strictly supporting K ;id est, for some positive scalar ǫ

〈x , β〉 ≥ ǫ‖x‖ ∀x∈K (172)

If closed convex cone K is not pointed, then it has no extreme point.2.30

Yet a pointed closed convex cone has only one extreme point [41, §3.3]: theexposed point residing at the origin; its vertex. Pointedness is invariantto Cartesian product by (170). And from the cone intersection theorem itfollows that an intersection of convex cones is pointed if at least one of thecones is; implying, each and every nonempty exposed face of a pointed closedconvex cone is a pointed closed convex cone.

2.7.2.2 Pointed closed convex cone induces partial order

Relation ¹ is a partial order on some set if the relation possesses2.31

reflexivity (x¹x)

antisymmetry (x¹z , z¹x ⇒ x=z)

transitivity (x¹ y , y¹z ⇒ x¹z), (x¹ y , y≺z ⇒ x≺z)

2.30 nor does it have extreme directions (§2.8.1).2.31A set is totally ordered if it further obeys a comparability property of the relation: foreach and every x and y from the set, x¹ y or y¹ x ; e.g., R is the smallest unboundedtotally ordered and connected set.

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2.7. CONES 101

C1

C2

x + K

y −K

x

y

(b)

(a)

R2

Figure 39: (confer Figure 68) (a) Point x is the minimum element of set C1

with respect to cone K because cone translated to x∈ C1 contains entire set.(Cones drawn truncated.) (b) Point y is a minimal element of set C2 withrespect to cone K because negative cone translated to y∈ C2 contains only y .These concepts, minimum/minimal, become equivalent under a total order.

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102 CHAPTER 2. CONVEX GEOMETRY

A pointed closed convex cone K induces partial order on Rn or Rm×n, [21, §1][321, p.7] essentially defined by vector or matrix inequality;

x ¹K

z ⇔ z − x ∈ K (173)

x ≺K

z ⇔ z − x ∈ rel intK (174)

Neither x or z is necessarily a member of K for these relations to hold. Onlywhen K is the nonnegative orthant do these inequalities reduce to ordinaryentrywise comparison. (§2.13.4.2.3) Inclusive of that special case, we ascribenomenclature generalized inequality to comparison with respect to a pointedclosed convex cone.

We say two points x and y are comparable when x¹ y or y¹ xwith respect to pointed closed convex cone K . Visceral mechanics ofactually comparing points, when cone K is not an orthant, are wellillustrated in the example of Figure 62 which relies on the equivalentmembership-interpretation in definition (173) or (174).

Comparable points and the minimum element of some vector- ormatrix-valued partially ordered set are thus well defined, so nonincreasingsequences with respect to cone K can therefore converge in this sense: Pointx ∈ C is the (unique) minimum element of set C with respect to cone K ifffor each and every z ∈ C we have x¹ z ; equivalently, iff C ⊆ x + K .2.32

A closely related concept, minimal element, is useful for partially orderedsets having no minimum element: Point x ∈ C is a minimal elementof set C with respect to pointed closed convex cone K if and only if(x −K) ∩ C = x . (Figure 39) No uniqueness is implied here, althoughimplicit is the assumption: dimK ≥ dim aff C . In words, a point that isa minimal element is smaller (with respect to K) than any other point in theset to which it is comparable.

Further properties of partial order with respect to pointed closed convexcone K are not defining:

homogeneity (x¹ y , λ≥0 ⇒ λx¹λz), (x≺ y , λ>0 ⇒ λx≺λz)

additivity (x¹z , u¹v ⇒ x+u¹ z+ v), (x≺z , u¹v ⇒ x+u≺ z+ v)

2.32Borwein & Lewis [53, §3.3, exer.21] ignore possibility of equality to x + K in thiscondition, and require a second condition: . . . and C ⊂ y + K for some y in Rn impliesx ∈ y + K .

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2.7.2.2.1 Definition. Proper cone: a cone that is pointed closed convex full-dimensional.

A proper cone remains proper under injective linear transformation.[88, §5.1] Examples of proper cones are the positive semidefinite cone SM

+ inthe ambient space of symmetric matrices (§2.9), the nonnegative real line R+

in vector space R , or any orthant in Rn.

2.8 Cone boundary

Every hyperplane supporting a convex cone contains the origin. [196, §A.4.2]Because any supporting hyperplane to a convex cone must therefore itself bea cone, then from the cone intersection theorem (§2.7.2.1.1) it follows:

2.8.0.0.1 Lemma. Cone faces. [26, §II.8]Each nonempty exposed face of a convex cone is a convex cone. ⋄

2.8.0.0.2 Theorem. Proper-cone boundary.Suppose a nonzero point Γ lies on the boundary ∂K of proper cone K in Rn.Then it follows that the ray ζΓ | ζ ≥ 0 also belongs to ∂K . ⋄

Proof. By virtue of its propriety, a proper cone guarantees existenceof a strictly supporting hyperplane at the origin. [303, cor.11.7.3]2.33 Hencethe origin belongs to the boundary of K because it is the zero-dimensionalexposed face. The origin belongs to the ray through Γ , and the ray belongsto K by definition (165). By the cone faces lemma, each and every nonemptyexposed face must include the origin. Hence the closed line segment 0Γ mustlie in an exposed face of K because both endpoints do by Definition 2.6.1.3.1.That means there exists a supporting hyperplane ∂H to K containing 0Γ .

2.33Rockafellar’s corollary yields a supporting hyperplane at the origin to any convex conein Rn not equal to Rn.

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104 CHAPTER 2. CONVEX GEOMETRY

So the ray through Γ belongs both to K and to ∂H . ∂H must thereforeexpose a face of K that contains the ray; id est,

ζΓ | ζ ≥ 0 ⊆ K ∩ ∂H ⊂ ∂K (175)

¨

Proper cone 0 in R0 has no boundary (24) because (12)

rel int0 = 0 (176)

The boundary of any proper cone in R is the origin.

The boundary of any convex cone whose dimension exceeds 1 can beconstructed entirely from an aggregate of rays emanating exclusively fromthe origin.

2.8.1 Extreme direction

The property extreme direction arises naturally in connection with thepointed closed convex cone K⊂ Rn, being analogous to extreme point.[303, §18, p.162]2.34 An extreme direction Γε of pointed K is a vectorcorresponding to an edge that is a ray emanating from the origin.2.35

Nonzero direction Γε in pointed K is extreme if and only if

ζ1 Γ1+ ζ2 Γ2 6= Γε ∀ ζ1 , ζ2 ≥ 0 , ∀ Γ1 , Γ2 ∈ K\ζΓε∈K | ζ≥0 (177)

In words, an extreme direction in a pointed closed convex cone is thedirection of a ray, called an extreme ray, that cannot be expressed as a coniccombination of directions of any rays in the cone distinct from it.

An extreme ray is a one-dimensional face of K . By (104), extremedirection Γε is not a point relatively interior to any line segment inK\ζΓε∈K | ζ≥0. Thus, by analogy, the corresponding extreme rayζΓε∈K | ζ≥0 is not a ray relatively interior to any plane segment 2.36

in K .

2.34We diverge from Rockafellar’s extreme direction: “extreme point at infinity”.2.35An edge (§2.6.0.0.3) of a convex cone is not necessarily a ray. A convex cone may

contain an edge that is a line; e.g., a wedge-shaped polyhedral cone (K∗in Figure 40).

2.36A planar fragment; in this context, a planar cone.

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2.8. CONE BOUNDARY 105

K

∂K∗

Figure 40: K is a pointed polyhedral cone not full-dimensional in R3 (drawntruncated in a plane parallel to the floor upon which you stand). Dual coneK∗

is a wedge whose truncated boundary is illustrated (drawn perpendicularto the floor). In this particular instance, K⊂ intK∗

(excepting the origin).Cartesian coordinate axes drawn for reference.

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106 CHAPTER 2. CONVEX GEOMETRY

2.8.1.1 extreme distinction, uniqueness

An extreme direction is unique, but its vector representation Γε is notbecause any positive scaling of it produces another vector in the same(extreme) direction. Hence an extreme direction is unique to within a positivescaling. When we say extreme directions are distinct, we are referring todistinctness of rays containing them. Nonzero vectors of various length inthe same extreme direction are therefore interpreted to be identical extremedirections.2.37

The extreme directions of the polyhedral cone in Figure 24 (p.71), forexample, correspond to its three edges. For any pointed polyhedral cone,there is a one-to-one correspondence of one-dimensional faces with extremedirections.

The extreme directions of the positive semidefinite cone (§2.9) comprisethe infinite set of all symmetric rank-one matrices. [21, §6] [192, §III] Itis sometimes prudent to instead consider the less infinite but completenormalized set, for M > 0 (confer (215))

zzT∈ SM | ‖z‖= 1 (178)

The positive semidefinite cone in one dimension M =1, S+ the nonnegativereal line, has one extreme direction belonging to its relative interior; anidiosyncrasy of dimension 1.

Pointed closed convex cone K= 0 has no extreme direction becauseextreme directions are nonzero by definition. If closed convex cone K is not pointed, then it has no extreme directions

and no vertex. [21, §1]

Conversely, pointed closed convex cone K is equivalent to the convex hullof its vertex and all its extreme directions. [303, §18, p.167] That is thepractical utility of extreme direction; to facilitate construction of polyhedralsets, apparent from the extremes theorem:

2.37Like vectors, an extreme direction can be identified with the Cartesian point at thevector’s head with respect to the origin.

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2.8.1.1.1 Theorem. (Klee) Extremes. [327, §3.6] [303, §18, p.166](confer §2.3.2, §2.12.2.0.1) Any closed convex set containing no lines can beexpressed as the convex hull of its extreme points and extreme rays. ⋄

It follows that any element of a convex set containing no lines maybe expressed as a linear combination of its extreme elements; e.g.,Example 2.9.2.4.1.

2.8.1.2 Generators

In the narrowest sense, generators for a convex set comprise any collectionof points and directions whose convex hull constructs the set.

When the extremes theorem applies, the extreme points and directionsare called generators of a convex set. An arbitrary collection of generatorsfor a convex set includes its extreme elements as a subset; the set of extremeelements of a convex set is a minimal set of generators for that convex set.Any polyhedral set has a minimal set of generators whose cardinality is finite.

When the convex set under scrutiny is a closed convex cone, coniccombination of generators during construction is implicit as shown inExample 2.8.1.2.1 and Example 2.10.2.0.1. So, a vertex at the origin (if itexists) becomes benign.

We can, of course, generate affine sets by taking the affine hull of anycollection of points and directions. We broaden, thereby, the meaning ofgenerator to be inclusive of all kinds of hulls.

Any hull of generators is loosely called a vertex-description. (§2.3.4)Hulls encompass subspaces, so any basis constitutes generators for avertex-description; span basisR(A).

2.8.1.2.1 Example. Application of extremes theorem.Given an extreme point at the origin and N extreme rays, denoting the ith

extreme direction by Γi∈ Rn, then the convex hull is (86)

P =

[0 Γ1 Γ2 · · · ΓN ] a ζ | aT1 = 1, a º 0, ζ ≥ 0

=

[Γ1 Γ2 · · · ΓN ] a ζ | aT1 ≤ 1, a º 0, ζ ≥ 0

=

[Γ1 Γ2 · · · ΓN ] b | b º 0

⊂ Rn(179)

a closed convex set that is simply a conic hull like (102). 2

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108 CHAPTER 2. CONVEX GEOMETRY

2.8.2 Exposed direction

2.8.2.0.1 Definition. Exposed point & direction of pointed convex cone.[303, §18] (confer §2.6.1.0.1) When a convex cone has a vertex, an exposed point, it resides at the

origin; there can be only one. In the closure of a pointed convex cone, an exposed direction is thedirection of a one-dimensional exposed face that is a ray emanatingfrom the origin. exposed directions ⊆ extreme directions

For a proper cone in vector space Rn with n≥ 2, we can say more:

exposed directions = extreme directions (180)

It follows from Lemma 2.8.0.0.1 for any pointed closed convex cone, thereis one-to-one correspondence of one-dimensional exposed faces with exposeddirections; id est, there is no one-dimensional exposed face that is not a raybase 0.

The pointed closed convex cone EDM2, for example, is a ray inisomorphic subspace R whose relative boundary (§2.6.1.3.1) is the origin.The conventionally exposed directions of EDM2 constitute the empty set∅ ⊂ extreme direction. This cone has one extreme direction belonging toits relative interior; an idiosyncrasy of dimension 1.

2.8.2.1 Connection between boundary and extremes

2.8.2.1.1 Theorem. Exposed. [303, §18.7] (confer §2.8.1.1.1)Any closed convex set C containing no lines (and whose dimension is atleast 2) can be expressed as closure of the convex hull of its exposed pointsand exposed rays. ⋄

From Theorem 2.8.1.1.1,

rel ∂ C = C \ rel int C (24)

= convexposed points and exposed rays \ rel int C= convextreme points and extreme rays \ rel int C

(181)

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2.8. CONE BOUNDARY 109

D

CB

A

0

Figure 41: Properties of extreme points carry over to extreme directions.[303, §18] Four rays (drawn truncated) on boundary of conic hull oftwo-dimensional closed convex set from Figure 31 lifted to R3. Raythrough point A is exposed hence extreme. Extreme direction B on coneboundary is not an exposed direction, although it belongs to the exposedface coneA , B. Extreme ray through C is exposed. Point D is neitheran exposed or extreme direction although it belongs to a two-dimensionalexposed face of the conic hull.

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110 CHAPTER 2. CONVEX GEOMETRY

Thus each and every extreme point of a convex set (that is not a point)resides on its relative boundary, while each and every extreme direction of aconvex set (that is not a halfline and contains no line) resides on its relativeboundary because extreme points and directions of such respective sets donot belong to relative interior by definition.

The relationship between extreme sets and the relative boundary actuallygoes deeper: Any face F of convex set C (that is not C itself) belongs torel ∂ C , so dimF < dim C . [303, §18.1.3]

2.8.2.2 Converse caveat

It is inconsequent to presume that each and every extreme point and directionis necessarily exposed, as might be erroneously inferred from the conventionalboundary definition (§2.6.1.3.1); although it can correctly be inferred: eachand every extreme point and direction belongs to some exposed face.

Arbitrary points residing on the relative boundary of a convex set are notnecessarily exposed or extreme points. Similarly, the direction of an arbitraryray, base 0, on the boundary of a convex cone is not necessarily an exposedor extreme direction. For the polyhedral cone illustrated in Figure 24, forexample, there are three two-dimensional exposed faces constituting theentire boundary, each composed of an infinity of rays. Yet there are onlythree exposed directions.

Neither is an extreme direction on the boundary of a pointed convex conenecessarily an exposed direction. Lift the two-dimensional set in Figure 31,for example, into three dimensions such that no two points in the set arecollinear with the origin. Then its conic hull can have an extreme directionB on the boundary that is not an exposed direction, illustrated in Figure 41.

2.9 Positive semidefinite (PSD) cone

The cone of positive semidefinite matrices studied in this sectionis arguably the most important of all non-polyhedral cones whosefacial structure we completely understand.

−Alexander Barvinok [26, p.78]

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2.9.0.0.1 Definition. Positive semidefinite cone.The set of all symmetric positive semidefinite matrices of particulardimension M is called the positive semidefinite cone:

SM+ ,

A ∈ SM | A º 0

=

A ∈ SM | yTAy≥ 0 ∀ ‖y‖= 1

=⋂

‖y‖=1

A ∈ SM | 〈yyT, A〉 ≥ 0

= A ∈ SM+ | rank A ≤ M

(182)

formed by the intersection of an infinite number of halfspaces (§2.4.1.1) invectorized variable2.38 A , each halfspace having partial boundary containingthe origin in isomorphic RM(M+1)/2. It is a unique immutable proper conein the ambient space of symmetric matrices SM .

The positive definite (full-rank) matrices comprise the cone interior

int SM+ =

A ∈ SM | A ≻ 0

=

A ∈ SM | yTAy> 0 ∀ ‖y‖= 1

=⋂

‖y‖=1

A ∈ SM | 〈yyT, A〉 > 0

= A ∈ SM+ | rank A = M

(183)

while all singular positive semidefinite matrices (having at least one0 eigenvalue) reside on the cone boundary (Figure 42); (§A.7.5)

∂SM+ =

A ∈ SM | A º 0 , A ⊁ 0

=

A ∈ SM | minλ(A)i , i=1 . . . M = 0

=

A ∈ SM+ | 〈yyT, A〉=0 for some ‖y‖= 1

=

A ∈ SM+ | rank A < M

(184)

where λ(A)∈RM holds the eigenvalues of A .

The only symmetric positive semidefinite matrix in SM+ having M

0-eigenvalues resides at the origin. (§A.7.3.0.1)

2.38 infinite in number when M >1. Because yTA y=yTATy , matrix A is almost alwaysassumed symmetric. (§A.2.1)

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112 CHAPTER 2. CONVEX GEOMETRY

α

√2β

γ

svec ∂ S2

+

[

α ββ γ

]

Minimal set of generators are the extreme directions: svecyyT | y∈RM

Figure 42: (d’Aspremont) Truncated boundary of PSD cone in S2 plotted inisometrically isomorphic R3 via svec (56); 0-contour of smallest eigenvalue(184). Lightest shading is closest, darkest shading is farthest and inside shell.Entire boundary can be constructed from an aggregate of rays (§2.7.0.0.1)emanating exclusively from the origin:

κ2[ z21

√2z1z2 z2

2 ]T | κ∈R

. Acircular cone in this dimension (§2.9.2.5), each and every ray on boundarycorresponds to an extreme direction but such is not the case in any higherdimension (confer Figure 24). PSD cone geometry is not as simple in higherdimensions [26, §II.12] although PSD cone is selfdual (355) in ambient realspace of symmetric matrices. [192, §II] PSD cone has no two-dimensionalface in any dimension, its only extreme point residing at 0.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 113

2.9.0.1 Membership

Observe the notation Aº 0 ; meaning (confer §2.3.1.1), matrix A belongs tothe positive semidefinite cone in the subspace of symmetric matrices, whereasA≻ 0 denotes membership to that cone’s interior. (§2.13.2) Notation A≻ 0can be read: symmetric matrix A is greater than the origin with respect tothe positive semidefinite cone. This notation further implies that coordinates[sic] for orthogonal expansion of a positive (semi)definite matrix must be its(nonnegative) positive eigenvalues (§2.13.7.1.1, §E.6.4.1.1) when expanded inits eigenmatrices (§A.5.0.3).

Generalizing comparison on the real line, the notation AºB denotescomparison with respect to the positive semidefinite cone; (§A.3.1) id est,AºB ⇔ A−B ∈ SM

+ but neither matrix A or B necessarily belongs tothe positive semidefinite cone. Yet, (1459) AºB , Bº 0 ⇒ Aº0 ; id est,A∈ SM

+ . (confer Figure 62)

2.9.0.1.1 Example. Equality constraints in semidefinite program (625).Employing properties of partial order (§2.7.2.2) for the pointed closed convexpositive semidefinite cone, it is easy to show, given A + S = C

S º 0 ⇔ A ¹ CS ≻ 0 ⇔ A ≺ C

(185)

2

2.9.1 Positive semidefinite cone is convex

The set of all positive semidefinite matrices forms a convex cone in theambient space of symmetric matrices because any pair satisfies definition(166); [199, §7.1] videlicet, for all ζ1 , ζ2 ≥ 0 and each and every A1 , A2 ∈ SM

ζ1 A1 + ζ2 A2 º 0 ⇐ A1 º 0 , A2 º 0 (186)

a fact easily verified by the definitive test for positive semidefiniteness of asymmetric matrix (§A):

A º 0 ⇔ xTAx ≥ 0 for each and every ‖x‖= 1 (187)

id est, for A1 , A2 º 0 and each and every ζ1 , ζ2 ≥ 0

ζ1 xTA1 x + ζ2 xTA2 x ≥ 0 for each and every normalized x ∈ RM (188)

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114 CHAPTER 2. CONVEX GEOMETRY

X

x

C

Figure 43: Convex set C=X∈ S × x∈R | Xº xxT drawn truncated.

The convex cone SM+ is more easily visualized in the isomorphic vector

space RM(M+1)/2 whose dimension is the number of free variables in asymmetric M×M matrix. When M = 2 the PSD cone is semiinfinite inexpanse in R3, having boundary illustrated in Figure 42. When M = 3 thePSD cone is six-dimensional, and so on.

2.9.1.0.1 Example. Sets from maps of positive semidefinite cone.The set

C = X∈ Sn× x∈Rn | Xº xxT (189)

is convex because it has Schur-form; (§A.4)

X − xxT º 0 ⇔ f(X , x) ,

[

X xxT 1

]

º 0 (190)

e.g., Figure 43. Set C is the inverse image (§2.1.9.0.1) of Sn+1+ under affine

mapping f . The set X∈ Sn× x∈Rn | X¹ xxT is not convex, in contrast,having no Schur-form. Yet for fixed x = xp , the set

X∈ Sn | X¹ xpxTp (191)

is simply the negative semidefinite cone shifted to xpxTp . 2

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 115

2.9.1.0.2 Example. Inverse image of positive semidefinite cone.Now consider finding the set of all matrices X∈ SN satisfying

AX + B º 0 (192)

given A ,B∈ SN . Define the set

X , X | AX + B º 0 ⊆ SN (193)

which is the inverse image of the positive semidefinite cone under affinetransformation g(X),AX+B . Set X must therefore be convex byTheorem 2.1.9.0.1.

Yet we would like a less amorphous characterization of this set, so insteadwe consider its vectorization (37) which is easier to visualize:

vec g(X) = vec(AX) + vec B = (I ⊗A) vec X + vec B (194)

whereI ⊗A , QΛQT ∈ SN 2

(195)

is block-diagonal formed by Kronecker product (§A.1.1 no.31, §D.1.2.1).Assign

x , vec X ∈ RN 2

b , vec B ∈ RN 2 (196)

then make the equivalent problem: Find

vecX = x∈RN 2 | (I ⊗A)x + b ∈ K (197)

whereK , vec SN

+ (198)

is a proper cone isometrically isomorphic with the positive semidefinite conein the subspace of symmetric matrices; the vectorization of every element ofSN

+ . Utilizing the diagonalization (195),

vecX = x | ΛQTx ∈ QT(K − b)= x | ΦQTx ∈ Λ†QT(K − b) ⊆ RN 2 (199)

where † denotes matrix pseudoinverse (§E) and

Φ , Λ†Λ (200)

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116 CHAPTER 2. CONVEX GEOMETRY

is a diagonal projection matrix whose entries are either 1 or 0 (§E.3). Wehave the complementary sum

ΦQTx + (I − Φ)QTx = QTx (201)

So, adding (I −Φ)QTx to both sides of the membership within (199) admits

vecX = x∈RN 2 | QTx ∈ Λ†QT(K − b) + (I − Φ)QTx= x | QTx ∈ Φ

(

Λ†QT(K − b))

⊕ (I − Φ)RN 2= x ∈ QΛ†QT(K − b) ⊕ Q(I − Φ)RN 2= (I ⊗A)†(K − b) ⊕ N (I ⊗A)

(202)

where we used the facts: linear function QTx in x on RN 2

is a bijection, andΦΛ†= Λ† .

vecX = (I ⊗A)† vec(SN+ − B) ⊕ N (I ⊗A) (203)

In words, set vecX is the vector sum of the translated PSD cone(linearly mapped onto the rowspace of I ⊗A (§E)) and the nullspace ofI ⊗A (synthesis of fact from §A.6.3 and §A.7.3.0.1). Should I ⊗A have nonullspace, then vecX =(I ⊗A)−1 vec(SN

+ − B) which is the expected result.2

2.9.2 Positive semidefinite cone boundary

For any symmetric positive semidefinite matrix A of rank ρ , there mustexist a rank ρ matrix Y such that A be expressible as an outer productin Y ; [328, §6.3]

A = Y Y T ∈ SM+ , rank A=ρ , Y ∈ RM×ρ (204)

Then the boundary of the positive semidefinite cone may be expressed

∂SM+ =

A ∈ SM+ | rank A<M

=

Y Y T | Y ∈ RM×M−1

(205)

Because the boundary of any convex body is obtained with closure of itsrelative interior (§2.1.7, §2.1.7.2), from (183) we must also have

SM+ =

A ∈ SM+ | rank A=M

=

Y Y T | Y ∈ RM×M , rank Y =M

=

Y Y T | Y ∈ RM×M

(206)

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 117

(a) (b)

√2β

α

[

α ββ γ

]

svec S2

+

Figure 44: (a) Projection of truncated PSD cone S2

+ , truncated above γ =1,on αβ-plane in isometrically isomorphic R3. View is from above with respectto Figure 42. (b) Truncated above γ =2. From these plots we might infer,for example, line

[ 0 1/√

2 γ ]T | γ∈R

intercepts PSD cone at some largevalue of γ ; in fact, γ =∞.

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118 CHAPTER 2. CONVEX GEOMETRY

2.9.2.1 rank ρ subset of the positive semidefinite cone

For the same reason (closure), this applies more generally; for 0≤ρ≤M

A ∈ SM+ | rank A= ρ

=

A ∈ SM+ | rank A≤ ρ

(207)

For easy reference, we give such generally nonconvex sets a name: rank ρsubset of a positive semidefinite cone. For ρ < M this subset, nonconvex forM > 1, resides on the positive semidefinite cone boundary.

2.9.2.1.1 Exercise. Closure and rank ρ subset.Prove equality in (207). H

For example,

∂SM+ =

A ∈ SM+ | rank A=M− 1

=

A ∈ SM+ | rank A≤M− 1

(208)

In S2, each and every ray on the boundary of the positive semidefinite conein isomorphic R3 corresponds to a symmetric rank-1 matrix (Figure 42), butthat does not hold in any higher dimension.

2.9.2.2 Subspace tangent to open rank ρ subset

When the positive semidefinite cone subset in (207) is left unclosed as in

M(ρ) ,

A ∈ SN+ | rank A= ρ

(209)

then we can specify a subspace tangent to the positive semidefinite coneat a particular member of manifold M(ρ). Specifically, the subspace RMtangent to manifold M(ρ) at B∈M(ρ) [183, §5, prop.1.1]

RM(B) , XB + BXT | X∈ RN×N ⊆ SN (210)

has dimension

dim svecRM(B) = ρ

(

N − ρ − 1

2

)

= ρ(N − ρ) +ρ(ρ + 1)

2(211)

Tangent subspace RM contains no member of the positive semidefinite coneSN

+ whose rank exceeds ρ .Subspace RM(B) is a hyperplane supporting SN

+ when B∈M(N−1).Another good example of tangent subspace is given in §E.7.2.0.2 by (1962);RM(11T) = SN⊥

c , orthogonal complement to the geometric center subspace.(Figure 146, p.522)

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 119

2.9.2.3 Faces of PSD cone, their dimension versus rank

Each and every face of the positive semidefinite cone, having dimension lessthan that of the cone, is exposed. [242, §6] [211, §2.3.4] Because each andevery face of the positive semidefinite cone contains the origin (§2.8.0.0.1),each face belongs to a subspace of dimension the same as the face.

Define F(SM+ ∋A) (162) as the smallest face, that contains a given

positive semidefinite matrix A , of positive semidefinite cone SM+ . Then

matrix A , having ordered diagonalization QΛQT (§A.5.1), is relativelyinterior to [26, §II.12] [110, §31.5.3] [235, §2.4] [236] (§A.7.4)

F(

SM+ ∋A

)

= X∈ SM+ | N (X) ⊇ N (A)

= X∈ SM+ | 〈Q(I − ΛΛ†)QT, X 〉 = 0

= QΛΛ†ΨΛΛ†QT |Ψ∈ SM+

= QΛΛ† SM+ ΛΛ†QT

≃ Srank A+

(212)

which is isomorphic with convex cone Srank A+ . Thus dimension of the smallest

face containing given matrix A is

dimF(

SM+ ∋A

)

= rank(A)(rank(A) + 1)/2 (213)

in isomorphic RM(M+1)/2, and each and every face of SM+ is isomorphic with

a positive semidefinite cone having dimension the same as the face. Observe:not all dimensions are represented, and the only zero-dimensional face is theorigin. The positive semidefinite cone has no facets, for example.

2.9.2.3.1 Table: Rank k versus dimension of S3

+ faces

k dimF(S3

+∋ rank-k matrix)0 0

boundary 1 12 3

interior 3 6

For the positive semidefinite cone S2

+ in isometrically isomorphic R3

depicted in Figure 42, for example, rank-2 matrices belong to the interior

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120 CHAPTER 2. CONVEX GEOMETRY

of that face having dimension 3 (the entire closed cone), rank-1 matricesbelong to relative interior of a face having dimension2.39 1, and the onlyrank-0 matrix is the point at the origin (the zero-dimensional face).

Any rank-k<M positive semidefinite matrix A belongs to a face, of thepositive semidefinite cone, described by intersection with a hyperplane: forA=QΛQT and 0≤k<M ,

F(

SM+ ∋A Ä rank(A)= k

)

= X∈ SM+ | 〈Q(I − ΛΛ†)QT, X 〉 = 0

=

X∈ SM+

Q

(

I −[

I∈ Sk 00T 0

])

QT, X

= 0

= SM+ ∩ ∂H+ (214)

Faces are doubly indexed: continuously indexed by orthogonal matrix Q ,and discretely indexed by rank k . Each and every orthogonal matrix Qmakes projectors Q(: , k+1:M)Q(: , k+1:M)T indexed by k , in other words,each projector describing a normal svec

(

Q(: , k+1:M)Q(: , k+1:M)T)

to asupporting hyperplane ∂H+ (containing the origin) exposing a face (§2.11)of the positive semidefinite cone containing only rank-k matrices.

2.9.2.3.2 Exercise. Simultaneously diagonalizable means commutative.Given diagonalization of rank-k≤M positive semidefinite matrix A = QΛQT

and any particular Ψº 0, both in SM from (212), show how I−ΛΛ† andΛΛ†ΨΛΛ† share a complete set of eigenvectors. H

2.9.2.4 Extreme directions of positive semidefinite cone

Because the positive semidefinite cone is pointed (§2.7.2.1.2), there is aone-to-one correspondence of one-dimensional faces with extreme directionsin any dimension M ; id est, because of the cone faces lemma (§2.8.0.0.1)and the direct correspondence of exposed faces to faces of SM

+ , it followsthere is no one-dimensional face of the positive semidefinite cone that is nota ray emanating from the origin.

Symmetric dyads constitute the set of all extreme directions: For M > 1

yyT∈ SM | y∈RM ⊂ ∂SM+ (215)

2.39The boundary constitutes all the one-dimensional faces, in R3, which are raysemanating from the origin.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 121

this superset of extreme directions (infinite in number, confer (178)) for thepositive semidefinite cone is, generally, a subset of the boundary. By theextremes theorem (§2.8.1.1.1), the convex hull of extreme rays and the originis the PSD cone:

convyyT∈ SM | y∈RM = SM+ (216)

For two-dimensional matrices (M =2, Figure 42)

yyT∈ S2 | y∈R2 = ∂S2

+ (217)

while for one-dimensional matrices, in exception, (M =1, §2.7)

yy∈ S | y 6=0 = int S+ (218)

Each and every extreme direction yyT makes the same angle with theidentity matrix in isomorphic RM(M+1)/2, dependent only on dimension;videlicet,2.40

Á(yyT, I ) = arccos〈yyT, I 〉

‖yyT‖F ‖I‖F

= arccos

(

1√M

)

∀ y ∈ RM (219)

2.9.2.4.1 Example. Positive semidefinite matrix from extreme directions.Diagonalizability (§A.5) of symmetric matrices yields the following results:

Any symmetric positive semidefinite matrix (1426) can be written in theform

A =∑

i

λi zizTi = AAT =

i

ai aTi º 0 , λ º 0 (220)

a conic combination of linearly independent extreme directions (ai aTi or ziz

Ti

where ‖zi‖=1), where λ is a vector of eigenvalues.If we limit consideration to all symmetric positive semidefinite matrices

bounded via unity trace

C , A º 0 | tr A = 1 (91)

then any matrix A from that set may be expressed as a convex combinationof linearly independent extreme directions;

A =∑

i

λi zizTi ∈ C , 1Tλ = 1 , λ º 0 (221)

Implications are:

2.40Analogy with respect to the EDM cone is considered in [181, p.162] where it is found:angle is not constant. Extreme directions of the EDM cone can be found in §6.5.3.2. Thecone’s axis is −E = 11T− I (1071).

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122 CHAPTER 2. CONVEX GEOMETRY

1. set C is convex (an intersection of PSD cone with hyperplane),

2. because the set of eigenvalues corresponding to a given square matrix Ais unique (§A.5.0.1), no single eigenvalue can exceed 1 ; id est, I ºA

3. and the converse holds: set C is an instance of Fantope (91). 2

2.9.2.5 Positive semidefinite cone is generally not circular

Extreme angle equation (219) suggests that the positive semidefinite conemight be invariant to rotation about its axis of revolution; id est, a circularcone. We investigate this now:

2.9.2.5.1 Definition. Circular cone:2.41

a pointed closed convex cone having hyperspherical sections orthogonal toits axis of revolution about which the cone is invariant to rotation.

A conic section is the intersection of a cone with any hyperplane. In threedimensions, an intersecting plane perpendicular to a circular cone’s axis ofrevolution produces a section bounded by a circle. (Figure 45) A prominentexample of a circular cone in convex analysis is Lorentz cone (169). Wealso find that the positive semidefinite cone and cone of Euclidean distancematrices are circular cones, but only in low dimension.

The positive semidefinite cone has axis of revolution that is the ray(base 0) through the identity matrix I . Consider a set of normalized extremedirections of the positive semidefinite cone: for some arbitrary positiveconstant a∈R+

yyT∈ SM | ‖y‖ =√

a ⊂ ∂SM+ (222)

The distance from each extreme direction to the axis of revolution is radius

R , infc‖yyT− cI‖F = a

1 − 1

M(223)

which is the distance from yyT to aM

I ; the length of vector yyT− aM

I .Because distance R (in a particular dimension) from the axis of revolution

to each and every normalized extreme direction is identical, the extreme

2.41A circular cone is assumed convex throughout, although not so by other authors. Wealso assume a right circular cone.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 123

0

R

Figure 45: This solid circular cone in R3 continues upward infinitely. Axis ofrevolution is illustrated as vertical line through origin. R represents radius:distance measured from an extreme direction to axis of revolution. Were thisa Lorentz cone, any plane slice containing axis of revolution would make aright angle.

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124 CHAPTER 2. CONVEX GEOMETRY

aM

Iθ a

M11T

yyT

R

Figure 46: Illustrated is a section, perpendicular to axis of revolution, ofcircular cone from Figure 45. Radius R is distance from any extremedirection to axis at a

MI . Vector a

M11T is an arbitrary reference by which

to measure angle θ .

directions lie on the boundary of a hypersphere in isometrically isomorphicRM(M+1)/2. From Example 2.9.2.4.1, the convex hull (excluding vertex atthe origin) of the normalized extreme directions is a conic section

C , convyyT | y∈RM , yTy = a = SM+ ∩ A∈ SM | 〈I , A〉 = a (224)

orthogonal to identity matrix I ;

〈C− a

MI , I 〉 = tr(C− a

MI ) = 0 (225)

Proof. Although the positive semidefinite cone possesses somecharacteristics of a circular cone, we can show it is not by demonstratingshortage of extreme directions; id est, some extreme directions correspondingto each and every angle of rotation about the axis of revolution arenonexistent: Referring to Figure 46, [377, §1-7]

cos θ =

aM

11T− aM

I , yyT− aM

I⟩

a2(1 − 1M

)(226)

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 125

Solving for vector y we get

a(1 + (M−1) cos θ) = (1Ty)2 (227)

which does not have real solution ∀ 0≤ θ≤ 2π in every matrix dimension M .¨

From the foregoing proof we can conclude that the positive semidefinitecone might be circular but only in matrix dimensions 1 and 2. Because of ashortage of extreme directions, conic section (224) cannot be hypersphericalby the extremes theorem (§2.8.1.1.1, Figure 41).

2.9.2.5.2 Exercise. Circular semidefinite cone.Prove the PSD cone to be circular in matrix dimensions 1 and 2 while it isa rotation of Lorentz cone (169) in matrix dimension 2 .2.42 H

2.9.2.5.3 Example. PSD cone inscription in three dimensions.

Theorem. Gersgorin discs. [199, §6.1] [361] [251, p.140]For p∈Rm

+ given A=[Aij]∈ Sm, then all eigenvalues of A belong to theunion of m closed intervals on the real line;

λ(A) ∈m⋃

i=1

ξ ∈ R |ξ − Aii| ≤ i ,1

pi

m∑

j=1

j 6= i

pj |Aij|

=m⋃

i=1

[Aii−i , Aii+i]

(228)

Furthermore, if a union of k of these m [intervals ] forms a connectedregion that is disjoint from all the remaining n−k [intervals ], thenthere are precisely k eigenvalues of A in this region. ⋄

To apply the theorem to determine positive semidefiniteness of symmetricmatrix A , we observe that for each i we must have

Aii ≥ i (229)

2.42Hint: Given cone

[

α β/√

2

β/√

2 γ

]

|√

α2+ β2 ≤ γ

, show 1√2

[

γ + α ββ γ − α

]

is

a vector rotation that is positive semidefinite under the same inequality.

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126 CHAPTER 2. CONVEX GEOMETRY

0

0.5

1

-1

-0.5

0

0.5

1

0

0.25

0.5

0.75

1

-1

-0.5

0

0.5

1

0

0.5

1

-1

-0.5

0

0.5

1

0

0.25

0.5

0.75

1

-1

-0.5

0

0.5

1

0

0.5

1

-1

-0.5

0

0.5

1

0

0.25

0.5

0.75

1

-1

-0.5

0

0.5

1

p =

[

1/21

]

p =

[

11

]

p =

[

21

]

A11

√2A12

A22

svec ∂S2

+

p1A11 ≥ ±p2A12

p2A22 ≥ ±p1A12

Figure 47: Proper polyhedral cone K , created by intersection of halfspaces,inscribes PSD cone in isometrically isomorphic R3 as predicted by Gersgorindiscs theorem for A=[Aij]∈ S2. Hyperplanes supporting K intersect alongboundary of PSD cone. Four extreme directions of K coincide with extremedirections of PSD cone.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 127

Suppose

m = 2 (230)

so A∈ S2. Vectorizing A as in (56), svec A belongs to isometricallyisomorphic R3. Then we have m2m−1 = 4 inequalities, in the matrix entriesAij with Gersgorin parameters p =[pi]∈ R2

+ ,

p1A11 ≥ ±p2A12

p2A22 ≥ ±p1A12(231)

which describe an intersection of four halfspaces in Rm(m+1)/2. Thatintersection creates the proper polyhedral cone K (§2.12.1) whoseconstruction is illustrated in Figure 47. Drawn truncated is the boundaryof the positive semidefinite cone svec S2

+ and the bounding hyperplanessupporting K .

Created by means of Gersgorin discs, K always belongs to the positivesemidefinite cone for any nonnegative value of p ∈ Rm

+ . Hence any point inK corresponds to some positive semidefinite matrix A . Only the extremedirections of K intersect the positive semidefinite cone boundary in thisdimension; the four extreme directions of K are extreme directions of thepositive semidefinite cone. As p1/p2 increases in value from 0, two extremedirections of K sweep the entire boundary of this positive semidefinite cone.Because the entire positive semidefinite cone can be swept by K , the systemof linear inequalities

Y Tsvec A ,

[

p1 ±p2/√

2 0

0 ±p1/√

2 p2

]

svec A º 0 (232)

when made dynamic can replace a semidefinite constraint Aº 0 ; id est, for

K = z | Y Tz º 0 ⊂ svec Sm+ (233)

given p where Y ∈ Rm(m+1)/2×m2m−1

svec A ∈ K ⇒ A ∈ Sm+ (234)

but

∃ p Ä Y Tsvec A º 0 ⇔ A º 0 (235)

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128 CHAPTER 2. CONVEX GEOMETRY

In other words, diagonal dominance [199, p.349, §7.2.3]

Aii ≥m

j=1

j 6= i

|Aij| , ∀ i = 1 . . . m (236)

is only a sufficient condition for membership to the PSD cone; but bydynamic weighting p in this dimension, it was made necessary andsufficient. 2

In higher dimension (m > 2), boundary of the positive semidefinite coneis no longer constituted completely by its extreme directions (symmetricrank-one matrices); its geometry becomes intricate. How all the extremedirections can be swept by an inscribed polyhedral cone,2.43 similarly to theforegoing example, remains an open question.

2.9.2.5.4 Exercise. Dual inscription.Find dual proper polyhedral cone K∗

from Figure 47. H

2.9.2.6 Boundary constituents of the positive semidefinite cone

2.9.2.6.1 Lemma. Sum of positive semidefinite matrices.For A,B∈ SM

+

rank(A + B) = rank(µA + (1−µ)B) (237)

over the open interval (0, 1) of µ . ⋄

Proof. Any positive semidefinite matrix belonging to the PSD conehas an eigenvalue decomposition that is a positively scaled sum of linearlyindependent symmetric dyads. By the linearly independent dyads definitionin §B.1.1.0.1, rank of the sum A+B is equivalent to the number of linearlyindependent dyads constituting it. Linear independence is insensitive tofurther positive scaling by µ . The assumption of positive semidefinitenessprevents annihilation of any dyad from the sum A +B . ¨

2.43It is not necessary to sweep the entire boundary in higher dimension.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 129

2.9.2.6.2 Example. Rank function quasiconcavity. (confer §3.9)For A,B∈ Rm×n [199, §0.4]

rank A + rankB ≥ rank(A + B) (238)

that follows from the fact [328, §3.6]

dimR(A) + dimR(B) = dimR(A + B) + dim(R(A) ∩R(B)) (239)

For A,B∈ SM+

rank A + rankB ≥ rank(A + B) ≥ minrank A , rank B (1442)

that follows from the fact

N (A + B) = N (A) ∩ N (B) , A,B∈ SM+ (157)

Rank is a quasiconcave function on SM+ because the right-hand inequality in

(1442) has the concave form (619); videlicet, Lemma 2.9.2.6.1. 2

From this example we see, unlike convex functions, quasiconvex functionsare not necessarily continuous. (§3.9) We also glean:

2.9.2.6.3 Theorem. Convex subsets of positive semidefinite cone.Subsets of the positive semidefinite cone SM

+ , for 0≤ρ≤M

SM+ (ρ) , X∈ SM

+ | rank X ≥ ρ (240)

are pointed convex cones, but not closed unless ρ = 0 ; id est, SM+ (0)= SM

+ .⋄

Proof. Given ρ , a subset SM+ (ρ) is convex if and only if

convex combination of any two members has rank at least ρ . That isconfirmed applying identity (237) from Lemma 2.9.2.6.1 to (1442); id est,for A,B∈ SM

+ (ρ) on the closed interval µ∈ [0 , 1]

rank(µA + (1−µ)B) ≥ minrank A , rank B (241)

It can similarly be shown, almost identically to proof of the lemma, any coniccombination of A,B in subset SM

+ (ρ) remains a member; id est, ∀ ζ , ξ≥ 0

rank(ζA + ξB) ≥ minrank(ζA) , rank(ξB) (242)

Therefore, SM+ (ρ) is a convex cone. ¨

Another proof of convexity can be made by projection arguments:

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130 CHAPTER 2. CONVEX GEOMETRY

2.9.2.7 Projection on SM+ (ρ)

Because these cones SM+ (ρ) indexed by ρ (240) are convex, projection on

them is straightforward. Given a symmetric matrix H having diagonalizationH , QΛQT∈ SM (§A.5.1) with eigenvalues Λ arranged in nonincreasingorder, then its Euclidean projection (minimum-distance projection) on SM

+ (ρ)

PSM+ (ρ)H = QΥ⋆QT (243)

corresponds to a map of its eigenvalues:

Υ⋆ii =

max ǫ , Λii , i=1 . . . ρmax 0 , Λii , i=ρ+1 . . . M

(244)

where ǫ is positive but arbitrarily close to 0.

2.9.2.7.1 Exercise. Projection on open convex cones.Prove (244) using Theorem E.9.2.0.1. H

Because each H∈ SM has unique projection on SM+ (ρ) (despite possibility

of repeated eigenvalues in Λ), we may conclude it is a convex set by theBunt-Motzkin theorem (§E.9.0.0.1).

Compare (244) to the well-known result regarding Euclidean projectionon a rank ρ subset of the positive semidefinite cone (§2.9.2.1)

SM+ \SM

+ (ρ + 1) = X∈ SM+ | rank X ≤ ρ (207)

PSM+ \SM

+ (ρ+1)H = QΥ⋆QT (245)

As proved in §7.1.4, this projection of H corresponds to the eigenvalue map

Υ⋆ii =

max 0 , Λii , i=1 . . . ρ0 , i=ρ+1 . . . M

(1317)

Together these two results (244) and (1317) mean: A higher-rank solutionto projection on the positive semidefinite cone lies arbitrarily close to anygiven lower-rank projection, but not vice versa. Were the number ofnonnegative eigenvalues in Λ known a priori not to exceed ρ , then thesetwo different projections would produce identical results in the limit ǫ→ 0.

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2.9. POSITIVE SEMIDEFINITE (PSD) CONE 131

2.9.2.8 Uniting constituents

Interior of the PSD cone int SM+ is convex by Theorem 2.9.2.6.3, for example,

because all positive semidefinite matrices having rank M constitute the coneinterior.

All positive semidefinite matrices of rank less than M constitute the coneboundary; an amalgam of positive semidefinite matrices of different rank.Thus each nonconvex subset of positive semidefinite matrices, for 0<ρ<M

Y ∈ SM+ | rank Y = ρ (246)

having rank ρ successively 1 lower than M , appends a nonconvexconstituent to the cone boundary; but only in their union is the boundarycomplete: (confer §2.9.2)

∂SM+ =

M−1⋃

ρ=0

Y ∈ SM+ | rank Y = ρ (247)

The composite sequence, the cone interior in union with each successiveconstituent, remains convex at each step; id est, for 0≤k≤M

M⋃

ρ=k

Y ∈ SM+ | rank Y = ρ (248)

is convex for each k by Theorem 2.9.2.6.3.

2.9.2.9 Peeling constituents

Proceeding the other way: To peel constituents off the complete positivesemidefinite cone boundary, one starts by removing the origin; the onlyrank-0 positive semidefinite matrix. What remains is convex. Next, theextreme directions are removed because they constitute all the rank-1 positivesemidefinite matrices. What remains is again convex, and so on. Proceedingin this manner eventually removes the entire boundary leaving, at last, theconvex interior of the PSD cone; all the positive definite matrices.

2.9.2.9.1 Exercise. Difference A − B .What about the difference of matrices A,B belonging to the positivesemidefinite cone? Show:

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132 CHAPTER 2. CONVEX GEOMETRY The difference of any two points on the boundary belongs to theboundary or exterior. The difference A−B , where A belongs to the boundary while B isinterior, belongs to the exterior. H

2.9.3 Barvinok’s proposition

Barvinok posits existence and quantifies an upper bound on rank of a positivesemidefinite matrix belonging to the intersection of the PSD cone with anaffine subset:

2.9.3.0.1 Proposition. (Barvinok) Affine intersection with PSD cone.[26, §II.13] [24, §2.2] Consider finding a matrix X∈ SN satisfying

X º 0 , 〈Aj , X 〉 = bj , j =1 . . . m (249)

given nonzero linearly independent (vectorized) Aj ∈ SN and real bj . Definethe affine subset

A , X | 〈Aj , X 〉= bj , j =1 . . . m ⊆ SN (250)

If the intersection A ∩ SN+ is nonempty given a number m of equalities,

then there exists a matrix X∈A ∩ SN+ such that

rank X (rank X + 1)/2 ≤ m (251)

whence the upper bound2.44

rank X ≤⌊√

8m + 1 − 1

2

(252)

Given desired rank instead, equivalently,

m < (rank X + 1)(rank X + 2)/2 (253)

An extreme point of A ∩ SN+ satisfies (252) and (253). (confer §4.1.1.3)

A matrix X ,RTR is an extreme point if and only if the smallest face, thatcontains X , of A ∩ SN

+ has dimension 0 ; [235, §2.4] [236] id est, iff (162)

2.44 §4.1.1.3 contains an intuitive explanation. This bound is itself limited above, of course,by N ; a tight limit corresponding to an interior point of SN

+ .

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2.10. CONIC INDEPENDENCE (C.I.) 133

dimF(

(A ∩ SN+ )∋X

)

= rank(X )(rank(X ) + 1)/2 − rank[

svec RA1RT svec RA2R

T · · · svec RAmRT]

(254)

equals 0 in isomorphic RN(N+1)/2.Now the intersection A ∩ SN

+ is assumed bounded: Assume a givennonzero upper bound ρ on rank, a number of equalities

m=(ρ + 1)(ρ + 2)/2 (255)

and matrix dimension N≥ ρ + 2≥ 3. If the intersection is nonempty andbounded, then there exists a matrix X∈A ∩ SN

+ such that

rank X ≤ ρ (256)

This represents a tightening of the upper bound; a reduction by exactly 1of the bound provided by (252) given the same specified number m (255) ofequalities; id est,

rank X ≤√

8m + 1 − 1

2− 1 (257)

When intersection A ∩ SN+ is known a priori to consist only of a single

point, then Barvinok’s proposition provides the greatest upper bound onits rank not exceeding N . The intersection can be a single nonzero pointonly if the number of linearly independent hyperplanes m constituting Asatisfies2.45

N(N + 1)/2 − 1 ≤ m ≤ N(N + 1)/2 (258)

2.10 Conic independence (c.i.)

In contrast to extreme direction, the property conically independentdirection is more generally applicable; inclusive of all closed convexcones (not only pointed closed convex cones). Arbitrary given directions

2.45For N >1, N(N+1)/2−1 independent hyperplanes in RN(N+1)/2 can make a linetangent to svec ∂SN

+ at a point because all one-dimensional faces of SN+ are exposed.

Because a pointed convex cone has only one vertex, the origin, there can be no intersectionof svec ∂SN

+ with any higher-dimensional affine subset A that will make a nonzero point.

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134 CHAPTER 2. CONVEX GEOMETRY

000

(a) (b) (c)

Figure 48: Vectors in R2 : (a) affinely and conically independent,(b) affinely independent but not conically independent, (c) conicallyindependent but not affinely independent. None of the examples exhibitslinear independence. (In general, a.i. < c.i.)

Γi∈ Rn, i=1 . . . N are conically independent if and only if (confer §2.1.2,§2.4.2.3)

Γi ζi + · · · + Γj ζj − Γℓ = 0 , i 6= · · · 6=j 6=ℓ = 1 . . . N (259)

has no solution ζ∈ RN+ ; in words, iff no direction from the given set can

be expressed as a conic combination of those remaining; e.g., Figure 48.[Matlab implementation of test (259) on Wıκımization] It is evident thatlinear independence (l.i.) of N directions implies their conic independence; l.i. ⇒ c.i.

Arranging any set of generators for a particular convex cone in a matrixcolumnar,

X , [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N (260)

then the relationship l.i. ⇒ c.i. suggests: number of l.i. generators in thecolumns of X cannot exceed number of c.i. generators. Denoting by k thenumber of conically independent generators contained in X , we have themost fundamental rank inequality for convex cones

dim aff K = dim aff[0 X ] = rank X ≤ k ≤ N (261)

Whereas N directions in n dimensions can no longer be linearly independentonce N exceeds n , conic independence remains possible:

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2.10. CONIC INDEPENDENCE (C.I.) 135

2.10.0.0.1 Table: Maximum number of c.i. directions

dimension n supk (pointed) supk (not pointed)

0 0 01 1 22 2 43 ∞ ∞...

......

Assuming veracity of this table, there is an apparent vastness between twoand three dimensions. The finite numbers of conically independent directionsindicate: Convex cones in dimensions 0, 1, and 2 must be polyhedral. (§2.12.1)

Conic independence is certainly one convex idea that cannot be completelyexplained by a two-dimensional picture as Barvinok suggests [26, p.vii].

From this table it is also evident that dimension of Euclidean space cannotexceed the number of conically independent directions possible; n ≤ supk

2.10.1 Preservation of conic independence

Independence in the linear (§2.1.2.1), affine (§2.4.2.4), and conic senses canbe preserved under linear transformation. Suppose a matrix X∈ Rn×N (260)holds a conically independent set columnar. Consider a transformation onthe domain of such matrices

T (X) : Rn×N → Rn×N , XY (262)

where fixed matrix Y , [ y1 y2 · · · yN ]∈ RN×N represents linear operator T .Conic independence of Xyi∈ Rn, i=1 . . . N demands, by definition (259),

Xyi ζi + · · · + Xyj ζj − Xyℓ = 0 , i 6= · · · 6=j 6=ℓ = 1 . . . N (263)

have no solution ζ∈ RN+ . That is ensured by conic independence of yi∈ RN

and by R(Y )∩N (X) = 0 ; seen by factoring out X .

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136 CHAPTER 2. CONVEX GEOMETRY

K

K

∂K∗

(a)

(b)

Figure 49: (a) A pointed polyhedral cone (drawn truncated) in R3 having sixfacets. The extreme directions, corresponding to six edges emanating fromthe origin, are generators for this cone; not linearly independent but theymust be conically independent. (b) The boundary of dual cone K∗

(drawntruncated) is now added to the drawing of same K . K∗

is polyhedral, proper,and has the same number of extreme directions as K has facets.

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2.10. CONIC INDEPENDENCE (C.I.) 137

2.10.1.1 linear maps of cones

[21, §7] If K is a convex cone in Euclidean space R and T is any linearmapping from R to Euclidean space M , then T (K) is a convex cone in Mand x ¹ y with respect to K implies T (x)¹ T (y) with respect to T (K).If K is full-dimensional in R , then so is T (K) in M .

If T is a linear bijection, then x ¹ y ⇔ T (x)¹ T (y). If K is pointed,then so is T (K). And if K is closed, so is T (K). If F is a face of K , thenT (F ) is a face of T (K).

Linear bijection is only a sufficient condition for pointedness andclosedness; convex polyhedra (§2.12) are invariant to any linear or inverselinear transformation [26, §I.9] [303, p.44, thm.19.3].

2.10.2 Pointed closed convex K & conic independence

The following bullets can be derived from definitions (177) and (259) inconjunction with the extremes theorem (§2.8.1.1.1):

The set of all extreme directions from a pointed closed convex cone K⊂Rn

is not necessarily a linearly independent set, yet it must be a conicallyindependent set; (compare Figure 24 on page 71 with Figure 49a) extreme directions ⇒ c.i.

When a conically independent set of directions from pointed closed convexcone K is known to comprise generators, conversely, then all directions fromthat set must be extreme directions of the cone; extreme directions ⇔ c.i. generators of pointed closed convex K

Barker & Carlson [21, §1] call the extreme directions a minimal generatingset for a pointed closed convex cone. A minimal set of generators is thereforea conically independent set of generators, and vice versa,2.46 for a pointedclosed convex cone.

An arbitrary collection of n or fewer distinct extreme directions frompointed closed convex cone K⊂ Rn is not necessarily a linearly independentset; e.g., dual extreme directions (461) from Example 2.13.11.0.3. ≤ n extreme directions in Rn ; l.i.2.46This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1implies; e.g., ponder four conically independent generators for a plane (n=2, Figure 48).

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138 CHAPTER 2. CONVEX GEOMETRY

0

H

x1

x2

x3

∂H

Figure 50: Minimal set of generators X = [ x1 x2 x3 ]∈ R2×3 (not extremedirections) for halfspace about origin; affinely and conically independent.

Linear dependence of few extreme directions is another convex idea thatcannot be explained by a two-dimensional picture as Barvinok suggests[26, p.vii]; indeed, it only first comes to light in four dimensions! But thereis a converse: [327, §2.10.9] extreme directions ⇐ l.i. generators of closed convex K

2.10.2.0.1 Example. Vertex-description of halfspace H about origin.From n + 1 points in Rn we can make a vertex-description of a convexcone that is a halfspace H , where xℓ∈Rn, ℓ=1 . . . n constitutes aminimal set of generators for a hyperplane ∂H through the origin. Anexample is illustrated in Figure 50. By demanding the augmented setxℓ∈Rn, ℓ=1 . . . n + 1 be affinely independent (we want vector xn+1 notparallel to ∂H ), then

H =⋃

ζ≥0

(ζ xn+1 + ∂H)

= ζ xn+1 + conexℓ∈Rn, ℓ=1 . . . n | ζ≥ 0

= conexℓ∈Rn, ℓ=1 . . . n + 1

(264)

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2.11. WHEN EXTREME MEANS EXPOSED 139

a union of parallel hyperplanes. Cardinality is one step beyond dimension ofthe ambient space, but xℓ ∀ ℓ is a minimal set of generators for this convexcone H which has no extreme elements. 2

2.10.2.0.2 Exercise. Enumerating conically independent directions.Describe a nonpointed polyhedral cone in three dimensions having more than8 conically independent generators. (confer Table 2.10.0.0.1) H

2.10.3 Utility of conic independence

Perhaps the most useful application of conic independence is determinationof the intersection of closed convex cones from their halfspace-descriptions,or representation of the sum of closed convex cones from theirvertex-descriptions.

⋂Ki A halfspace-description for the intersection of any number of closedconvex cones Ki can be acquired by pruning normals; specifically,only the conically independent normals from the aggregate of all thehalfspace-descriptions need be retained.

∑Ki Generators for the sum of any number of closed convex cones Ki canbe determined by retaining only the conically independent generatorsfrom the aggregate of all the vertex-descriptions.

Such conically independent sets are not necessarily unique or minimal.

2.11 When extreme means exposed

For any convex full-dimensional polyhedral set in Rn, distinction betweenthe terms extreme and exposed vanishes [327, §2.4] [110, §2.2] for faces ofall dimensions except n ; their meanings become equivalent as we saw inFigure 20 (discussed in §2.6.1.2). In other words, each and every face of anypolyhedral set (except the set itself) can be exposed by a hyperplane, andvice versa; e.g., Figure 24.

Lewis [242, §6] [211, §2.3.4] claims nonempty extreme proper subsets andthe exposed subsets coincide for Sn

+ ; id est, each and every face of the positivesemidefinite cone, whose dimension is less than dimension of the cone, isexposed. A more general discussion of cones having this property can befound in [338]; e.g., Lorentz cone (169) [20, §II.A].

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140 CHAPTER 2. CONVEX GEOMETRY

2.12 Convex polyhedra

Every polyhedron, such as the convex hull (86) of a bounded list X , canbe expressed as the solution set of a finite system of linear equalities andinequalities, and vice versa. [110, §2.2]

2.12.0.0.1 Definition. Convex polyhedra, halfspace-description.A convex polyhedron is the intersection of a finite number of halfspaces andhyperplanes;

P = y | Ay º b , Cy = d ⊆ Rn (265)

where coefficients A and C generally denote matrices. Each row of C is avector normal to a hyperplane, while each row of A is a vector inward-normalto a hyperplane partially bounding a halfspace.

By the halfspaces theorem in §2.4.1.1.1, a polyhedron thus described isa closed convex set possibly not full-dimensional; e.g., Figure 20. Convexpolyhedra2.47 are finite-dimensional comprising all affine sets (§2.3.1),polyhedral cones, line segments, rays, halfspaces, convex polygons, solids[219, def.104/6, p.343], polychora, polytopes,2.48 etcetera.

It follows from definition (265) by exposure that each face of a convexpolyhedron is a convex polyhedron.

The projection of any polyhedron on a subspace remains a polyhedron.More generally, image and inverse image of a convex polyhedron under anylinear transformation remains a convex polyhedron; [26, §I.9] [303, thm.19.3]the foremost consequence being, invariance of polyhedral set closedness.

When b and d in (265) are 0, the resultant is a polyhedral cone. Theset of all polyhedral cones is a subset of convex cones:

2.12.1 Polyhedral cone

From our study of cones, we see: the number of intersecting hyperplanesand halfspaces constituting a convex cone is possibly but not necessarilyinfinite. When the number is finite, the convex cone is termed polyhedral.That is the primary distinguishing feature between the set of all convex

2.47We consider only convex polyhedra throughout, but acknowledge the existence ofconcave polyhedra. [370, Kepler-Poinsot Solid ]2.48Some authors distinguish bounded polyhedra via the designation polytope. [110, §2.2]

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2.12. CONVEX POLYHEDRA 141

cones and polyhedra; all polyhedra, including polyhedral cones, are finitelygenerated [303, §19]. (Figure 51) We distinguish polyhedral cones in the setof all convex cones for this reason, although all convex cones of dimension 2or less are polyhedral.

2.12.1.0.1 Definition. Polyhedral cone, halfspace-description.2.49

(confer (102)) A polyhedral cone is the intersection of a finite number ofhalfspaces and hyperplanes about the origin;

K = y | Ay º 0 , Cy = 0 ⊆ Rn (a)

= y | Ay º 0 , Cy º 0 , Cy ¹ 0 (b)

=

y |

AC

−C

y º 0

(c)

(266)

where coefficients A and C generally denote matrices of finite dimension.Each row of C is a vector normal to a hyperplane containing the origin,while each row of A is a vector inward-normal to a hyperplane containingthe origin and partially bounding a halfspace.

A polyhedral cone thus defined is closed, convex (§2.4.1.1), has onlya finite number of generators (§2.8.1.2), and can be not full-dimensional.(Minkowski) Conversely, any finitely generated convex cone is polyhedral.(Weyl) [327, §2.8] [303, thm.19.1]

From the definition it follows that any single hyperplane through theorigin, or any halfspace partially bounded by a hyperplane through the originis a polyhedral cone. The most familiar example of polyhedral cone is anyquadrant (or orthant, §2.1.3) generated by Cartesian half-axes. Esotericexamples of polyhedral cone include the point at the origin, any line throughthe origin, any ray having the origin as base such as the nonnegative realline R+ in subspace R , polyhedral flavor (proper) Lorentz cone (295), anysubspace, and Rn. More polyhedral cones are illustrated in Figure 49 andFigure 24.

2.49Rockafellar [303, §19] proposes affine sets be handled via complementary pairs of affineinequalities; e.g., Cyºd and Cy¹d which, when taken together, can present severedifficulties to some interior-point methods of numerical solution.

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142 CHAPTER 2. CONVEX GEOMETRY

boundedpolyhedra

polyhedra

polyhedralcones

convex cones

Figure 51: Polyhedral cones are finitely generated, unbounded, and convex.

2.12.2 Vertices of convex polyhedra

By definition, a vertex (§2.6.1.0.1) always lies on the relative boundary of aconvex polyhedron. [219, def.115/6, p.358] In Figure 20, each vertex of thepolyhedron is located at the intersection of three or more facets, and everyedge belongs to precisely two facets [26, §VI.1, p.252]. In Figure 24, the onlyvertex of that polyhedral cone lies at the origin.

The set of all polyhedral cones is clearly a subset of convex polyhedraand a subset of convex cones (Figure 51). Not all convex polyhedra arebounded; evidently, neither can they all be described by the convex hull of abounded set of points as defined in (86). Hence a universal vertex-descriptionof polyhedra in terms of that same finite-length list X (76):

2.12.2.0.1 Definition. Convex polyhedra, vertex-description.(confer §2.8.1.1.1) Denote the truncated a-vector

ai:ℓ =

[

ai...aℓ

]

(267)

By discriminating a suitable finite-length generating list (or set) arrangedcolumnar in X∈ Rn×N , then any particular polyhedron may be described

P =

Xa | aT1:k1 = 1 , am:N º 0 , 1 . . . k ∪ m . . . N = 1 . . . N

(268)

where 0≤k≤N and 1≤m≤N+1. Setting k=0 removes the affineequality condition. Setting m=N+1 removes the inequality.

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2.12. CONVEX POLYHEDRA 143

Coefficient indices in (268) may or may not be overlapping, but allthe coefficients are assumed constrained. From (78), (86), and (102), wesummarize how the coefficient conditions may be applied;

affine sets −→ aT1:k1 = 1

polyhedral cones −→ am:N º 0

←− convex hull (m ≤ k) (269)

It is always possible to describe a convex hull in the region of overlappingindices because, for 1 ≤ m ≤ k ≤ N

am:k | aTm:k1 = 1, am:k º 0 ⊆ am:k | aT

1:k1 = 1, am:N º 0 (270)

Members of a generating list are not necessarily vertices of thecorresponding polyhedron; certainly true for (86) and (268), some subsetof list members reside in the polyhedron’s relative interior. Conversely, whenboundedness (86) applies, convex hull of the vertices is a polyhedron identicalto convex hull of the generating list.

2.12.2.1 Vertex-description of polyhedral cone

Given closed convex cone K in a subspace of Rn having any set of generatorsfor it arranged in a matrix X∈ Rn×N as in (260), then that cone is describedsetting m=1 and k=0 in vertex-description (268):

K = cone X = Xa | a º 0 ⊆ Rn (102)

a conic hull of N generators.

2.12.2.2 Pointedness

(§2.7.2.1.2) [327, §2.10] Assuming all generators constituting the columns ofX∈ Rn×N are nonzero, polyhedral cone K is pointed if and only if there isno nonzero aº 0 that solves Xa=0 ; id est, iff

find asubject to Xa = 0

1Ta = 1a º 0

(271)

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144 CHAPTER 2. CONVEX GEOMETRY

1

S = s | s º 0 , 1Ts ≤ 1

Figure 52: Unit simplex S in R3 is a unique solid tetrahedron but not regular.

is infeasible or iff N (X) ∩ RN+ = 0 .2.50 Otherwise, the cone will contain at

least one line and there can be no vertex; id est, the cone cannot otherwisebe pointed. Euclidean vector space Rn or any halfspace are examples ofnonpointed polyhedral cone; hence, no vertex. This null-pointedness criterionXa=0 means that a pointed polyhedral cone is invariant to linear injectivetransformation.

Examples of pointed polyhedral cone K include: the origin, any 0-basedray in a subspace, any two-dimensional V-shaped cone in a subspace, anyorthant in Rn or Rm×n ; e.g., nonnegative real line R+ in vector space R .

2.50If rankX = n , then the dual cone K∗(§2.13.1) is pointed. (288)

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2.12. CONVEX POLYHEDRA 145

2.12.3 Unit simplex

A peculiar subset of the nonnegative orthant with halfspace-description

S , s | s º 0 , 1Ts ≤ 1 ⊆ Rn+ (272)

is a unique bounded convex full-dimensional polyhedron called unit simplex(Figure 52) having n + 1 facets, n + 1 vertices, and dimension

dimS = n (273)

The origin supplies one vertex while heads of the standard basis [199][328] ei , i=1 . . . n in Rn constitute those remaining;2.51 thus itsvertex-description:

S = conv 0, ei , i=1 . . . n=

[0 e1 e2 · · · en ] a | aT1 = 1 , a º 0 (274)

2.12.3.1 Simplex

The unit simplex comes from a class of general polyhedra called simplex,having vertex-description: [92] [303] [368] [110]

convxℓ ∈ Rn | ℓ = 1 . . . k+1 , dim affxℓ= k , n≥ k (275)

So defined, a simplex is a closed bounded convex set possibly notfull-dimensional. Examples of simplices, by increasing affine dimension, are:a point, any line segment, any triangle and its relative interior, a generaltetrahedron, any five-vertex polychoron, and so on.

2.12.3.1.1 Definition. Simplicial cone.A proper polyhedral (§2.7.2.2.1) cone K in Rn is called simplicial iff K hasexactly n extreme directions; [20, §II.A] equivalently, iff proper K has exactlyn linearly independent generators contained in any given set of generators.

simplicial cone ⇒ proper polyhedral cone

There are an infinite variety of simplicial cones in Rn ; e.g., Figure 24,Figure 53, Figure 63. Any orthant is simplicial, as is any rotation thereof.

2.51In R0 the unit simplex is the point at the origin, in R the unit simplex is the linesegment [0, 1], in R2 it is a triangle and its relative interior, in R3 it is the convex hull ofa tetrahedron (Figure 52), in R4 it is the convex hull of a pentatope [370], and so on.

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146 CHAPTER 2. CONVEX GEOMETRY

Figure 53: Two views of a simplicial cone and its dual in R3 (second view onnext page). Semiinfinite boundary of each cone is truncated for illustration.Each cone has three facets (confer §2.13.11.0.3). Cartesian axes are drawnfor reference.

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2.12. CONVEX POLYHEDRA 147

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148 CHAPTER 2. CONVEX GEOMETRY

2.12.4 Converting between descriptions

Conversion between halfspace- (265) (266) and vertex-description (86) (268)is nontrivial, in general, [15] [110, §2.2] but the conversion is easy forsimplices. [59, §2.2.4] Nonetheless, we tacitly assume the two descriptions tobe equivalent. [303, §19, thm.19.1] We explore conversions in §2.13.4, §2.13.9,and §2.13.11:

2.13 Dual cone & generalized inequality

& biorthogonal expansion

These three concepts, dual cone, generalized inequality, and biorthogonalexpansion, are inextricably melded; meaning, it is difficult to completelydiscuss one without mentioning the others. The dual cone is critical in testsfor convergence by contemporary primal/dual methods for numerical solutionof conic problems. [388] [273, §4.5] For unique minimum-distance projectionon a closed convex cone K , the negative dual cone −K∗

plays the role thatorthogonal complement plays for subspace projection.2.52 (§E.9.2.1) Indeed,−K∗

is the algebraic complement in Rn ;

K ⊞ −K∗= Rn (1985)

where ⊞ denotes unique orthogonal vector sum.One way to think of a pointed closed convex cone is as a new kind of

coordinate system whose basis is generally nonorthogonal; a conic system,very much like the familiar Cartesian system whose analogous cone is thefirst quadrant (the nonnegative orthant). Generalized inequality ºK is aformalized means to determine membership to any pointed closed convexcone K (§2.7.2.2) whereas biorthogonal expansion is, fundamentally, anexpression of coordinates in a pointed conic system whose axes are linearlyindependent but not necessarily orthogonal. When cone K is the nonnegativeorthant, then these three concepts come into alignment with the Cartesianprototype: biorthogonal expansion becomes orthogonal expansion, the dualcone becomes identical to the orthant, and generalized inequality obeys atotal order entrywise.

2.52Namely, projection on a subspace is ascertainable from its projection on the orthogonalcomplement.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 149

2.13.1 Dual cone

For any set K (convex or not), the dual cone [106, §4.2]

K∗, y∈ Rn | 〈y , x〉 ≥ 0 for all x ∈ K (276)

is a unique cone2.53 that is always closed and convex because it is anintersection of halfspaces (§2.4.1.1.1). Each halfspace has inward-normal x ,belonging to K , and boundary containing the origin; e.g., Figure 54a.

When cone K is convex, there is a second and equivalent construction:Dual cone K∗

is the union of each and every vector y inward-normalto a hyperplane supporting K (§2.4.2.6.1); e.g., Figure 54b. When K isrepresented by a halfspace-description such as (266), for example, where

A ,

aT1...

aTm

∈ Rm×n , C ,

cT1...

cTp

∈ Rp×n (277)

then the dual cone can be represented as the conic hull

K∗= conea1 , . . . , am , ±c1 , . . . , ±cp (278)

a vertex-description, because each and every conic combination of normalsfrom the halfspace-description of K yields another inward-normal to ahyperplane supporting K .

K∗can also be constructed pointwise using projection theory from §E.9.2:

for PKx the Euclidean projection of point x on closed convex cone K

−K∗= x − PKx | x∈Rn = x∈Rn | PKx = 0 (1986)

2.13.1.0.1 Exercise. Manual dual cone construction.Perhaps the most instructive graphical method of dual cone construction iscut-and-try. Find the dual of each polyhedral cone from Figure 55 by usingdual cone equation (276). H

2.53The dual cone is the negative polar cone defined by many authors; K∗= −K

.[196, §A.3.2] [303, §14] [40] [26] [327, §2.7]

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150 CHAPTER 2. CONVEX GEOMETRY

0

K(a)

K∗

K∗

0

K (b)

y

Figure 54: Two equivalent constructions of dual cone K∗in R2 : (a) Showing

construction by intersection of halfspaces about 0 (drawn truncated). Onlythose two halfspaces whose bounding hyperplanes have inward-normalcorresponding to an extreme direction of this pointed closed convex coneK⊂R2 need be drawn; by (347). (b) Suggesting construction by union ofinward-normals y to each and every hyperplane ∂H+ supporting K . Thisinterpretation is valid when K is convex because existence of a supportinghyperplane is then guaranteed (§2.4.2.6).

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2.13. DUAL CONE & GENERALIZED INEQUALITY 151

−0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

KK

K∗

R2 R3

∂K∗∂K∗

(a) (b)q

x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ G(K∗) (344)

Figure 55: Dual cone construction by right angle. Each extreme direction ofa proper polyhedral cone is orthogonal to a facet of its dual cone, and viceversa, in any dimension. (§2.13.6.1) (a) This characteristic guides graphicalconstruction of dual cone in two dimensions: It suggests finding dual-coneboundary ∂ by making right angles with extreme directions of polyhedralcone. The construction is then pruned so that each dual boundary vector doesnot exceed π/2 radians in angle with each and every vector from polyhedralcone. Were dual cone in R2 to narrow, Figure 56 would be reached in limit.(b) Same polyhedral cone and its dual continued into three dimensions.(confer Figure 63)

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152 CHAPTER 2. CONVEX GEOMETRY

K

K∗

0

Figure 56: Polyhedral cone K is a halfspace about origin in R2. Dual cone K∗

is a ray base 0, hence not full-dimensional in R2 ; so K cannot be pointed,hence has no extreme directions. (Both convex cones appear truncated.)

2.13.1.0.2 Exercise. Dual cone definitions.What is x∈Rn | xTz≥0 ∀ z∈Rn ?What is x∈Rn | xTz≥1 ∀ z∈Rn ?What is x∈Rn | xTz≥1 ∀ z∈Rn

+ ? H

As defined, dual cone K∗exists even when the affine hull of the

original cone is a proper subspace; id est, even when the original coneis not full-dimensional. Rockafellar formulates dimension of K and K∗

.[303, §14]2.54

To further motivate our understanding of the dual cone, consider theease with which convergence can be ascertained in the following optimizationproblem (p):

2.13.1.0.3 Example. Dual problem. (confer §4.1)Duality is a powerful and widely employed tool in applied mathematics fora number of reasons. First, the dual program is always convex even if theprimal is not. Second, the number of variables in the dual is equal to the

2.54His monumental work Convex Analysis has not one figure or illustration. See[26, §II.16] for a good illustration of Rockafellar’s recession cone [41].

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2.13. DUAL CONE & GENERALIZED INEQUALITY 153

f(xp , z) or g(z)

f(x , zp) or f(x)

xz

Figure 57: (Drawing by Lucas V. Barbosa) This serves as mnemonic icon forprimal and dual problems, although objective functions from conic problems(281p) (281d) are linear. When problems are strong duals, duality gap is 0 ;meaning, functions f(x) , g(z) (dotted) kiss at saddle value as depicted atcenter. Otherwise, dual functions never meet (f(x) > g(z)) by (279).

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154 CHAPTER 2. CONVEX GEOMETRY

number of constraints in the primal which is often less than the number ofvariables in the primal program. Third, the maximum value achieved bythe dual problem is often equal to the minimum of the primal. [295, §2.1.3]When not equal, the dual always provides a bound on the primal optimalobjective. For convex problems, the dual variables provide necessary andsufficient optimality conditions:

Essentially, Lagrange duality theory concerns representation of a givenoptimization problem as half of a minimax problem. [303, §36] [59, §5.4] Givenany real function f(x, z)

minimizex

maximizez

f(x, z) ≥ maximizez

minimizex

f(x, z) (279)

always holds. When

minimizex

maximizez

f(x, z) = maximizez

minimizex

f(x, z) (280)

we have strong duality and then a saddle value [149] exists. (Figure 57)[300, p.3] Consider primal conic problem (p) (over cone K) and itscorresponding dual problem (d): [290, §3.3.1] [235, §2.1] [236] givenvectors α , β and matrix constant C

(p)

minimizex

αTx

subject to x ∈ KCx = β

maximizey , z

βTz

subject to y ∈ K∗

CTz + y = α

(d) (281)

Observe: the dual problem is also conic, and its objective function valuenever exceeds that of the primal;

αTx ≥ βTz

xT(CTz + y) ≥ (Cx)Tz

xTy ≥ 0

(282)

which holds by definition (276). Under the sufficient condition: (281p) isa convex problem2.55 and satisfies Slater’s condition (p.277), then equalityxTy=0 is achieved; each problem (p) and (d) attains the same optimal valueof its objective and each problem is called a strong dual to the other because

2.55A convex problem, essentially, has convex objective function optimized over a convexset. (§4) In this context, problem (p) is convex if K is a convex cone.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 155

the duality gap (optimal primal−dual objective difference) becomes 0. Then(p) and (d) are together equivalent to the minimax problem

minimizex,y,z

αTx − βTz

subject to x ∈ K , y ∈ K∗

Cx = β , CTz + y = α

(p)−(d) (283)

whose optimal objective always has the saddle value 0 (regardless of theparticular convex cone K and other problem parameters). [356, §3.2] Thusdetermination of convergence for either primal or dual problem is facilitated.

Were convex cone K polyhedral (§2.12.1), then problems (p) and (d)would be linear programs.2.56 Were K a positive semidefinite cone, thenproblem (p) has the form of prototypical semidefinite program2.57 (625) with(d) its dual. It is sometimes possible to solve a primal problem by way of itsdual; advantageous when the dual problem is easier to solve than the primalproblem, for example, because it can be solved analytically, or has some specialstructure that can be exploited. [59, §5.5.5] (§4.2.3.1) 2

2.13.1.1 Key properties of dual cone For any cone, (−K)∗

= −K∗ For any cones K1 and K2 , K1 ⊆ K2 ⇒ K∗1 ⊇ K∗

2 [327, §2.7] (Cartesian product) For closed convex cones K1 and K2 , theirCartesian product K = K1 × K2 is a closed convex cone, and

K∗= K∗

1 × K∗2 (284)

where each dual is determined with respect to a cone’s ambient space. (conjugation) [303, §14] [106, §4.5] [327, p.52] When K is any convexcone, dual of the dual cone equals closure of the original cone;

K∗∗= K (285)

2.56Selfdual nonnegative orthant yields primal prototypical linear program and its dual.2.57A semidefinite program is any convex program constraining a variable to any subset ofa positive semidefinite cone. The qualifier prototypical conventionally means: a programhaving linear objective, affine equality constraints, but no inequality constraints exceptfor cone membership.

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156 CHAPTER 2. CONVEX GEOMETRY

is the intersection of all halfspaces about the origin that contain K .Because K∗∗∗

= K∗always holds,

K∗= (K)

∗(286)

When convex cone K is closed, then dual of the dual cone is the originalcone; K∗∗

= K ⇔ K is a closed convex cone: [327, p.53, p.95]

K = x∈ Rn | 〈y , x〉 ≥ 0 ∀ y ∈ K∗ (287) If any cone K is full-dimensional, then K∗is pointed;

K full-dimensional ⇒ K∗pointed (288)

If the closure of any convex cone K is pointed, conversely, then K∗is

full-dimensional;

K pointed ⇒ K∗full-dimensional (289)

Given that a cone K⊂Rn is closed and convex, K is pointed if and onlyif K∗−K∗

= Rn ; id est, iff K∗is full-dimensional. [53, §3.3 exer.20] (vector sum) [303, thm.3.8] For convex cones K1 and K2

K1 + K2 = conv(K1 ∪ K2) (290)

is a convex cone. (dual vector-sum) [303, §16.4.2] [106, §4.6] For convex cones K1 and K2

K∗1 ∩ K∗

2 = (K1 + K2)∗

= (K1 ∪ K2)∗

(291) (closure of vector sum of duals)2.58 For closed convex cones K1 and K2

(K1 ∩ K2)∗

= K∗1 + K∗

2 = conv(K∗1 ∪ K∗

2) (292)

[327, p.96] where operation closure becomes superfluous under thesufficient condition K1 ∩ intK2 6= ∅ [53, §3.3 exer.16, §4.1 exer.7].

2.58These parallel analogous results for subspaces R1 ,R2⊆Rn ; [106, §4.6]

(R1+ R2)⊥ = R⊥

1 ∩R⊥2

(R1∩R2)⊥ = R⊥

1 + R⊥2

R⊥⊥=R for any subspace R .

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2.13. DUAL CONE & GENERALIZED INEQUALITY 157 (Krein-Rutman) Given closed convex cones K1⊆ Rm and K2⊆ Rn

and any linear map A : Rn→ Rm, then provided intK1 ∩ AK2 6= ∅[53, §3.3.13, confer §4.1 exer.9]

(A−1K1 ∩ K2)∗

= ATK∗1 + K∗

2 (293)

where dual of cone K1 is with respect to its ambient space Rm and dualof cone K2 is with respect to Rn, where A−1K1 denotes inverse image(§2.1.9.0.1) of K1 under mapping A , and where AT denotes adjointoperation. The particularly important case K2 = Rn is easy to show:for ATA = I

(ATK1)∗

, y∈ Rn | xTy ≥ 0 ∀x∈ATK1= y∈ Rn | (ATz)Ty ≥ 0 ∀ z∈K1= ATw | zTw ≥ 0 ∀ z∈K1= ATK∗

1

(294) K is proper if and only if K∗is proper. K is polyhedral if and only if K∗

is polyhedral. [327, §2.8] K is simplicial if and only if K∗is simplicial. (§2.13.9.2) A simplicial

cone and its dual are proper polyhedral cones (Figure 63, Figure 53),but not the converse. K ⊞ −K∗

= Rn ⇔ K is closed and convex. (1985) Any direction in a proper cone K is normal to a hyperplane separatingK from −K∗

.

2.13.1.2 Examples of dual cone

When K is Rn, K∗is the point at the origin, and vice versa.

When K is a subspace, K∗is its orthogonal complement, and vice versa.

(§E.9.2.1, Figure 58)When cone K is a halfspace in Rn with n > 0 (Figure 56 for example),

the dual cone K∗is a ray (base 0) belonging to that halfspace but orthogonal

to its bounding hyperplane (that contains the origin), and vice versa.

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158 CHAPTER 2. CONVEX GEOMETRY

K∗

K

0

K∗

K0

R3

R2

Figure 58: When convex cone K is any one Cartesian axis, its dual coneis the convex hull of all axes remaining. In R3, dual cone K∗

(drawn tiledand truncated) is a hyperplane through origin; its normal belongs to line K .In R2, dual cone K∗

is a line through origin while convex cone K is that lineorthogonal.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 159

When convex cone K is a closed halfplane in R3 (Figure 59), it isneither pointed or full-dimensional; hence, the dual cone K∗

can be neitherfull-dimensional or pointed.

When K is any particular orthant in Rn, the dual cone is identical; id est,K=K∗

.When K is any quadrant in subspace R2, K∗

is a wedge-shaped polyhedral

cone in R3 ; e.g., for K equal to quadrant I , K∗=

[

R2

+

R

]

.

When K is a polyhedral flavor Lorentz cone (confer (169))

Kℓ =

[

xt

]

∈ Rn× R | ‖x‖ℓ ≤ t

, ℓ∈1, ∞ (295)

its dual is the proper cone [59, exmp.2.25]

Kq = K∗ℓ =

[

xt

]

∈ Rn× R | ‖x‖q ≤ t

, ℓ∈1, 2, ∞ (296)

where ‖x‖∗ℓ = ‖x‖q is that norm dual to ‖x‖ℓ determined via solution to1/ℓ + 1/q = 1. Figure 61 illustrates K=K1 and K∗

=K∗1 =K∞ in R2× R .

2.13.2 Abstractions of Farkas’ lemma

2.13.2.0.1 Corollary. Generalized inequality and membership relation.[196, §A.4.2] Let K be any closed convex cone and K∗

its dual, and let xand y belong to a vector space Rn. Then

y ∈ K∗ ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (297)

which is, merely, a statement of fact by definition of dual cone (276). Byclosure we have conjugation: [303, thm.14.1]

x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K∗(298)

which may be regarded as a simple translation of Farkas’ lemma [130] asin [303, §22] to the language of convex cones, and a generalization of thewell-known Cartesian cone fact

x º 0 ⇔ 〈y , x〉 ≥ 0 for all y º 0 (299)

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160 CHAPTER 2. CONVEX GEOMETRY

K

K∗

Figure 59: K and K∗are halfplanes in R3 ; blades. Both semiinfinite convex

cones appear truncated. Each cone is like K from Figure 56 but embeddedin a two-dimensional subspace of R3. Cartesian coordinate axes drawn forreference.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 161

for which implicitly K = K∗= Rn

+ the nonnegative orthant.Membership relation (298) is often written instead as dual generalized

inequalities, when K and K∗are pointed closed convex cones,

x ºK

0 ⇔ 〈y , x〉 ≥ 0 for all y ºK∗

0 (300)

meaning, coordinates for biorthogonal expansion of x (§2.13.7.1.2, §2.13.8)[362] must be nonnegative when x belongs to K . Conjugating,

y ºK∗

0 ⇔ 〈y , x〉 ≥ 0 for all x ºK

0 (301)

When pointed closed convex cone K is not polyhedral, coordinate axesfor biorthogonal expansion asserted by the corollary are taken from extremedirections of K ; expansion is assured by Caratheodory’s theorem (§E.6.4.1.1).

We presume, throughout, the obvious:

x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K∗(298)

⇔x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K∗

, ‖y‖= 1

(302)

2.13.2.0.2 Exercise. Test of dual generalized inequalities.Test Corollary 2.13.2.0.1 and (302) graphically on the two-dimensionalpolyhedral cone and its dual in Figure 55. H

(confer §2.7.2.2) When pointed closed convex cone K is implicit from context:

x º 0 ⇔ x ∈ Kx ≻ 0 ⇔ x ∈ rel intK

(303)

Strict inequality x≻ 0 means coordinates for biorthogonal expansion of xmust be positive when x belongs to rel intK . Strict membership relationsare useful; e.g., for any proper cone2.59 K and its dual K∗

x ∈ intK ⇔ 〈y , x〉 > 0 for all y ∈ K∗, y 6= 0 (304)

x ∈ K , x 6= 0 ⇔ 〈y , x〉 > 0 for all y ∈ intK∗(305)

2.59An open cone K is admitted to (304) and (307) by (19).

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162 CHAPTER 2. CONVEX GEOMETRY

Conjugating, we get the dual relations:

y ∈ intK∗ ⇔ 〈y , x〉 > 0 for all x ∈ K , x 6= 0 (306)

y ∈ K∗, y 6= 0 ⇔ 〈y , x〉 > 0 for all x ∈ intK (307)

Boundary-membership relations for proper cones are also useful:

x ∈ ∂K ⇔ ∃ y 6= 0 Ä 〈y , x〉 = 0 , y ∈ K∗, x ∈ K (308)

y ∈ ∂K∗ ⇔ ∃ x 6= 0 Ä 〈y , x〉 = 0 , x ∈ K , y ∈ K∗(309)

which are consistent; e.g., x∈∂K ⇔ x /∈ intK and x∈K .

2.13.2.0.3 Example. Linear inequality. [334, §4](confer §2.13.5.1.1) Consider a given matrix A and closed convex cone K .By membership relation we have

Ay ∈ K∗ ⇔ xTAy≥ 0 ∀x ∈ K⇔ yTz≥ 0 ∀ z ∈ ATx | x ∈ K⇔ y ∈ ATx | x ∈ K∗

(310)

This impliesy | Ay ∈ K∗ = ATx | x ∈ K∗ (311)

When K is the selfdual nonnegative orthant (§2.13.5.1), for example, then

y | Ay º 0 = ATx | x º 0∗ (312)

and the dual relation

y | Ay º 0∗ = ATx | x º 0 (313)

comes by a theorem of Weyl (p.141) that yields closedness for any vertexdescription of a polyhedral cone. 2

2.13.2.1 Null certificate, Theorem of the alternative

If in particular xp /∈K a closed convex cone, then construction in Figure 54bsuggests there exists a supporting hyperplane (having inward-normalbelonging to dual cone K∗

) separating xp from K ; indeed, (298)

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2.13. DUAL CONE & GENERALIZED INEQUALITY 163

xp /∈ K ⇔ ∃ y ∈ K∗Ä 〈y , xp〉 < 0 (314)

Existence of any one such y is a certificate of null membership. From adifferent perspective,

xp ∈ Kor in the alternative

∃ y ∈ K∗Ä 〈y , xp〉 < 0

(315)

By alternative is meant: these two systems are incompatible; one system isfeasible while the other is not.

2.13.2.1.1 Example. Theorem of the alternative for linear inequality.Myriad alternative systems of linear inequality can be explained in terms ofpointed closed convex cones and their duals.

Beginning from the simplest Cartesian dual generalized inequalities (299)(with respect to nonnegative orthant Rm

+ ),

y º 0 ⇔ xTy ≥ 0 for all x º 0 (316)

Given A∈Rn×m, we make vector substitution y ← ATy

ATy º 0 ⇔ xTATy ≥ 0 for all x º 0 (317)

Introducing a new vector by calculating b , Ax we get

ATy º 0 ⇔ bTy ≥ 0 , b = Ax for all x º 0 (318)

By complementing sense of the scalar inequality:

ATy º 0

or in the alternative

bTy < 0 , ∃ b = Ax , x º 0

(319)

If one system has a solution, then the other does not; define a convexcone K=y | ATyº 0 , then y ∈ K or in the alternative y /∈ K . Scalarinequality bTy< 0 can be moved to the other side of the alternative, but

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164 CHAPTER 2. CONVEX GEOMETRY

that requires some explanation: From the results of Example 2.13.2.0.3, thedual cone is K∗

=Ax | xº 0. From (298) we have

y ∈ K ⇔ bTy ≥ 0 for all b ∈ K∗

ATy º 0 ⇔ bTy ≥ 0 for all b ∈ Ax | xº 0 (320)

Given some b vector and y ∈ K , then bTy< 0 can only mean b /∈ K∗. An

alternative system is therefore simply b ∈ K∗: [196, p.59] (Farkas/Tucker)

ATy º 0 , bTy < 0

or in the alternative

b = Ax , x º 0

(321)

Geometrically this means, either vector b belongs to convex cone K∗or it

does not. When b /∈K∗, then there is a vector y∈K normal to a hyperplane

separating point b from cone K∗.

For another example, from membership relation (297) with affinetransformation of dual variable we may write: Given A∈Rn×m and b∈Rn

b − Ay ∈ K∗ ⇔ xT(b − Ay)≥ 0 ∀x ∈ K (322)

ATx=0 , b − Ay ∈ K∗ ⇒ xTb≥ 0 ∀x ∈ K (323)

From membership relation (322), conversely, suppose we allow any y∈Rm.Then because −xTAy is unbounded below, xT(b−Ay)≥ 0 implies ATx=0 :for y∈Rm

ATx=0 , b − Ay ∈ K∗ ⇐ xT(b − Ay)≥ 0 ∀x ∈ K (324)

In toto,b − Ay ∈ K∗ ⇔ xTb≥ 0 , ATx=0 ∀x ∈ K (325)

Vector x belongs to cone K but is also constrained to lie in a subspaceof Rn specified by an intersection of hyperplanes through the originx∈Rn |ATx=0. From this, alternative systems of generalized inequalitywith respect to pointed closed convex cones K and K∗

Ay ¹K∗

b

or in the alternative

xTb < 0 , ATx=0 , x ºK

0

(326)

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2.13. DUAL CONE & GENERALIZED INEQUALITY 165

derived from (325) simply by taking the complementary sense of theinequality in xTb . These two systems are alternatives; if one system hasa solution, then the other does not.2.60 [303, p.201]

By invoking a strict membership relation between proper cones (304),we can construct a more exotic alternative strengthened by demand for aninterior point;

b − Ay ≻K∗

0 ⇔ xTb > 0 , ATx=0 ∀x ºK

0 , x 6= 0 (327)

From this, alternative systems of generalized inequality [59, pages:50,54,262]

Ay ≺K∗

b

or in the alternative

xTb≤ 0 , ATx=0 , x ºK

0 , x 6= 0

(328)

derived from (327) by taking the complementary sense of the inequalityin xTb .

And from this, alternative systems with respect to the nonnegativeorthant attributed to Gordan in 1873: [156] [53, §2.2] substituting A←AT

and setting b = 0ATy ≺ 0

or in the alternative

Ax = 0 , x º 0 , ‖x‖1 = 1

(329)

Ben-Israel collects related results from Farkas, Motzkin, Gordan, and Stiemkein Motzkin transposition theorem. [182] 2

2.60If solutions at ±∞ are disallowed, then the alternative systems become insteadmutually exclusive with respect to nonpolyhedral cones. Simultaneous infeasibility of thetwo systems is not precluded by mutual exclusivity; called a weak alternative. Ye providesan example illustrating simultaneous infeasibility with respect to the positive semidefinite

cone: x∈S2, y∈R , A =

[

1 00 0

]

, and b =

[

0 11 0

]

where xTb means 〈x , b〉 .

A better strategy than disallowing solutions at ±∞ is to demand an interior point asin (328) or Lemma 4.2.1.1.2. Then question of simultaneous infeasibility is moot.

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166 CHAPTER 2. CONVEX GEOMETRY

2.13.3 Optimality condition

(confer §2.13.10.1) The general first-order necessary and sufficient conditionfor optimality of solution x⋆ to a minimization problem ((281p) for example)with real differentiable convex objective function f(x) : Rn→R is [302, §3]

∇f(x⋆)T(x − x⋆) ≥ 0 ∀x ∈ C , x⋆∈ C (330)

where C is a convex feasible set,2.61 and where ∇f(x⋆) is the gradient (§3.7) off with respect to x evaluated at x⋆. In words, negative gradient is normal toa hyperplane supporting the feasible set at a point of optimality. (Figure 66)

Direct solution to variation inequality (330), instead of the correspondingminimization, spawned from calculus of variations. [246, p.178] [129, p.37]One solution method solves an equivalent fixed point-of-projection problem

x = PC(x −∇f(x)) (331)

that follows from a necessary and sufficient condition for projection on convexset C (Theorem E.9.1.0.2)

P (x⋆−∇f(x⋆)) ∈ C , 〈x⋆−∇f(x⋆)−x⋆, x−x⋆〉 ≤ 0 ∀x ∈ C (1972)

Proof of equivalence is provided by Sandor Zoltan Nemeth on Wıκımization.Given minimum-distance projection problem

minimizex

12‖x − y‖2

subject to x ∈ C (332)

on convex feasible set C for example, the equivalent fixed point problem

x = PC(x −∇f(x)) = PC(y) (333)

is solved in one step.In the unconstrained case (C=Rn), optimality condition (330) reduces to

the classical rule (p.243)

∇f(x⋆) = 0 , x⋆∈ dom f (334)

which can be inferred from the following application:

2.61 presumably nonempty set of all variable values satisfying all given problem constraints;the set of feasible solutions.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 167

2.13.3.0.1 Example. Optimality for equality constrained problem.Given a real differentiable convex function f(x) : Rn→R defined ondomain Rn, a fat full-rank matrix C∈ Rp×n, and vector d∈Rp, the convexoptimization problem

minimizex

f(x)

subject to Cx = d(335)

is characterized by the well-known necessary and sufficient optimalitycondition [59, §4.2.3]

∇f(x⋆) + CTν = 0 (336)

where ν∈Rp is the eminent Lagrange multiplier. [301] [246, p.188] [225] Inother words, solution x⋆ is optimal if and only if ∇f(x⋆) belongs to R(CT).

Via membership relation, we now derive condition (336) from the generalfirst-order condition for optimality (330): For problem (335)

C , x∈Rn | Cx = d = Zξ + xp | ξ∈ Rn−rank C (337)

is the feasible set where Z∈ Rn×n−rank C holds basisN (C) columnar, andxp is any particular solution to Cx = d . Since x⋆∈ C , we arbitrarily choosexp = x⋆ which yields an equivalent optimality condition

∇f(x⋆)TZξ ≥ 0 ∀ ξ∈ Rn−rank C (338)

when substituted into (330). But this is simply half of a membership relationwhere the cone dual to Rn−rank C is the origin in Rn−rank C . We must thereforehave

ZT∇f(x⋆) = 0 ⇔ ∇f(x⋆)TZξ ≥ 0 ∀ ξ∈ Rn−rank C (339)

meaning, ∇f(x⋆) must be orthogonal to N (C). These conditions

ZT∇f(x⋆) = 0 , Cx⋆ = d (340)

are necessary and sufficient for optimality. 2

2.13.4 Discretization of membership relation

2.13.4.1 Dual halfspace-description

Halfspace-description of dual cone K∗is equally simple as vertex-description

K = cone(X) = Xa | a º 0 ⊆ Rn (102)

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168 CHAPTER 2. CONVEX GEOMETRY

for corresponding closed convex cone K : By definition (276), for X∈ Rn×N

as in (260), (confer (266))

K∗=

y ∈ Rn | zTy ≥ 0 for all z ∈ K

=

y ∈ Rn | zTy ≥ 0 for all z = Xa , a º 0

=

y ∈ Rn | aTXTy ≥ 0 , a º 0

=

y ∈ Rn | XTy º 0

(341)

that follows from the generalized inequality and membership corollary (299).The semi-infinity of tests specified by all z ∈ K has been reduced to a setof generators for K constituting the columns of X ; id est, the test has beendiscretized.

Whenever cone K is known to be closed and convex, the conjugatestatement must also hold; id est, given any set of generators for dual coneK∗

arranged columnar in Y , then the consequent vertex-description of dualcone connotes a halfspace-description for K : [327, §2.8]

K∗= Y a | a º 0 ⇔ K∗∗

= K =

z | Y Tz º 0

(342)

2.13.4.2 First dual-cone formula

From these two results (341) and (342) we deduce a general principle: From any [sic] given vertex-description (102) of closed convex cone K ,a halfspace-description of the dual cone K∗

is immediate by matrixtransposition (341); conversely, from any given halfspace-description(266) of K , a dual vertex-description is immediate (342). [303, p.122]

Various other converses are just a little trickier. (§2.13.9, §2.13.11)We deduce further: For any polyhedral cone K , the dual cone K∗

is alsopolyhedral and K∗∗

= K . [327, p.56]The generalized inequality and membership corollary is discretized in the

following theorem inspired by (341) and (342):

2.13.4.2.1 Theorem. Discretized membership. (confer §2.13.2.0.1)2.62

Given any set of generators (§2.8.1.2) denoted by G(K) for closed convexcone K⊆Rn and any set of generators denoted G(K∗

) for its dual such that

K = coneG(K) , K∗= coneG(K∗

) (343)

2.62Stated in [21, §1] without proof for pointed closed convex case.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 169

then discretization of the generalized inequality and membership corollary(p.159) is necessary and sufficient for certifying cone membership: for x andy in vector space Rn

x ∈ K ⇔ 〈γ∗, x〉 ≥ 0 for all γ∗ ∈ G(K∗) (344)

y ∈ K∗ ⇔ 〈γ , y〉 ≥ 0 for all γ ∈ G(K) (345)

Proof. K∗= G(K∗

)a | aº 0. y∈K∗⇔ y=G(K∗)a for some aº 0.

x∈K ⇔ 〈y , x〉≥ 0 ∀ y ∈K∗⇔ 〈G(K∗)a , x〉≥ 0 ∀ aº 0 (298). a,

i αi ei

where ei is the ith member of a standard basis of possibly infinitecardinality. 〈G(K∗

)a , x〉≥ 0 ∀ aº 0 ⇔ ∑

i αi〈G(K∗)ei , x〉≥ 0 ∀αi ≥ 0 ⇔

〈G(K∗)ei , x〉≥ 0 ∀ i . Conjugate relation (345) is similarly derived. ¨

2.13.4.2.2 Exercise. Test of discretized dual generalized inequalities.Test Theorem 2.13.4.2.1 on Figure 55a using extreme directions there asgenerators. H

From the discretized membership theorem we may further deduce a moresurgical description of closed convex cone that prescribes only a finite numberof halfspaces for its construction when polyhedral: (Figure 54a)

K = x ∈ Rn | 〈γ∗, x〉 ≥ 0 for all γ∗∈ G(K∗) (346)

K∗= y ∈ Rn | 〈γ , y〉 ≥ 0 for all γ ∈ G(K) (347)

2.13.4.2.3 Exercise. Partial order induced by orthant.When comparison is with respect to the nonnegative orthant K= Rn

+ , thenfrom the discretized membership theorem it directly follows:

x ¹ z ⇔ xi ≤ zi ∀ i (348)

Generate simple counterexamples demonstrating that this equivalence withentrywise inequality holds only when the underlying cone inducing partialorder is the nonnegative orthant; e.g., explain Figure 60. H

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170 CHAPTER 2. CONVEX GEOMETRY

Kx

Figure 60: xº 0 with respect to K but not with respect to nonnegativeorthant R2

+ (pointed convex cone K drawn truncated).

2.13.4.2.4 Example. Boundary membership to proper polyhedral cone.For a polyhedral cone, test (308) of boundary membership can be formulatedas a linear program. Say proper polyhedral cone K is specified completelyby generators that are arranged columnar in

X = [ Γ1 · · · ΓN ] ∈ Rn×N (260)

id est, K = Xa | a º 0. Then membership relation

c ∈ ∂K ⇔ ∃ y 6= 0 Ä 〈y , c〉 = 0 , y ∈ K∗, c ∈ K (308)

may be expressedfinda , y

y 6= 0

subject to cTy = 0XTy º 0Xa = ca º 0

(349)

This linear feasibility problem has a solution iff c∈∂K . 2

2.13.4.3 smallest face of pointed closed convex cone

Given nonempty convex subset C of a convex set K , the smallest face of Kcontaining C is equivalent to the intersection of all faces of K containing C .

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2.13. DUAL CONE & GENERALIZED INEQUALITY 171

[303, p.164] By (287), membership relation (308) means that each andevery point on boundary ∂K of proper cone K belongs to a hyperplanesupporting K whose normal y belongs to dual cone K∗

. It follows that thesmallest face F , containing C ⊂ ∂K⊂ Rn on boundary of proper cone K ,is the intersection of all hyperplanes containing C whose normals are in K∗

;

F(K⊃C) = x ∈ K | x ⊥ K∗∩ C⊥ (350)

where

C⊥ , y∈Rn | 〈z , y〉=0 ∀ z∈ C (351)

When C ∩ intK 6= ∅ then F(K⊃C)=K .

2.13.4.3.1 Example. Finding smallest face of cone.Suppose polyhedral cone K is completely specified by generators arrangedcolumnar in

X = [ Γ1 · · · ΓN ] ∈ Rn×N (260)

To find its smallest face F(K∋ c) containing a given point c∈K , by thediscretized membership theorem 2.13.4.2.1, it is necessary and sufficient tofind generators for the smallest face. We may do so one generator at a time:Consider generator Γi . If there exists a vector z∈K∗

orthogonal to c butnot to Γi , then Γi cannot belong to the smallest face of K containing c .Such a vector z can be realized by a linear feasibility problem:

find z∈ Rn

subject to cTz = 0XTz º 0ΓT

i z = 1

(352)

If there exists a solution z for which ΓTi z=1, then

Γi 6⊥ K∗∩ c⊥ = z∈ Rn | XTzº 0 , cTz=0 (353)

so Γi /∈ F(K∋ c) ; solution z is a certificate of null membership. If thisproblem is infeasible for generator Γi∈K , conversely, then Γi ∈ F(K∋ c)by (350) and (341) because Γi ⊥ K∗∩ c⊥ ; in that case, Γi is a generator ofF(K∋ c).

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172 CHAPTER 2. CONVEX GEOMETRY

Since the constant in constraint ΓTi z =1 is arbitrary positively, then by

theorem of the alternative there is correspondence between (352) and (326)admitting the alternative linear problem: for a given point c∈K

finda∈RN , t∈R

a , t

subject to tc − Γi = Xaa º 0

(354)

Now if this problem is feasible (bounded) for generator Γi∈K , then (352) isinfeasible and Γi ∈ F(K∋ c) is a generator of the smallest face containing c .

2

2.13.4.3.2 Exercise. Finding smallest face of pointed closed convex cone.Show that formula (350) and algorithms (352) and (354) apply more broadly;id est, a full-dimensional cone K is an unnecessary condition.2.63 H

2.13.4.3.3 Exercise. Smallest face of positive semidefinite cone.Derive (212) from (350). H

2.13.5 Dual PSD cone and generalized inequality

The dual positive semidefinite cone K∗is confined to SM by convention;

SM ∗+ , Y ∈ SM | 〈Y , X 〉 ≥ 0 for all X∈ SM

+ = SM+ (355)

The positive semidefinite cone is selfdual in the ambient space of symmetricmatrices [59, exmp.2.24] [38] [192, §II]; K=K∗

.Dual generalized inequalities with respect to the positive semidefinite cone

in the ambient space of symmetric matrices can therefore be simply stated:(Fejer)

X º 0 ⇔ tr(Y TX) ≥ 0 for all Y º 0 (356)

Membership to this cone can be determined in the isometrically isomorphicEuclidean space RM 2

via (38). (§2.2.1) By the two interpretations in §2.13.1,positive semidefinite matrix Y can be interpreted as inward-normal to ahyperplane supporting the positive semidefinite cone.

2.63Hint: A hyperplane, with normal in K∗, containing cone K is admissible.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 173

The fundamental statement of positive semidefiniteness, yTXy≥0 ∀ y(§A.3.0.0.1), evokes a particular instance of these dual generalizedinequalities (356):

X º 0 ⇔ 〈yyT, X 〉 ≥ 0 ∀ yyT(º 0) (1418)

Discretization (§2.13.4.2.1) allows replacement of positive semidefinitematrices Y with this minimal set of generators comprising the extremedirections of the positive semidefinite cone (§2.9.2.4).

2.13.5.1 selfdual cones

From (130) (a consequence of the halfspaces theorem, §2.4.1.1.1), where theonly finite value of the support function for a convex cone is 0 [196, §C.2.3.1],or from discretized definition (347) of the dual cone we get a ratherself-evident characterization of selfdual cones:

K = K∗ ⇔ K =⋂

γ∈G(K)

y | γTy ≥ 0

(357)

In words: Cone K is selfdual iff its own extreme directions are inward-normalsto a (minimal) set of hyperplanes bounding halfspaces whose intersectionconstructs it. This means each extreme direction of K is normal to ahyperplane exposing one of its own faces; a necessary but insufficientcondition for selfdualness (Figure 61, for example).

Selfdual cones are necessarily full-dimensional. [30, §I] Their mostprominent representatives are the orthants (Cartesian cones), the positivesemidefinite cone SM

+ in the ambient space of symmetric matrices (355), andLorentz cone (169) [20, §II.A] [59, exmp.2.25]. In three dimensions, a planecontaining the axis of revolution of a selfdual cone (and the origin) willproduce a slice whose boundary makes a right angle.

2.13.5.1.1 Example. Linear matrix inequality. (confer §2.13.2.0.3)Consider a peculiar vertex-description for a convex cone K defined overa positive semidefinite cone (instead of a nonnegative orthant as indefinition (102)): for X∈ Sn

+ given Aj ∈ Sn, j =1 . . . m

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174 CHAPTER 2. CONVEX GEOMETRY

K0 ∂K∗

K

∂K∗

(a)

(b)

Figure 61: Two (truncated) views of a polyhedral cone K and its dual in R3.Each of four extreme directions from K belongs to a face of dual cone K∗

.Cone K , shrouded by its dual, is symmetrical about its axis of revolution.Each pair of diametrically opposed extreme directions from K makes a rightangle. An orthant (or any rotation thereof; a simplicial cone) is not theonly selfdual polyhedral cone in three or more dimensions; [20, §2.A.21] e.g.,consider an equilateral having five extreme directions. In fact, every selfdualpolyhedral cone in R3 has an odd number of extreme directions. [22, thm.3]

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2.13. DUAL CONE & GENERALIZED INEQUALITY 175

K =

〈A1 , X 〉...

〈Am , X 〉

| Xº 0

⊆ Rm

=

svec(A1)T

...svec(Am)T

svec X | Xº 0

, A svec X | Xº 0

(358)

where A∈Rm×n(n+1)/2, and where symmetric vectorization svec is definedin (56). Cone K is indeed convex because, by (166)

A svec Xp1 , A svec Xp2 ∈ K ⇒ A(ζ svec Xp1+ξ svec Xp2)∈K for all ζ , ξ ≥ 0(359)

since a nonnegatively weighted sum of positive semidefinite matrices must bepositive semidefinite. (§A.3.1.0.2) Although matrix A is finite-dimensional, Kis generally not a polyhedral cone (unless m=1 or 2) simply because X∈ Sn

+ .

Theorem. Inverse image closedness. [196, prop.A.2.1.12][303, thm.6.7] Given affine operator g : Rm→ Rp, convex set D⊆ Rm,and convex set C ⊆ Rp Ä g−1(rel int C) 6= ∅ , then

rel int g(D)= g(rel intD) , rel int g−1C= g−1(rel int C) , g−1C= g−1C(360)⋄

By this theorem, relative interior of K may always be expressed

rel intK = A svec X | X≻ 0 (361)

Because dim(aff K)=dim(A svec Sn) (126) then, provided the vectorized Aj

matrices are linearly independent,

rel intK = intK (14)

meaning, cone K is full-dimensional ⇒ dual cone K∗is pointed by (288).

Convex cone K can be closed, by this corollary:

Corollary. Cone closedness invariance. [54, §3] [49, §3]Given linear operator A : Rp→ Rm and closed convex cone X ⊆ Rp,

convex cone K= A(X ) is closed(

A(X ) = A(X ))

if and only if

N (A) ∩ X = 0 or N (A) ∩ X * rel ∂X (362)

Otherwise, K = A(X ) ⊇ A(X ) ⊇ A(X ). [303, thm.6.6] ⋄

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176 CHAPTER 2. CONVEX GEOMETRY

If matrix A has no nontrivial nullspace, then A svec X is an isomorphismin X between cone Sn

+ and range R(A) of matrix A ; (§2.2.1.0.1, §2.10.1.1)sufficient for convex cone K to be closed and have relative boundary

rel ∂K = A svec X | Xº 0 , X ⊁ 0 (363)

Now consider the (closed convex) dual cone:

K∗= y | 〈z , y〉 ≥ 0 for all z∈K ⊆ Rm

= y | 〈z , y〉 ≥ 0 for all z = A svec X , Xº 0= y | 〈A svec X , y〉 ≥ 0 for all Xº 0=

y | 〈svec X , ATy〉 ≥ 0 for all Xº 0

=

y | svec−1(ATy) º 0

(364)

that follows from (356) and leads to an equally peculiar halfspace-description

K∗= y∈Rm |

m∑

j=1

yjAj º 0 (365)

The summation inequality with respect to positive semidefinite cone Sn+ is

known as linear matrix inequality. [57] [147] [258] [359] Although we alreadyknow that the dual cone is convex (§2.13.1), inverse image theorem 2.1.9.0.1certifies convexity of K∗

which is the inverse image of positive semidefinitecone Sn

+ under linear transformation g(y),∑

yjAj . And although wealready know that the dual cone is closed, it is certified by (360). By theinverse image closedness theorem, dual cone relative interior may always beexpressed

rel intK∗= y∈Rm |

m∑

j=1

yjAj ≻ 0 (366)

Function g(y) on Rm is an isomorphism when the vectorized Aj matricesare linearly independent; hence, uniquely invertible. Inverse image K∗

musttherefore have dimension equal to dim

(

R(AT)∩ svec Sn+

)

(49) and relativeboundary

rel ∂K∗= y∈Rm |

m∑

j=1

yjAj º 0 ,m

j=1

yjAj ⊁ 0 (367)

When this dimension equals m , then dual cone K∗is full-dimensional

rel intK∗= intK∗

(14)

which implies: closure of convex cone K is pointed (288). 2

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2.13. DUAL CONE & GENERALIZED INEQUALITY 177

2.13.6 Dual of pointed polyhedral cone

In a subspace of Rn, now we consider a pointed polyhedral cone K given interms of its extreme directions Γi arranged columnar in

X = [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N (260)

The extremes theorem (§2.8.1.1.1) provides the vertex-description of apointed polyhedral cone in terms of its finite number of extreme directionsand its lone vertex at the origin:

2.13.6.0.1 Definition. Pointed polyhedral cone, vertex-description.Given pointed polyhedral cone K in a subspace of Rn, denoting its ith extremedirection by Γi∈ Rn arranged in a matrix X as in (260), then that cone maybe described: (86) (confer (179) (272))

K =

[0 X ] a ζ | aT1 = 1, a º 0, ζ ≥ 0

=

Xaζ | aT1 ≤ 1, a º 0, ζ ≥ 0

=

Xb | b º 0

⊆ Rn(368)

that is simply a conic hull (like (102)) of a finite number N of directions.Relative interior may always be expressed

rel intK = Xb | b ≻ 0 ⊂ Rn (369)

but identifying the cone’s relative boundary in this manner

rel ∂K = Xb | b º 0 , b ⊁ 0 (370)

holds only when matrix X represents a bijection onto its range.

Whenever cone K is pointed closed and convex (not only polyhedral), thendual cone K∗

has a halfspace-description in terms of the extreme directionsΓi of K :

K∗=

y | γTy ≥ 0 for all γ ∈ Γi , i=1 . . . N ⊆ rel ∂K

(371)

because when Γi constitutes any set of generators for K , the discretizationresult in §2.13.4.1 allows relaxation of the requirement ∀x∈K in (276) to

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178 CHAPTER 2. CONVEX GEOMETRY

∀ γ∈Γi directly.2.64 That dual cone so defined is unique, identical to (276),polyhedral whenever the number of generators N is finite

K∗=

y | XTy º 0

⊆ Rn (341)

and is full-dimensional because K is assumed pointed. But K∗is not

necessarily pointed unless K is full-dimensional (§2.13.1.1).

2.13.6.1 Facet normal & extreme direction

We see from (341) that the conically independent generators of cone K(namely, the extreme directions of pointed closed convex cone K constitutingthe N columns of X) each define an inward-normal to a hyperplanesupporting dual cone K∗

(§2.4.2.6.1) and exposing a dual facet when N isfinite. Were K∗

pointed and finitely generated, then by closure the conjugatestatement would also hold; id est, the extreme directions of pointed K∗

eachdefine an inward-normal to a hyperplane supporting K and exposing a facetwhen N is finite. Examine Figure 55 or Figure 61, for example.

We may conclude, the extreme directions of proper polyhedral K arerespectively orthogonal to the facets of K∗

; likewise, the extreme directionsof proper polyhedral K∗

are respectively orthogonal to the facets of K .

2.13.7 Biorthogonal expansion by example

2.13.7.0.1 Example. Relationship to dual polyhedral cone.Simplicial cone K illustrated in Figure 62 induces a partial order on R2. Allpoints greater than x with respect to K , for example, are contained in thetranslated cone x + K . The extreme directions Γ1 and Γ2 of K do not makean orthogonal set; neither do extreme directions Γ3 and Γ4 of dual cone K∗

;rather, we have the biorthogonality condition, [362]

ΓT4 Γ1 = ΓT

3 Γ2 = 0

ΓT3 Γ1 6= 0 , ΓT

4 Γ2 6= 0(372)

Biorthogonal expansion of x ∈K is then

x = Γ1ΓT

3 x

ΓT3 Γ1

+ Γ2ΓT

4 x

ΓT4 Γ2

(373)

2.64The extreme directions of K constitute a minimal set of generators. Formulae andconversions to vertex-description of the dual cone are in §2.13.9 and §2.13.11.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 179

x

y

z

w

w −Ku0

Γ1 ⊥ Γ4

Γ2 ⊥ Γ3

Γ1

Γ2

Γ3

Γ4

K

K∗

K∗

Figure 62: (confer Figure 161) Simplicial cone K in R2 and its dual K∗drawn

truncated. Conically independent generators Γ1 and Γ2 constitute extremedirections of K while Γ3 and Γ4 constitute extreme directions of K∗

. Dottedray-pairs bound translated cones K . Point x is comparable to point z(and vice versa) but not to y ; z ºK x ⇔ z − x ∈K ⇔ z − x ºK 0 iff ∃nonnegative coordinates for biorthogonal expansion of z − x . Point y is notcomparable to z because z does not belong to y ±K . Translating a negatedcone is quite helpful for visualization: u ¹K w ⇔ u ∈ w −K ⇔ u − w ¹K 0.Points need not belong to K to be comparable; e.g., all points less than w(with respect to K) belong to w −K .

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180 CHAPTER 2. CONVEX GEOMETRY

where ΓT3 x/(ΓT

3 Γ1) is the nonnegative coefficient of nonorthogonal projection(§E.6.1) of x on Γ1 in the direction orthogonal to Γ3 (y in Figure 161, p.705),and where ΓT

4 x/(ΓT4 Γ2) is the nonnegative coefficient of nonorthogonal

projection of x on Γ2 in the direction orthogonal to Γ4 (z in Figure 161);they are coordinates in this nonorthogonal system. Those coefficients mustbe nonnegative x ºK 0 because x ∈ K (303) and K is simplicial.

If we ascribe the extreme directions of K to the columns of a matrix

X , [ Γ1 Γ2 ] (374)

then we find that the pseudoinverse transpose matrix

X†T =

[

Γ31

ΓT3 Γ1

Γ41

ΓT4 Γ2

]

(375)

holds the extreme directions of the dual cone. Therefore,

x = XX†x (381)

is the biorthogonal expansion (373) (§E.0.1), and the biorthogonalitycondition (372) can be expressed succinctly (§E.1.1)2.65

X†X = I (382)

Expansion w=XX†w for any w∈ R2 is unique if and only if the extremedirections of K are linearly independent; id est, iff X has no nullspace. 2

2.13.7.1 Pointed cones and biorthogonality

Biorthogonality condition X†X = I from Example 2.13.7.0.1 means Γ1 andΓ2 are linearly independent generators of K (§B.1.1.1); generators becauseevery x∈K is their conic combination. From §2.10.2 we know that meansΓ1 and Γ2 must be extreme directions of K .

A biorthogonal expansion is necessarily associated with a pointed closedconvex cone; pointed, otherwise there can be no extreme directions (§2.8.1).We will address biorthogonal expansion with respect to a pointed polyhedralcone not full-dimensional in §2.13.8.

2.65Possibly confusing is the fact that formula XX†x is simultaneously the orthogonalprojection of x on R(X) (1873), and a sum of nonorthogonal projections of x∈R(X) onthe range of each and every column of full-rank X skinny-or-square (§E.5.0.0.2).

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2.13. DUAL CONE & GENERALIZED INEQUALITY 181

2.13.7.1.1 Example. Expansions implied by diagonalization.(confer §6.5.3.2.1) When matrix X∈ RM×M is diagonalizable (§A.5),

X = SΛS−1 = [ s1 · · · sM ] Λ

wT1...

wTM

=M

i=1

λi siwTi (1513)

coordinates for biorthogonal expansion are its eigenvalues λi (contained indiagonal matrix Λ) when expanded in S ;

X = SS−1X = [ s1 · · · sM ]

wT1 X...

wTMX

=M

i=1

λi siwTi (376)

Coordinate value depend upon the geometric relationship of X to its linearlyindependent eigenmatrices siw

Ti . (§A.5.0.3, §B.1.1) Eigenmatrices siw

Ti are linearly independent dyads constituted by right

and left eigenvectors of diagonalizable X and are generators of somepointed polyhedral cone K in a subspace of RM×M .

When S is real and X belongs to that polyhedral cone K , for example,then coordinates of expansion (the eigenvalues λi ) must be nonnegative.

When X = QΛQT is symmetric, coordinates for biorthogonal expansionare its eigenvalues when expanded in Q ; id est, for X∈ SM

X = QQTX =M

i=1

qi qTi X =

M∑

i=1

λi qiqTi ∈ SM (377)

becomes an orthogonal expansion with orthonormality condition QTQ=Iwhere λi is the ith eigenvalue of X , qi is the corresponding ith eigenvectorarranged columnar in orthogonal matrix

Q = [ q1 q2 · · · qM ] ∈ RM×M (378)

and where eigenmatrix qiqTi is an extreme direction of some pointed

polyhedral cone K⊂ SM and an extreme direction of the positive semidefinitecone SM

+ . Orthogonal expansion is a special case of biorthogonal expansion ofX∈ aff K occurring when polyhedral cone K is any rotation about theorigin of an orthant belonging to a subspace.

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182 CHAPTER 2. CONVEX GEOMETRY

Similarly, when X = QΛQT belongs to the positive semidefinite cone inthe subspace of symmetric matrices, coordinates for orthogonal expansionmust be its nonnegative eigenvalues (1426) when expanded in Q ; id est, forX∈ SM

+

X = QQTX =M

i=1

qi qTi X =

M∑

i=1

λi qiqTi ∈ SM

+ (379)

where λi≥ 0 is the ith eigenvalue of X . This means matrix Xsimultaneously belongs to the positive semidefinite cone and to the pointedpolyhedral cone K formed by the conic hull of its eigenmatrices. 2

2.13.7.1.2 Example. Expansion respecting nonpositive orthant.Suppose x ∈ K any orthant in Rn .2.66 Then coordinates for biorthogonalexpansion of x must be nonnegative; in fact, absolute value of the Cartesiancoordinates.

Suppose, in particular, x belongs to the nonpositive orthant K = Rn− .

Then the biorthogonal expansion becomes an orthogonal expansion

x = XXTx =n

i=1

−ei(−eTi x) =

n∑

i=1

−ei|eTi x| ∈ Rn

− (380)

and the coordinates of expansion are nonnegative. For this orthant K we haveorthonormality condition XTX = I where X =−I , ei∈Rn is a standardbasis vector, and −ei is an extreme direction (§2.8.1) of K .

Of course, this expansion x=XXTx applies more broadly to domain Rn,but then the coordinates each belong to all of R . 2

2.13.8 Biorthogonal expansion, derivation

Biorthogonal expansion is a means for determining coordinates in a pointedconic coordinate system characterized by a nonorthogonal basis. Studyof nonorthogonal bases invokes pointed polyhedral cones and their duals;extreme directions of a cone K are assumed to constitute the basis whilethose of the dual cone K∗

determine coordinates.Unique biorthogonal expansion with respect to K depends upon existence

of its linearly independent extreme directions: Polyhedral cone K must be

2.66An orthant is simplicial and selfdual.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 183

pointed; then it possesses extreme directions. Those extreme directions mustbe linearly independent to uniquely represent any point in their span.

We consider nonempty pointed polyhedral cone K possibly notfull-dimensional; id est, we consider a basis spanning a subspace. Then weneed only observe that section of dual cone K∗

in the affine hull of K because,by expansion of x , membership x∈ aff K is implicit and because any breachof the ordinary dual cone into ambient space becomes irrelevant (§2.13.9.3).Biorthogonal expansion

x = XX†x ∈ aff K = aff cone(X) (381)

is expressed in the extreme directions Γi of K arranged columnar in

X = [ Γ1 Γ2 · · · ΓN ] ∈ Rn×N (260)

under assumption of biorthogonality

X†X = I (382)

where † denotes matrix pseudoinverse (§E). We therefore seek, in thissection, a vertex-description for K∗∩ aff K in terms of linearly independentdual generators Γ∗

i ⊂ aff K in the same finite quantity2.67 as the extremedirections Γi of

K = cone(X) = Xa | a º 0 ⊆ Rn (102)

We assume the quantity of extreme directions N does not exceed thedimension n of ambient vector space because, otherwise, the expansion couldnot be unique; id est, assume N linearly independent extreme directionshence N ≤ n (X skinny2.68-or-square full-rank). In other words, fat full-rankmatrix X is prohibited by uniqueness because of existence of an infinity ofright inverses; polyhedral cones whose extreme directions number in excess of the

ambient space dimension are precluded in biorthogonal expansion.

2.67When K is contained in a proper subspace of Rn, the ordinary dual cone K∗will have

more generators in any minimal set than K has extreme directions.2.68“Skinny” meaning thin; more rows than columns.

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184 CHAPTER 2. CONVEX GEOMETRY

2.13.8.1 x ∈ KSuppose x belongs to K⊆Rn. Then x =Xa for some aº0. Vector a isunique only when Γi is a linearly independent set.2.69 Vector a∈RN cantake the form a =Bx if R(B)= RN . Then we require Xa =XBx = x andBx=BXa = a . The pseudoinverse B =X†∈ RN×n (§E) is suitable when Xis skinny-or-square and full-rank. In that case rankX =N , and for all c º 0and i=1 . . . N

a º 0 ⇔ X†Xa º 0 ⇔ aTXTX†Tc ≥ 0 ⇔ ΓTi X†Tc ≥ 0 (383)

The penultimate inequality follows from the generalized inequality andmembership corollary, while the last inequality is a consequence of thatcorollary’s discretization (§2.13.4.2.1).2.70 From (383) and (371) we deduce

K∗∩ aff K = cone(X†T) = X†Tc | c º 0 ⊆ Rn (384)

is the vertex-description for that section of K∗in the affine hull of K because

R(X†T) = R(X) by definition of the pseudoinverse. From (288), we knowK∗∩ aff K must be pointed if rel intK is logically assumed nonempty withrespect to aff K .

Conversely, suppose full-rank skinny-or-square matrix (N ≤ n)

X†T ,[

Γ∗1 Γ

∗2 · · · Γ

∗N

]

∈ Rn×N (385)

comprises the extreme directions Γ∗i ⊂ aff K of the dual-cone intersection

with the affine hull of K .2.71 From the discretized membership theorem and

2.69Conic independence alone (§2.10) is insufficient to guarantee uniqueness.2.70

a º 0 ⇔ aTXTX†Tc ≥ 0 ∀ (c º 0 ⇔ aTXTX†Tc ≥ 0 ∀ a º 0)∀ (c º 0 ⇔ ΓT

i X†Tc ≥ 0 ∀ i ) ¨

Intuitively, any nonnegative vector a is a conic combination of the standard basisei∈RN; aº 0 ⇔ ai eiº 0 for all i . The last inequality in (383) is a consequence of thefact that x=Xa may be any extreme direction of K , in which case a is a standard basisvector; a = eiº 0. Theoretically, because cº 0 defines a pointed polyhedral cone (in fact,the nonnegative orthant in RN), we can take (383) one step further by discretizing c :

a º 0 ⇔ ΓTi Γ

∗j ≥ 0 for i, j =1 . . . N ⇔ X†X ≥ 0

In words, X†X must be a matrix whose entries are each nonnegative.2.71When closed convex cone K is not full-dimensional, K∗

has no extreme directions.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 185

(292) we get a partial dual to (371); id est, assuming x∈ aff cone X

x ∈ K ⇔ γ∗Tx ≥ 0 for all γ∗∈

Γ∗i , i=1 . . . N

⊂ ∂K∗∩ aff K (386)

⇔ X†x º 0 (387)

that leads to a partial halfspace-description,

K =

x∈aff cone X | X†x º 0

(388)

For γ∗=X†Tei , any x =Xa , and for all i we have eTi X

†Xa = eTi a ≥ 0

only when a º 0. Hence x∈K .

When X is full-rank, then the unique biorthogonal expansion of x ∈ Kbecomes (381)

x = XX†x =N

i=1

Γi Γ∗T

i x (389)

whose coordinates Γ∗T

i x must be nonnegative because K is assumed pointedclosed and convex. Whenever X is full-rank, so is its pseudoinverse X†.(§E) In the present case, the columns of X†T are linearly independentand generators of the dual cone K∗∩ aff K ; hence, the columns constituteits extreme directions. (§2.10.2) That section of the dual cone is itself apolyhedral cone (by (266) or the cone intersection theorem, §2.7.2.1.1) havingthe same number of extreme directions as K .

2.13.8.2 x ∈ aff K

The extreme directions of K and K∗∩ aff K have a distinct relationship;because X†X = I , then for i,j = 1 . . . N , ΓT

i Γ∗i = 1, while for i 6= j ,

ΓTi Γ

∗j = 0. Yet neither set of extreme directions, Γi nor Γ∗

i , isnecessarily orthogonal. This is a biorthogonality condition, precisely,[362, §2.2.4] [199] implying each set of extreme directions is linearlyindependent. (§B.1.1.1)

The biorthogonal expansion therefore applies more broadly; meaning,for any x ∈ aff K , vector x can be uniquely expressed x =Xb whereb∈RN because aff K contains the origin. Thus, for any such x∈R(X)(confer §E.1.1), biorthogonal expansion (389) becomes x =XX†Xb =Xb .

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186 CHAPTER 2. CONVEX GEOMETRY

2.13.9 Formulae finding dual cone

2.13.9.1 Pointed K , dual, X skinny-or-square full-rank

We wish to derive expressions for a convex cone and its ordinary dualunder the general assumptions: pointed polyhedral K denoted by its linearlyindependent extreme directions arranged columnar in matrix X such that

rank(X∈ Rn×N) = N , dim aff K ≤ n (390)

The vertex-description is given:

K = Xa | a º 0 ⊆ Rn (391)

from which a halfspace-description for the dual cone follows directly:

K∗= y∈Rn | XTy º 0 (392)

By defining a matrixX⊥ , basisN (XT) (393)

(a columnar basis for the orthogonal complement of R(X)), we can say

aff cone X = aff K = x | X⊥Tx = 0 (394)

meaning K lies in a subspace, perhaps Rn. Thus a halfspace-description

K = x∈Rn | X†x º 0 , X⊥Tx = 0 (395)

and a vertex-description2.72 from (292)

K∗= [ X†T X⊥ −X⊥ ]b | b º 0 ⊆ Rn (396)

These results are summarized for a pointed polyhedral cone, havinglinearly independent generators, and its ordinary dual:

Cone Table 1 K K∗

vertex-description X X†T, ±X⊥

halfspace-description X† , X⊥T XT

2.72These descriptions are not unique. A vertex-description of the dual cone, for example,might use four conically independent generators for a plane (§2.10.0.0.1, Figure 48) whenonly three would suffice.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 187

2.13.9.2 Simplicial case

When a convex cone is simplicial (§2.12.3), Cone Table 1 simplifies becausethen aff cone X = Rn : For square X and assuming simplicial K such that

rank(X∈ Rn×N) = N , dim aff K = n (397)

we have

Cone Table S K K∗

vertex-description X X†T

halfspace-description X† XT

For example, vertex-description (396) simplifies to

K∗= X†Tb | b º 0 ⊂ Rn (398)

Now, because dimR(X)= dimR(X†T) , (§E) the dual cone K∗is simplicial

whenever K is.

2.13.9.3 Cone membership relations in a subspace

It is obvious by definition (276) of ordinary dual cone K∗, in ambient vector

space R , that its determination instead in subspace S ⊆ R is identicalto its intersection with S ; id est, assuming closed convex cone K⊆S andK∗⊆R

(K∗were ambient S) ≡ (K∗

in ambient R) ∩ S (399)

because

y ∈ S | 〈y , x〉 ≥ 0 for all x ∈ K = y ∈R | 〈y , x〉 ≥ 0 for all x ∈ K ∩ S(400)

From this, a constrained membership relation for the ordinary dual coneK∗⊆R , assuming x, y∈S and closed convex cone K⊆S

y ∈ K∗∩ S ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (401)

By closure in subspace S we have conjugation (§2.13.1.1):

x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K∗∩ S (402)

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188 CHAPTER 2. CONVEX GEOMETRY

This means membership determination in subspace S requires knowledge ofdual cone only in S . For sake of completeness, for proper cone K withrespect to subspace S (confer (304))

x ∈ intK ⇔ 〈y , x〉 > 0 for all y ∈ K∗∩ S , y 6= 0 (403)

x ∈ K , x 6= 0 ⇔ 〈y , x〉 > 0 for all y ∈ intK∗∩ S (404)

(By closure, we also have the conjugate relations.) Yet when S equals aff Kfor K a closed convex cone

x ∈ rel intK ⇔ 〈y , x〉 > 0 for all y ∈ K∗∩ aff K , y 6= 0 (405)

x ∈ K , x 6= 0 ⇔ 〈y , x〉 > 0 for all y ∈ rel int(K∗∩ aff K) (406)

2.13.9.4 Subspace S = aff KAssume now a subspace S that is the affine hull of cone K : Consider againa pointed polyhedral cone K denoted by its extreme directions arrangedcolumnar in matrix X such that

rank(X∈ Rn×N) = N , dim aff K ≤ n (390)

We want expressions for the convex cone and its dual in subspace S=aff K :

Cone Table A K K∗∩ aff Kvertex-description X X†T

halfspace-description X† , X⊥T XT, X⊥T

When dim aff K = n , this table reduces to Cone Table S. These descriptionsfacilitate work in a proper subspace. The subspace of symmetric matrices SN ,for example, often serves as ambient space.2.73

2.13.9.4.1 Exercise. Conically independent columns and rows.We suspect the number of conically independent columns (rows) of X tobe the same for X†T, where † denotes matrix pseudoinverse (§E). Provewhether it holds that the columns (rows) of X are c.i. ⇔ the columns (rows)of X†T are c.i. H

2.73The dual cone of positive semidefinite matrices SN∗+ = SN

+ remains in SN by convention,

whereas the ordinary dual cone would venture into RN×N .

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2.13. DUAL CONE & GENERALIZED INEQUALITY 189

−0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

KM+

K∗M+

K∗M+

KM+

∂K∗M+

(a)

(b)

X†T(: , 1) =

1−1

0

X†T(: , 2) =

01

−1

X†T(: , 3) =

001

X =

1 1 10 1 10 0 1

Figure 63: Simplicial cones. (a) Monotone nonnegative cone KM+ and itsdual K∗

M+ (drawn truncated) in R2. (b) Monotone nonnegative cone andboundary of its dual (both drawn truncated) in R3. Extreme directions ofK∗

M+ are indicated.

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190 CHAPTER 2. CONVEX GEOMETRY

2.13.9.4.2 Example. Monotone nonnegative cone.[59, exer.2.33] [350, §2] Simplicial cone (§2.12.3.1.1) KM+ is the cone of allnonnegative vectors having their entries sorted in nonincreasing order:

KM+ , x | x1 ≥ x2 ≥ · · · ≥ xn ≥ 0 ⊆ Rn+

= x | (ei − ei+1)Tx ≥ 0, i = 1 . . . n−1, eT

nx ≥ 0= x | X†x º 0

(407)

a halfspace-description where ei is the ith standard basis vector, and where2.74

X†T , [ e1−e2 e2−e3 · · · en ] ∈ Rn×n (408)

For any vectors x and y , simple algebra demands

xTy =n

i=1

xi yi = (x1 − x2)y1 + (x2 − x3)(y1 + y2) + (x3 − x4)(y1 + y2 + y3) + · · ·

+ (xn−1 − xn)(y1 + · · · + yn−1) + xn(y1 + · · · + yn) (409)

Because xi − xi+1 ≥ 0 ∀ i by assumption whenever x∈KM+ , we can employdual generalized inequalities (301) with respect to the selfdual nonnegativeorthant Rn

+ to find the halfspace-description of dual monotone nonnegative

cone K∗M+ . We can say xTy≥ 0 for all X†xº 0 [sic] if and only if

y1 ≥ 0 , y1 + y2 ≥ 0 , . . . , y1 + y2 + · · · + yn ≥ 0 (410)

id est,xTy ≥ 0 ∀X†x º 0 ⇔ XTy º 0 (411)

whereX = [ e1 e1+ e2 e1+ e2+ e3 · · · 1 ] ∈ Rn×n (412)

Because X†xº 0 connotes membership of x to pointed KM+ , then by(276) the dual cone we seek comprises all y for which (411) holds; thus itshalfspace-description

K∗M+ = y º

K∗

M+

0 = y | ∑ki=1 yi ≥ 0 , k = 1 . . . n = y | XTy º 0 ⊂ Rn

(413)

2.74With X† in hand, we might concisely scribe the remaining vertex- andhalfspace-descriptions from the tables for KM+ and its dual. Instead we use dualgeneralized inequalities in their derivation.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 191

−1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

x1

x2

KM

KMK∗

M

Figure 64: Monotone cone KM and its dual K∗M (drawn truncated) in R2.

The monotone nonnegative cone and its dual are simplicial, illustrated fortwo Euclidean spaces in Figure 63.

From §2.13.6.1, the extreme directions of proper KM+ are respectivelyorthogonal to the facets of K∗

M+ . Because K∗M+ is simplicial, the

inward-normals to its facets constitute the linearly independent rows of XT

by (413). Hence the vertex-description for KM+ employs the columns of Xin agreement with Cone Table S because X†=X−1. Likewise, the extremedirections of proper K∗

M+ are respectively orthogonal to the facets of KM+

whose inward-normals are contained in the rows of X† by (407). So thevertex-description for K∗

M+ employs the columns of X†T. 2

2.13.9.4.3 Example. Monotone cone.(Figure 64, Figure 65) Full-dimensional but not pointed, the monotone coneis polyhedral and defined by the halfspace-description

KM , x ∈ Rn | x1 ≥ x2 ≥ · · · ≥ xn = x ∈ Rn | X∗Tx º 0 (414)

Its dual is therefore pointed but not full-dimensional;

K∗M = X∗ b , [ e1− e2 e2− e3 · · · en−1− en ] b | b º 0 ⊂ Rn (415)

the dual cone vertex-description where the columns of X∗ comprise itsextreme directions. Because dual monotone cone K∗

M is pointed and satisfies

rank(X∗∈ Rn×N) = N , dim aff K∗ ≤ n (416)

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192 CHAPTER 2. CONVEX GEOMETRY

-x2

x3

∂KM

K∗M

x2

x3

∂KM

K∗M

Figure 65: Two views of monotone cone KM and its dual K∗M (drawn

truncated) in R3. Monotone cone is not pointed. Dual monotone coneis not full-dimensional. Cartesian coordinate axes are drawn for reference.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 193

where N = n−1, and because KM is closed and convex, we may adapt ConeTable 1 (p.186) as follows:

Cone Table 1* K∗ K∗∗= K

vertex-description X∗ X∗†T, ±X∗⊥

halfspace-description X∗† , X∗⊥T X∗T

The vertex-description for KM is therefore

KM = [ X∗†T X∗⊥ −X∗⊥ ]a | a º 0 ⊂ Rn (417)

where X∗⊥= 1 and

X∗† =1

n

n − 1 −1 −1 · · · −1 −1 −1

n − 2 n − 2 −2. . . · · · −2 −2

... n − 3 n − 3. . . −(n − 4)

... −3

3... n − 4

. . . −(n − 3) −(n − 3)...

2 2 · · · . . . 2 −(n − 2) −(n − 2)

1 1 1 · · · 1 1 −(n − 1)

∈ Rn−1×n

(418)while

K∗M = y ∈ Rn | X∗†y º 0 , X∗⊥Ty = 0 (419)

is the dual monotone cone halfspace-description. 2

2.13.9.4.4 Exercise. Inside the monotone cones.Mathematically describe the respective interior of the monotone nonnegativecone and monotone cone. In three dimensions, also describe the relativeinterior of each face. H

2.13.9.5 More pointed cone descriptions with equality condition

Consider pointed polyhedral cone K having a linearly independent set ofgenerators and whose subspace membership is explicit; id est, we are giventhe ordinary halfspace-description

K = x | Ax º 0 , Cx = 0 ⊆ Rn (266a)

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194 CHAPTER 2. CONVEX GEOMETRY

where A∈ Rm×n and C ∈ Rp×n. This can be equivalently written in termsof nullspace of C and vector ξ :

K = Zξ ∈ Rn | AZξ º 0 (420)

where R(Z∈ Rn×n−rank C ),N (C). Assuming (390) is satisfied

rank X , rank(

(AZ)†∈ Rn−rank C×m)

= m − ℓ = dim aff K ≤ n − rank C(421)

where ℓ is the number of conically dependent rows in AZ which mustbe removed to make AZ before the Cone Tables become applicable.2.75

Then results collected there admit assignment X , (AZ)†∈ Rn−rank C×m−ℓ,where A∈Rm−ℓ×n, followed with linear transformation by Z . So we get thevertex-description, for full-rank (AZ)† skinny-or-square,

K = Z(AZ)† b | b º 0 (422)

From this and (341) we get a halfspace-description of the dual cone

K∗= y ∈ Rn | (ZTAT)†ZTy º 0 (423)

From this and Cone Table 1 (p.186) we get a vertex-description, (1836)

K∗= [ Z†T(AZ)T CT −CT ]c | c º 0 (424)

Yet because

K = x | Ax º 0 ∩ x | Cx = 0 (425)

then, by (292), we get an equivalent vertex-description for the dual cone

K∗= x | Ax º 0∗ + x | Cx = 0∗

= [ AT CT −CT ]b | b º 0(426)

from which the conically dependent columns may, of course, be removed.

2.75When the conically dependent rows (§2.10) are removed, the rows remaining must belinearly independent for the Cone Tables (p.19) to apply.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 195

2.13.10 Dual cone-translate

First-order optimality condition (330) inspires a dual-cone variant: For anyset K , the negative dual of its translation by any a∈Rn is

−(K − a)∗ , y ∈ Rn | 〈y , x − a〉≤ 0 for all x ∈ K= y ∈ Rn | 〈y , x〉≤ 0 for all x ∈ K − a (427)

a closed convex cone called normal cone to K at point a . (K⊥(a) §E.10.3.2.1)From this, a new membership relation like (298) for closed convex cone K :

y ∈ −(K − a)∗ ⇔ 〈y , x − a〉≤ 0 for all x ∈ K (428)

and by closure the conjugate

x ∈ K ⇔ 〈y , x − a〉≤ 0 for all y ∈ −(K − a)∗ (429)

2.13.10.1 first-order optimality condition - restatement

(confer §2.13.3) The general first-order necessary and sufficient condition foroptimality of solution x⋆ to a minimization problem with real differentiableconvex objective function f(x) : Rn→R over convex feasible set C is [302, §3]

−∇f(x⋆) ∈ −(C − x⋆)∗ , x⋆∈ C (430)

id est, the negative gradient (§3.7) belongs to the normal cone to C at x⋆ asin Figure 66.

2.13.10.1.1 Example. Normal cone to orthant.Consider proper cone K= Rn

+ , the selfdual nonnegative orthant in Rn. Thenormal cone to Rn

+ at a∈K is (2053)

K⊥Rn

+(a∈Rn

+) = −(Rn+ − a)∗ = −Rn

+ ∩ a⊥ , a∈Rn+ (431)

where −Rn+ =−K∗

is the algebraic complement of Rn+ , and a⊥ is the

orthogonal complement to range of vector a . This means: When point ais interior to Rn

+ , the normal cone is the origin. If np represents numberof nonzero entries in vector a∈∂Rn

+ , then dim(−Rn+ ∩ a⊥)= n − np and

there is a complementary relationship between the nonzero entries in vector aand the nonzero entries in any vector x∈−Rn

+ ∩ a⊥. 2

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196 CHAPTER 2. CONVEX GEOMETRY

α

C

β

γ

x⋆

−∇f(x⋆)

y | ∇f(x⋆)T(y − x⋆) = 0 , f(x⋆)= γ

z | f(z) = α

α ≥ β ≥ γ

Figure 66: (confer Figure 77) Shown is a plausible contour plot in R2 ofsome arbitrary differentiable convex real function f(x) at selected levels α ,β , and γ ; id est, contours of equal level f (level sets) drawn dashed infunction’s domain. From results in §3.7.2 (p.251), gradient ∇f(x⋆) is normalto γ-sublevel set Lγf (537) by Definition E.9.1.0.1. From §2.13.10.1, functionis minimized over convex set C at point x⋆ iff negative gradient −∇f(x⋆)belongs to normal cone to C there. In circumstance depicted, normal coneis a ray whose direction is coincident with negative gradient. So, gradient isnormal to a hyperplane supporting both C and the γ-sublevel set.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 197

2.13.10.1.2 Example. Optimality conditions for conic problem.Consider a convex optimization problem having real differentiable convexobjective function f(x) : Rn→R defined on domain Rn

minimizex

f(x)

subject to x ∈ K (432)

Let’s first suppose that the feasible set is a pointed polyhedral cone Kpossessing a linearly independent set of generators and whose subspacemembership is made explicit by fat full-rank matrix C∈ Rp×n ; id est, weare given the halfspace-description, for A∈Rm×n

K = x | Ax º 0 , Cx = 0 ⊆ Rn (266a)

(We’ll generalize to any convex cone K shortly.) Vertex-description of thiscone, assuming (AZ)† skinny-or-square full-rank, is

K = Z(AZ)† b | b º 0 (422)

where A∈Rm−ℓ×n, ℓ is the number of conically dependent rows in AZ (§2.10)which must be removed, and Z∈ Rn×n−rank C holds basisN (C) columnar.

From optimality condition (330),

∇f(x⋆)T(Z(AZ)† b − x⋆)≥ 0 ∀ b º 0 (433)

−∇f(x⋆)TZ(AZ)†(b − b⋆)≤ 0 ∀ b º 0 (434)

becausex⋆ , Z(AZ)† b⋆∈ K (435)

From membership relation (428) and Example 2.13.10.1.1

〈−(ZTAT)†ZT∇f(x⋆) , b − b⋆〉 ≤ 0 for all b ∈ Rm−ℓ+

⇔−(ZTAT)†ZT∇f(x⋆) ∈ −Rm−ℓ

+ ∩ b⋆⊥(436)

Then equivalent necessary and sufficient conditions for optimality of conicproblem (432) with feasible set K are: (confer (340))

(ZTAT)†ZT∇f(x⋆) ºR

m−ℓ+

0 , b⋆ ºR

m−ℓ+

0 , ∇f(x⋆)TZ(AZ)† b⋆ = 0 (437)

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198 CHAPTER 2. CONVEX GEOMETRY

expressible, by (423),

∇f(x⋆) ∈ K∗, x⋆ ∈ K , ∇f(x⋆)Tx⋆ = 0 (438)

This result (438) actually applies more generally to any convex cone Kcomprising the feasible set: Necessary and sufficient optimality conditionsare in terms of objective gradient

−∇f(x⋆) ∈ −(K − x⋆)∗ , x⋆∈ K (430)

whose membership to normal cone, assuming only cone K convexity,

−(K − x⋆)∗ = K⊥K(x⋆∈ K) = −K∗∩ x⋆⊥ (2053)

equivalently expresses conditions (438).When K= Rn

+ , in particular, then C =0, A=Z = I∈ Sn ; id est,

minimizex

f(x)

subject to x ºRn

+

0 (439)

Necessary and sufficient optimality conditions become (confer [59, §4.2.3])

∇f(x⋆) ºRn

+

0 , x⋆ ºRn

+

0 , ∇f(x⋆)Tx⋆ = 0 (440)

equivalent to condition (308)2.76 (under nonzero gradient) for membershipto the nonnegative orthant boundary ∂Rn

+ . 2

2.13.10.1.3 Example. Complementarity problem. [206]A complementarity problem in nonlinear function f is nonconvex

find z ∈ Ksubject to f(z) ∈ K∗

〈z , f(z)〉 = 0(441)

yet bears strong resemblance to (438) and to (1989) Moreau’s decompositiontheorem on page 728 for projection P on mutually polar cones K and −K∗

.

2.76 and equivalent to well-known Karush-Kuhn-Tucker (KKT) optimality conditions[59, §5.5.3] because the dual variable becomes gradient ∇f(x).

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2.13. DUAL CONE & GENERALIZED INEQUALITY 199

Identify a sum of mutually orthogonal projections x , z−f(z) ; in Moreau’sterms, z=PKx and −f(z)=P−K∗x . Then f(z)∈K∗

(§E.9.2.2 no.4) and zis a solution to the complementarity problem iff it is a fixed point of

z = PKx = PK(z − f(z)) (442)

Given that a solution exists, existence of a fixed point would be guaranteedby theory of contraction. [223, p.300] But because only nonexpansivity(Theorem E.9.3.0.1) is achievable by a projector, uniqueness cannot beassured. [200, p.155] Elegant proofs of equivalence between complementarityproblem (441) and fixed point problem (442) are provided by Sandor ZoltanNemeth on Wıκımization. 2

2.13.10.1.4 Example. Linear complementarity problem. [85] [268] [307]Given matrix B∈Rn×n and vector q∈Rn, a prototypical complementarityproblem on the nonnegative orthant K= Rn

+ is linear in w = f(z) :

find z º 0subject to w º 0

wTz = 0w = q + Bz

(443)

This problem is not convex when both vectors w and z are variable.2.77

Notwithstanding, this linear complementarity problem can be solved byidentifying w←∇f(z)= q + Bz then substituting that gradient into (441)

find z ∈ Ksubject to ∇f(z) ∈ K∗

〈z , ∇f(z)〉 = 0(444)

2.77But if one of them is fixed, then the problem becomes convex with a very simplegeometric interpretation: Define the affine subset

A , y∈Rn | By = w − q

For wTz to vanish, there must be a complementary relationship between the nonzeroentries of vectors w and z ; id est, wizi =0 ∀ i . Given wº 0, then z belongs to theconvex set of solutions:

z ∈ −K⊥Rn

+(w∈Rn

+) ∩ A = Rn+ ∩ w⊥ ∩ A

where K⊥Rn

+(w) is the normal cone to Rn

+ at w (431). If this intersection is nonempty, then

the problem is solvable.

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200 CHAPTER 2. CONVEX GEOMETRY

which is simply a restatement of optimality conditions (438) for conic problem(432). Suitable f(z) is the quadratic objective from convex problem

minimizez

12zTBz + qTz

subject to z º 0(445)

which means B∈ Sn+ should be (symmetric) positive semidefinite for solution

of (443) by this method. Then (443) has solution iff (445) does. 2

2.13.10.1.5 Exercise. Optimality for equality constrained conic problem.Consider a conic optimization problem like (432) having real differentiableconvex objective function f(x) : Rn→R

minimizex

f(x)

subject to Cx = dx ∈ K

(446)

minimized over convex cone K but, this time, constrained to affine setA = x | Cx = d. Show, by means of first-order optimality condition(330) or (430), that necessary and sufficient optimality conditions are:(confer (438))

x⋆ ∈ KCx⋆ = d

∇f(x⋆) + CTν⋆ ∈ K∗

〈∇f(x⋆) + CTν⋆, x⋆〉 = 0

(447)

where ν⋆ is any vector2.78 satisfying these conditions. H

2.13.11 Proper nonsimplicial K , dual, X fat full-rank

Since conically dependent columns can always be removed from X toconstruct K or to determine K∗

[Wıκımization], then assume we are givena set of N conically independent generators (§2.10) of an arbitrary properpolyhedral cone K in Rn arranged columnar in X∈ Rn×N such that N > n(fat) and rankX = n . Having found formula (398) to determine the dual ofa simplicial cone, the easiest way to find a vertex-description of proper dual

2.78 an optimal dual variable, these optimality conditions are equivalent to the KKTconditions [59, §5.5.3].

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2.13. DUAL CONE & GENERALIZED INEQUALITY 201

cone K∗is to first decompose K into simplicial parts Ki so that K=

⋃Ki .2.79

Each component simplicial cone in K corresponds to some subset of nlinearly independent columns from X . The key idea, here, is how theextreme directions of the simplicial parts must remain extreme directions ofK . Finding the dual of K amounts to finding the dual of each simplicial part:

2.13.11.0.1 Theorem. Dual cone intersection. [327, §2.7]Suppose proper cone K⊂Rn equals the union of M simplicial cones Ki whoseextreme directions all coincide with those of K . Then proper dual cone K∗

is the intersection of M dual simplicial cones K∗i ; id est,

K =M⋃

i=1

Ki ⇒ K∗=

M⋂

i=1

K∗i (448)

Proof. For Xi∈ Rn×n, a complete matrix of linearly independentextreme directions (p.137) arranged columnar, corresponding simplicial Ki

(§2.12.3.1.1) has vertex-description

Ki = Xi c | c º 0 (449)

Now suppose,

K =M⋃

i=1

Ki =M⋃

i=1

Xi c | c º 0 (450)

The union of all Ki can be equivalently expressed

K =

[ X1 X2 · · · XM ]

ab...c

| a , b . . . c º 0

(451)

Because extreme directions of the simplices Ki are extreme directions of Kby assumption, then by the extremes theorem (§2.8.1.1.1),

2.79That proposition presupposes, of course, that we know how to perform simplicialdecomposition efficiently; also called “triangulation”. [299] [169, §3.1] [170, §3.1] Existenceof multiple simplicial parts means expansion of x∈K like (389) can no longer be uniquebecause the number N of extreme directions in K exceeds dimension n of the space.

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202 CHAPTER 2. CONVEX GEOMETRY

K = [ X1 X2 · · · XM ] d | d º 0 (452)

Defining X , [ X1 X2 · · · XM ] (with any redundant [sic] columns optionallyremoved from X), then K∗

can be expressed, (341) (Cone Table S, p.187)

K∗= y | XTy º 0 =

M⋂

i=1

y | XTi y º 0 =

M⋂

i=1

K∗i (453)

¨

To find the extreme directions of the dual cone, first we observe that somefacets of each simplicial part Ki are common to facets of K by assumption,and the union of all those common facets comprises the set of all facets ofK by design. For any particular proper polyhedral cone K , the extremedirections of dual cone K∗

are respectively orthogonal to the facets of K .(§2.13.6.1) Then the extreme directions of the dual cone can be foundamong inward-normals to facets of the component simplicial cones Ki ; thosenormals are extreme directions of the dual simplicial cones K∗

i . From thetheorem and Cone Table S (p.187),

K∗=

M⋂

i=1

K∗i =

M⋂

i=1

X†Ti c | c º 0 (454)

The set of extreme directions Γ∗i for proper dual cone K∗

is thereforeconstituted by those conically independent generators, from the columnsof all the dual simplicial matrices X†T

i , that do not violate discretedefinition (341) of K∗

;

Γ∗1 , Γ

∗2 . . . Γ

∗N

= c.i.

X†Ti (:,j) , i=1 . . . M , j =1 . . . n | X†

i (j, :)Γℓ ≥ 0, ℓ =1 . . . N

(455)

where c.i. denotes selection of only the conically independent vectors fromthe argument set, argument (:,j) denotes the j th column while (j, :) denotesthe j th row, and Γℓ constitutes the extreme directions of K . Figure 49b(p.136) shows a cone and its dual found via this algorithm.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 203

2.13.11.0.2 Example. Dual of K nonsimplicial in subspace aff K .Given conically independent generators for pointed closed convex cone K inR4 arranged columnar in

X = [ Γ1 Γ2 Γ3 Γ4 ] =

1 1 0 0−1 0 1 0

0 −1 0 10 0 −1 −1

(456)

having dim aff K= rank X = 3, (261) then performing the most inefficientsimplicial decomposition in aff K we find

X1 =

1 1 0−1 0 1

0 −1 00 0 −1

, X2 =

1 1 0−1 0 0

0 −1 10 0 −1

X3 =

1 0 0−1 1 0

0 0 10 −1 −1

, X4 =

1 0 00 1 0

−1 0 10 −1 −1

(457)

The corresponding dual simplicial cones in aff K have generators respectivelycolumnar in

4X†T1 =

2 1 1−2 1 1

2 −3 1−2 1 −3

, 4X†T2 =

1 2 1−3 2 1

1 −2 11 −2 −3

4X†T3 =

3 2 −1−1 2 −1−1 −2 3−1 −2 −1

, 4X†T4 =

3 −1 2−1 3 −2−1 −1 2−1 −1 −2

(458)

Applying algorithm (455) we get

[

Γ∗1 Γ

∗2 Γ

∗3 Γ

∗4

]

=1

4

1 2 3 21 2 −1 −21 −2 −1 2

−3 −2 −1 −2

(459)

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204 CHAPTER 2. CONVEX GEOMETRY

whose rank is 3, and is the known result;2.80 a conically independent setof generators for that pointed section of the dual cone K∗

in aff K ; id est,K∗∩ aff K . 2

2.13.11.0.3 Example. Dual of proper polyhedral K in R4.Given conically independent generators for a full-dimensional pointed closedconvex cone K

X = [ Γ1 Γ2 Γ3 Γ4 Γ5 ] =

1 1 0 1 0−1 0 1 0 1

0 −1 0 1 00 0 −1 −1 0

(460)

we count 5!/((5−4)! 4!)=5 component simplices.2.81 Applying algorithm(455), we find the six extreme directions of dual cone K∗

(with Γ2 = Γ∗5)

X∗ =[

Γ∗1 Γ

∗2 Γ

∗3 Γ

∗4 Γ

∗5 Γ

∗6

]

=

1 0 0 1 1 11 0 0 1 0 01 0 −1 0 −1 11 −1 −1 1 0 0

(461)

which means, (§2.13.6.1) this proper polyhedral K= cone(X) has six(three-dimensional) facets generated G by its extreme directions:

G

F1

F2

F3

F4

F5

F6

=

Γ1 Γ2 Γ3

Γ1 Γ2 Γ5

Γ1 Γ4 Γ5

Γ1 Γ3 Γ4

Γ3 Γ4 Γ5

Γ2 Γ3 Γ5

(462)

2.80These calculations proceed so as to be consistent with [111, §6]; as if the ambientvector space were proper subspace aff K whose dimension is 3. In that ambient space, Kmay be regarded as a proper cone. Yet that author (from the citation) erroneously statesdimension of the ordinary dual cone to be 3 ; it is, in fact, 4.2.81There are no linearly dependent combinations of three or four extreme directions inthe primal cone.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 205

whereas dual proper polyhedral cone K∗has only five:

G

F∗1

F∗2

F∗3

F∗4

F∗5

=

Γ∗1 Γ∗

2 Γ∗3 Γ∗

4

Γ∗1 Γ∗

2 Γ∗6

Γ∗1 Γ∗

4 Γ∗5 Γ∗

6

Γ∗3 Γ∗

4 Γ∗5

Γ∗2 Γ∗

3 Γ∗5 Γ∗

6

(463)

Six two-dimensional cones, having generators respectively Γ∗1 Γ

∗3 Γ∗

2 Γ∗4

Γ∗1 Γ

∗5 Γ∗

4 Γ∗6 Γ∗

2 Γ∗5 Γ∗

3 Γ∗6 , are relatively interior to dual facets;

so cannot be two-dimensional faces of K∗(by Definition 2.6.0.0.3).

We can check this result (461) by reversing the process; we find6!/((6−4)! 4!)− 3=12 component simplices in the dual cone.2.82 Applyingalgorithm (455) to those simplices returns a conically independent set ofgenerators for K equivalent to (460). 2

2.13.11.0.4 Exercise. Reaching proper polyhedral cone interior.Name two extreme directions Γi of cone K from Example 2.13.11.0.3 whoseconvex hull passes through that cone’s interior. Explain why. Are there twosuch extreme directions of dual cone K∗

? H

A natural question pertains to whether a theory of unique coordinates,like biorthogonal expansion, is extensible to proper cones whose extremedirections number in excess of ambient spatial dimensionality.

2.13.11.0.5 Theorem. Conic coordinates.With respect to vector v in some finite-dimensional Euclidean space Rn,define a coordinate t⋆v of point x in full-dimensional pointed closed convexcone K

t⋆v(x) , supt∈R | x − tv∈K (464)

Given points x and y in cone K , if t⋆v(x)= t⋆v(y) for each and every extremedirection v of K then x = y . ⋄2.82Three combinations of four dual extreme directions are linearly dependent; they belongto the dual facets. But there are no linearly dependent combinations of three dual extremedirections.

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206 CHAPTER 2. CONVEX GEOMETRY

2.13.11.0.6 Proof. Vector x− t⋆v must belong to the coneboundary ∂K by definition (464). So there must exist a nonzero vector λ thatis inward-normal to a hyperplane supporting cone K and containing x− t⋆v ;id est, by boundary membership relation for full-dimensional pointed closedconvex cones (§2.13.2)

x− t⋆v ∈ ∂K ⇔ ∃ λ 6= 0 Ä 〈λ , x− t⋆v〉 = 0 , λ ∈ K∗, x− t⋆v ∈ K (308)

where

K∗= w∈ Rn | 〈v , w〉 ≥ 0 for all v ∈ G(K) (347)

is the full-dimensional pointed closed convex dual cone. The set G(K) , ofpossibly infinite cardinality N , comprises generators for cone K ; e.g., itsextreme directions which constitute a minimal generating set. If x− t⋆vis nonzero, any such vector λ must belong to the dual cone boundary byconjugate boundary membership relation

λ ∈ ∂K∗ ⇔ ∃ x−t⋆v 6= 0 Ä 〈λ , x−t⋆v〉 = 0 , x−t⋆v ∈ K , λ ∈ K∗(309)

where

K = z∈ Rn | 〈λ , z〉 ≥ 0 for all λ ∈ G(K∗) (346)

This description of K means: cone K is an intersection of halfspaces whoseinward-normals are generators of the dual cone. Each and every face ofcone K (except the cone itself) belongs to a hyperplane supporting K . Eachand every vector x− t⋆v on the cone boundary must therefore be orthogonalto an extreme direction constituting generators G(K∗

) of the dual cone.To the ith extreme direction v = Γi∈ Rn of cone K , ascribe a coordinate

t⋆i (x)∈ R of x from definition (464). On domain K , the mapping

t⋆(x) =

t⋆1(x)...

t⋆N(x)

: Rn→ RN (465)

has no nontrivial nullspace. Because x− t⋆v must belong to ∂K by definition,the mapping t⋆(x) is equivalent to a convex problem (separable in index i)whose objective (by (308)) is tightly bounded below by 0 :

t⋆(x) ≡ arg minimizet∈RN

N∑

i=1

Γ∗Tj(i)(x − tiΓi)

subject to x − tiΓi ∈ K , i=1 . . . N(466)

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2.13. DUAL CONE & GENERALIZED INEQUALITY 207

where index j∈ I is dependent on i and where (by (346)) λ = Γ∗j ∈ Rn is an

extreme direction of dual cone K∗that is normal to a hyperplane supporting

K and containing x − t⋆i Γi . Because extreme-direction cardinality N forcone K is not necessarily the same as for dual cone K∗

, index j must bejudiciously selected from a set I .

To prove injectivity when extreme-direction cardinality N > n exceedsspatial dimension, we need only show mapping t⋆(x) to be invertible;[125, thm.9.2.3] id est, x is recoverable given t⋆(x) :

x = arg minimizex∈R

n

N∑

i=1

Γ∗Tj(i)(x − t⋆i Γi)

subject to x − t⋆i Γi ∈ K , i=1 . . . N(467)

The feasible set of this nonseparable convex problem is an intersection oftranslated full-dimensional pointed closed convex cones

iK + t⋆i Γi . Theobjective function’s linear part describes movement in normal-direction −Γ∗

j

for each of N hyperplanes. The optimal point of hyperplane intersection isthe unique solution x when Γ∗

j comprises n linearly independent normalsthat come from the dual cone and make the objective vanish. Because thedual cone K∗

is full-dimensional, pointed, closed, and convex by assumption,there exist N extreme directions Γ∗

j from K∗⊂ Rn that span Rn. Sowe need simply choose N spanning dual extreme directions that make theoptimal objective vanish. Because such dual extreme directions preexistby (308), t⋆(x) is invertible.

Otherwise, in the case N≤ n , t⋆(x) holds coordinates for biorthogonalexpansion. Reconstruction of x is therefore unique. ¨

2.13.11.1 conic coordinate computation

The foregoing proof of the conic coordinates theorem is not constructive; itestablishes existence of dual extreme directions Γ∗

j that will reconstructa point x from its coordinates t⋆(x) via (467), but does not prescribe theindex set I . There are at least two computational methods for specifyingΓ∗

j(i) : one is combinatorial but sure to succeed, the other is a geometricmethod that searches for a minimum of a nonconvex function. We describethe latter:

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208 CHAPTER 2. CONVEX GEOMETRY

Convex problem (P)

(P)maximize

t∈R

t

subject to x − tv∈K

minimizeλ∈R

nλTx

subject to λTv = 1

λ ∈ K∗(D) (468)

is equivalent to definition (464) whereas convex problem (D) is its dual;2.83

meaning, primal and dual optimal objectives are equal t⋆ = λ⋆Tx assumingSlater’s condition (p.277) is satisfied. Under this assumption of strongduality, λ⋆T(x − t⋆v)= t⋆(1 − λ⋆Tv)=0 ; which implies, the primal problemis equivalent to

minimizet∈R

λ⋆T(x − tv)

subject to x − tv∈K(P) (469)

while the dual problem is equivalent to

minimizeλ∈R

nλT(x − t⋆v)

subject to λTv = 1

λ ∈ K∗(D) (470)

Instead given coordinates t⋆(x) and a description of cone K , we proposeinversion by alternating solution of primal and dual problems

minimizex∈R

n

N∑

i=1

Γ∗Ti (x − t⋆i Γi)

subject to x − t⋆i Γi ∈ K , i=1 . . . N(471)

minimizeΓ∗

i∈Rn

N∑

i=1

Γ∗Ti (x⋆− t⋆i Γi)

subject to Γ∗Ti Γi = 1 , i=1 . . . N

Γ∗i ∈ K∗

, i=1 . . . N

(472)

2.83Form a Lagrangian associated with primal problem (P):

L(t , λ) = t + λT(x − tv) = λTx + t(1 − λTv) , λ ºK∗

0

supt

L(t , λ) = λTx , 1 − λTv = 0

Dual variable (Lagrange multiplier [246, p.216]) λ generally has a nonnegative sense forprimal maximization with any cone membership constraint, whereas λ would have anonpositive sense were the primal instead a minimization problem having a membershipconstraint.

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2.13. DUAL CONE & GENERALIZED INEQUALITY 209

where dual extreme directions Γ∗i are initialized arbitrarily and ultimately

ascertained by the alternation. Convex problems (471) and (472) areiterated until convergence which is guaranteed by virtue of a monotonicallynonincreasing real sequence of objective values. Convergence can be fast.The mapping t⋆(x) is uniquely inverted when the necessarily nonnegativeobjective vanishes; id est, when Γ∗T

i (x⋆− t⋆i Γi)=0 ∀ i . Here, a zeroobjective can occur only at the true solution x . But this global optimalitycondition cannot be guaranteed by the alternation because the commonobjective function, when regarded in both primal x and dual Γ∗

i variablessimultaneously, is generally neither quasiconvex or monotonic. (§3.9.0.0.3)

Conversely, a nonzero objective at convergence is a certificate thatinversion was not performed properly. A nonzero objective indicates thatthe global minimum of a multimodal objective function could not be foundby this alternation. That is a flaw in this particular iterative algorithm forinversion; not in theory.2.84 A numerical remedy is to reinitialize the Γ∗

i todifferent values.

2.84The Proof 2.13.11.0.6, that suitable dual extreme directions Γ∗j always exist, means

that a global optimization algorithm would always find the zero objective of alternation(471) (472); hence, the unique inversion x . But such an algorithm can be combinatorial.

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210 CHAPTER 2. CONVEX GEOMETRY

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Chapter 3

Geometry of convex functions

The link between convex sets and convex functions is via theepigraph: A function is convex if and only if its epigraph is aconvex set.

−Werner Fenchel

We limit our treatment of multidimensional functions3.1 to finite-dimensionalEuclidean space. Then an icon for a one-dimensional (real) convex functionis bowl-shaped (Figure 76), whereas the concave icon is the inverted bowl;respectively characterized by a unique global minimum and maximum whoseexistence is assumed. Because of this simple relationship, usage of the termconvexity is often implicitly inclusive of concavity. Despite iconic imagery,the reader is reminded that the set of all convex, concave, quasiconvex, andquasiconcave functions contains the monotonic functions [200] [211, §2.3.5];e.g., [59, §3.6, exer.3.46].

3.1 vector- or matrix-valued functions including the real functions. Appendix D, with itstables of first- and second-order gradients, is the practical adjunct to this chapter.© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

211

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212 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.1 Convex function

3.1.1 real and vector-valued function

Vector-valued function

f(X) : Rp×k→RM =

f1(X)...

fM(X)

(473)

assigns each X in its domain dom f (a subset of ambient vector space Rp×k)to a specific element [254, p.3] of its range (a subset of RM). Function f(X)is linear in X on its domain if and only if, for each and every Y, Z∈dom fand α , β∈R

f(αY + βZ) = αf(Y ) + βf(Z) (474)

A vector-valued function f(X) : Rp×k→RM is convex in X if and only ifdom f is a convex set and, for each and every Y, Z∈dom f and 0≤µ≤1

f(µY + (1 − µ)Z) ¹RM

+

µf(Y ) + (1 − µ)f(Z) (475)

As defined, continuity is implied but not differentiability (nor smoothness).3.2

Apparently some, but not all, nonlinear functions are convex. Reversing senseof the inequality flips this definition to concavity. Linear (and affine §3.5)functions attain equality in this definition. Linear functions are thereforesimultaneously convex and concave.

Vector-valued functions are most often compared (173) as in (475) withrespect to the M -dimensional selfdual nonnegative orthant RM

+ , a propercone.3.3 In this case, the test prescribed by (475) is simply a comparisonon R of each entry fi of a vector-valued function f . (§2.13.4.2.3) Thevector-valued function case is therefore a straightforward generalization ofconventional convexity theory for a real function. This conclusion followsfrom theory of dual generalized inequalities (§2.13.2.0.1) which asserts

f convex w.r.t RM+ ⇔ wTf convex ∀w∈ G(RM∗

+ ) (476)

3.2Figure 67b illustrates a nondifferentiable convex function. Differentiability is certainlynot a requirement for optimization of convex functions by numerical methods; e.g., [239].3.3Definition of convexity can be broadened to other (not necessarily proper) cones.

Referred to in the literature as K-convexity, [291] RM∗+ (476) generalizes to K∗

.

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3.1. CONVEX FUNCTION 213

(a) (b)

f1(x) f2(x)

Figure 67: Convex real functions here have a unique minimizer x⋆. Forx∈R , f1(x)=x2 =‖x‖2

2 is strictly convex whereas nondifferentiable functionf2(x)=

√x2 = |x|=‖x‖2 is convex but not strictly. Strict convexity of a real

function is only a sufficient condition for minimizer uniqueness.

shown by substitution of the defining inequality (475). Discretization allows

relaxation (§2.13.4.2.1) of a semiinfinite number of conditions w∈ RM∗+ to:

w ∈ G(RM∗+ ) ≡ ei∈ RM , i=1 . . . M (477)

(the standard basis for RM and a minimal set of generators (§2.8.1.2) for RM+ )

from which the stated conclusion follows; id est, the test for convexity of avector-valued function is a comparison on R of each entry.

3.1.2 strict convexity

When f(X) instead satisfies, for each and every distinct Y and Z in dom fand all 0<µ<1

f(µY + (1 − µ)Z) ≺RM

+

µf(Y ) + (1 − µ)f(Z) (478)

then we shall say f is a strictly convex function.Similarly to (476)

f strictly convex w.r.t RM+ ⇔ wTf strictly convex ∀w∈ G(RM∗

+ ) (479)

discretization allows relaxation of the semiinfinite number of conditionsw∈ RM∗

+ , w 6= 0 (304) to a finite number (477). More tests for strictconvexity are given in §3.7.1.0.2, §3.7.4, and §3.8.3.0.2.

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214 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Any convex real function f(X) has unique minimum value over anyconvex subset of its domain. [298, p.123] Yet solution to some convexoptimization problem is, in general, not unique; e.g., given minimizationof a convex real function over some convex feasible set C

minimizeX

f(X)

subject to X∈ C(480)

any optimal solution X⋆ comes from a convex set of optimal solutions

X⋆ ∈ X | f(X) = infY ∈C

f(Y ) ⊆ C (481)

But a strictly convex real function has a unique minimizer X⋆ ; id est, for theoptimal solution set in (481) to be a single point, it is sufficient (Figure 67)that f(X) be a strictly convex real3.4 function and set C convex. [318]

Quadratic real functions xTAx + bTx + c are convex in x iff Aº0.Quadratics characterized by positive definite matrix A≻0 are strictly convex.The vector 2-norm square ‖x‖2 (Euclidean norm square) and Frobenius’norm square ‖X‖2

F , for example, are strictly convex functions of theirrespective argument (each absolute norm is convex but not strictly convex).Figure 67a illustrates a strictly convex real function.

3.1.2.1 minimum/minimal element, dual cone characterization

f(X⋆) is the minimum element of its range if and only if, for each and every

w∈ int RM∗+ , it is the unique minimizer of wTf . (Figure 68) [59, §2.6.3]

If f(X⋆) is a minimal element of its range, then there exists a nonzero

w∈ RM∗+ such that f(X⋆) minimizes wTf . If f(X⋆) minimizes wTf for some

w∈ int RM∗+ , conversely, then f(X⋆) is a minimal element of its range.

3.1.2.1.1 Exercise. Cone of convex functions.Prove that relation (476) implies: the set of all convex vector-valued functionsin RM is a convex cone. So, the trivial function f = 0 is convex. Indeed, anynonnegatively weighted sum of (strictly) convex functions remains (strictly)convex.3.5 Interior to the cone are the strictly convex functions. H

3.4It is more customary to consider only a real function for the objective of a convexoptimization problem because vector- or matrix-valued functions can introduce ambiguityinto the optimal objective value. (§2.7.2.2, §3.1.2.1) Study of multidimensional objectivefunctions is called multicriteria- [321] or multiobjective- or vector-optimization.3.5Strict case excludes cone’s point at origin. By these definitions (476) (479), positively

weighted sums mixing convex and strictly convex real functions are not strictly convex

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3.1. CONVEX FUNCTION 215

Rf

Rf

f

f

f(X⋆)

f(X⋆)

f(X)1

f(X)2

w

w

(b)

(a)

Figure 68: (confer Figure 39) Function range is convex for a convex problem.(a) Point f(X⋆) is the unique minimum element of function range Rf .(b) Point f(X⋆) is a minimal element of depicted range.(Cartesian axes drawn for reference.)

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216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.2 Conic origins of Lagrangian

The cone of convex functions, membership relation (476), provides afoundation for what is known as a Lagrangian function. [248, p.398] [276]

Consider a conic optimization problem, for proper cone K and affinesubset A

minimizex

f(x)

subject to g(x) ∈ Kh(x) ∈ A

(482)

A Cartesian product of convex functions remains convex, so we may write

[

fgh

]

convex w.r.t

RM+

KA

⇔ [wT λT νT ]

[

fgh

]

convex ∀

wλν

∈ G

RM∗+

K∗

A⊥

(483)

where A⊥ is the normal cone to A . This holds because of equality for hin convexity criterion (475) and because membership relation (429), givenpoint a∈A , becomes

h ∈ A ⇔ 〈ν , h − a〉=0 for all ν ∈ G(A⊥) (484)

When A= 0, for example, a minimal set of generators G for A⊥ is a supersetof the standard basis for RM (Example E.5.0.0.7) the ambient space of A .

A real Lagrangian arises from the scalar term in (483); id est,

L , [ wT λT νT ]

[

fgh

]

= wTf + λTg + νTh (485)

3.3 Practical norm functions, absolute value

To some mathematicians, “all norms on Rn are equivalent” [155, p.53];meaning, ratios of different norms are bounded above and below by finitepredeterminable constants. But to statisticians and engineers, all normsare certainly not created equal; as evidenced by the compressed sensingrevolution, begun in 2004, whose focus is predominantly 1-norm.

because each summand is considered to be an entry from a vector-valued function.

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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 217

A norm on Rn is a convex function f : Rn→ R satisfying: for x, y∈ Rn,α∈R [223, p.59] [155, p.52]

1. f(x) ≥ 0 (f(x) = 0 ⇔ x = 0) (nonnegativity)

2. f(x + y) ≤ f(x) + f(y) (triangle inequality)3.6

3. f(αx) = |α|f(x) (nonnegative homogeneity)

Convexity follows by properties 2 and 3. Most useful are 1-, 2-, and ∞-norm:

‖x‖1 = minimizet∈R

n1Tt

subject to −t ¹ x ¹ t(486)

where |x|= t⋆ (entrywise absolute value equals optimal t ).3.7

‖x‖2 = minimizet∈R

t

subject to

[

tI xxT t

]

ºS

n+1+

0(487)

where ‖x‖2 = ‖x‖ ,√

xTx = t⋆.

‖x‖∞ = minimizet∈R

t

subject to −t1 ¹ x ¹ t1(488)

where max|xi| , i=1 . . . n = t⋆.

‖x‖1 = minimizeα∈R

n, β∈R

n1T(α + β)

subject to α, β º 0x = α − β

(489)

where |x| = α⋆ + β⋆ because of complementarity α⋆Tβ⋆ = 0 at optimality.(486) (488) (489) represent linear programs, (487) is a semidefinite program.

3.6 with equality iff x = κy where κ≥ 0.3.7Vector 1 may be replaced with any positive [sic ] vector to get absolute value,

theoretically, although 1 provides the 1-norm.

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218 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Over some convex set C given vector constant y or matrix constant Y

arg infx∈C

‖x − y‖2 = arg infx∈C

‖x − y‖22 (490)

arg infX∈C

‖X − Y ‖F = arg infX∈C

‖X − Y ‖2F (491)

are unconstrained convex quadratic problems. Equality does not hold for asum of norms. (§5.4.2.3.2) Optimal solution is norm dependent: [59, p.297]

minimizex∈R

n‖x‖1

subject to x ∈ C≡

minimizex∈R

n, t∈R

n1Tt

subject to −t ¹ x ¹ t

x ∈ C(492)

minimizex∈R

n‖x‖2

subject to x ∈ C≡

minimizex∈R

n, t∈R

t

subject to

[

tI xxT t

]

ºS

n+1+

0

x ∈ C

(493)

minimizex∈R

n‖x‖∞

subject to x ∈ C≡

minimizex∈R

n, t∈R

t

subject to −t1 ¹ x ¹ t1

x ∈ C(494)

In Rn these norms represent: ‖x‖1 is length measured along a grid(taxicab distance), ‖x‖2 is Euclidean length, ‖x‖∞ is maximum |coordinate|.

minimizex∈R

n‖x‖1

subject to x ∈ C≡

minimizeα∈R

n, β∈R

n1T(α + β)

subject to α, β º 0x = α − βx ∈ C

(495)

These foregoing problems (486)-(495) are convex whenever set C is. Theirequivalence transformations make objectives smooth.

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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 219

A= x∈R3 |Ax=b

R3

B1 = x∈ R3 | ‖x‖1≤ 1

Figure 69: 1-norm ball B1 is convex hull of all cardinality-1 vectors of unitnorm (its vertices). Ball boundary contains all points equidistant from originin 1-norm. Cartesian axes drawn for reference. Plane A is overhead (drawntruncated). If 1-norm ball is expanded until it kisses A (intersects ballonly at boundary), then distance (in 1-norm) from origin to A is achieved.Euclidean ball would be spherical in this dimension. Only were A parallel totwo axes could there be a minimum cardinality least Euclidean norm solution.Yet 1-norm ball offers infinitely many, but not all, A-orientations resultingin a minimum cardinality solution. (1-norm ball is an octahedron in thisdimension while ∞-norm ball is a cube.)

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220 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.3.0.0.1 Example. Projecting the origin on an affine subset, in 1-norm.In (1847) we interpret least norm solution to linear system Ax = b asorthogonal projection of the origin 0 on affine subset A= x∈Rn |Ax=bwhere A∈ Rm×n is fat full-rank. Suppose, instead of the Euclidean metric,we use taxicab distance to do projection. Then the least 1-norm problem isstated, for b ∈R(A)

minimizex

‖x‖1

subject to Ax = b(496)

Optimal solution can be interpreted as an oblique projection on A simplybecause the Euclidean metric is not employed. This problem statementsometimes returns optimal x⋆ having minimum cardinality; which can beexplained intuitively with reference to Figure 69: [19]

Projection of the origin, in 1-norm, on affine subset A is equivalent tomaximization (in this case) of the 1-norm ball until it kisses A ; rather, akissing point in A achieves the distance in 1-norm from the origin to A . Forthe example illustrated (m=1, n=3), it appears that a vertex of the ball willbe first to touch A . 1-norm ball vertices in R3 represent nontrivial points ofminimum cardinality 1, whereas edges represent cardinality 2, while relativeinteriors of facets represent maximum cardinality 3. By reorienting affinesubset A so it were parallel to an edge or facet, it becomes evident as weexpand or contract the ball that a kissing point is not necessarily unique.3.8

The 1-norm ball in Rn has 2n facets and 2n vertices.3.9 For n > 0

B1 = x∈ Rn | ‖x‖1≤ 1 = conv±ei∈ Rn, i=1 . . . n (497)

is a vertex-description of the unit 1-norm ball. Maximization of the 1-normball until it kisses A is equivalent to minimization of the 1-norm ball until itno longer intersects A . Then projection of the origin on affine subset A is

minimizex∈R

n‖x‖1

subject to Ax = b≡

minimizec∈R , x∈R

nc

subject to x ∈ cB1

Ax = b

(498)

wherecB1 = [ I −I ]a | aT1= c , aº0 (499)

3.8This is unlike the case for the Euclidean ball (1847) where minimum-distanceprojection on a convex set is unique (§E.9); all kissable faces of the Euclidean ball aresingle points (vertices).3.9The ∞-norm ball in Rn has 2n facets and 2n vertices.

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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 221

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

positivesigned

m/n

k/m

(496)minimize

x‖x‖1

subject to Ax = b

minimizex

‖x‖1

subject to Ax = bx º 0

(501)

Figure 70: Exact recovery transition: Respectively signed [114] [116] orpositive [121] [119] [120] solutions x to Ax=b with sparsity k below thickcurve are recoverable. For Gaussian random matrix A∈Rm×n, thick curvedemarcates phase transition in ability to find sparsest solution x by linearprogramming. These results were empirically reproduced in [37].

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

f2(x) f3(x) f4(x)

Figure 71: Under 1-norm f2(x) , histogram (hatched) of residual amplitudesAx− b exhibits predominant accumulation of zero-residuals. Nonnegativelyconstrained 1-norm f3(x) from (501) accumulates more zero-residualsthan f2(x). Under norm f4(x) (not discussed), histogram would exhibitpredominant accumulation of (nonzero) residuals at gradient discontinuities.

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222 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Then (498) is equivalent to

minimizec∈R , x∈R

n , a∈R2n

c

subject to x = [ I −I ]aaT1 = ca º 0Ax = b

≡minimize

a∈R2n

‖a‖1

subject to [ A −A ]a = ba º 0

(500)

where x⋆ = [ I −I ]a⋆. (confer (495)) Significance of this result: (confer p.375) Any vector 1-norm minimization problem may have itsvariable replaced with a nonnegative variable of twice the length.

All other things being equal, nonnegative variables are easier to solve forsparse solutions. (Figure 70, Figure 71, Figure 98) The compressed sensingproblem becomes easier to interpret; e.g., for A∈ Rm×n

minimizex

‖x‖1

subject to Ax = bx º 0

≡minimize

x1Tx

subject to Ax = bx º 0

(501)

movement of a hyperplane (Figure 26, Figure 30) over a bounded polyhedronalways has a vertex solution [92, p.22]. Or vector b might lie on the relativeboundary of a pointed polyhedral cone K= Ax | xº 0. In the lattercase, we find practical application of the smallest face F containing b from§2.13.4.3 to remove all columns of matrix A not belonging to F ; becausethose columns correspond to 0-entries in vector x . 2

3.3.0.0.2 Exercise. Combinatorial optimization.A device commonly employed to relax combinatorial problems is to arrangedesirable solutions at vertices of bounded polyhedra; e.g., the permutationmatrices of dimension n , which are factorial in number, are the extremepoints of a polyhedron in the nonnegative orthant described by anintersection of 2n hyperplanes (§2.3.2.0.4). Minimizing a linear objectivefunction over a bounded polyhedron is a convex problem (a linear program)that always has an optimal solution residing at a vertex.

What about minimizing other functions? Given some nonsingularmatrix A , geometrically describe three circumstances under which there are

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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 223

likely to exist vertex solutions to

minimizex∈R

n‖Ax‖1

subject to x ∈ P(502)

optimized over some bounded polyhedron P .3.10 H

3.3.1 k smallest entries

Sum of the k smallest entries of x∈Rn is the optimal objective value from:for 1≤k≤n

n∑

i=n−k+1

π(x)i = minimizey∈R

nxTy

subject to 0 ¹ y ¹ 1

1Ty = k

n∑

i=n−k+1

π(x)i = maximizez∈R

n, t∈R

k t + 1Tz

subject to x º t1 + zz ¹ 0

(503)

which are dual linear programs, where π(x)1 = maxxi , i=1 . . . n whereπ is a nonlinear permutation-operator sorting its vector argument intononincreasing order. Finding k smallest entries of an n-length vector x isexpressible as an infimum of n!/(k!(n − k)!) linear functions of x . The sum∑

π(x)i is therefore a concave function of x ; in fact, monotonic (§3.7.1.0.1)in this instance.

3.3.2 k largest entries

Sum of the k largest entries of x∈Rn is the optimal objective value from:[59, exer.5.19]

k∑

i=1

π(x)i = maximizey∈R

nxTy

subject to 0 ¹ y ¹ 1

1Ty = k

k∑

i=1

π(x)i = minimizez∈R

n , t∈R

k t + 1Tz

subject to x ¹ t1 + zz º 0

(504)

3.10Hint: Suppose, for example, P belongs to an orthant and A were orthogonal. Beginwith A=I and apply level sets of the objective, as in Figure 66 and Figure 69. Or rewrite

the problem as a linear program like (492) and (494) but in a composite variable

[

xt

]

← y .

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224 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

which are dual linear programs. Finding k largest entries of an n-lengthvector x is expressible as a supremum of n!/(k!(n − k)!) linear functionsof x . (Figure 73) The summation is therefore a convex function (andmonotonic in this instance, §3.7.1.0.1).

3.3.2.1 k-largest norm

Let Πx be a permutation of entries xi such that their absolute valuebecomes arranged in nonincreasing order: |Πx|1 ≥ |Πx|2 ≥ · · · ≥ |Πx|n .Sum of the k largest entries of |x|∈ Rn is a norm, by properties of vectornorm (§3.3), and is the optimal objective value of a linear program:

‖x‖nk

,k

i=1

|Πx|i =k

i=1

π(|x|)i = minimizez∈R

n , t∈R

k t + 1Tz

subject to −t1 − z ¹ x ¹ t1 + zz º 0

= supi∈I

aTi x

aij ∈ −1, 0, 1card ai = k

= maximizey1 , y2∈R

n(y1 − y2)

Tx

subject to 0 ¹ y1 ¹ 10 ¹ y2 ¹ 1(y1 + y2)

T1 = k

(505)

where the norm subscript derives from a binomial coefficient

(

nk

)

, and

‖x‖nn

= ‖x‖1

‖x‖n1

= ‖x‖∞‖x‖n

k= ‖π(|x|)1:k‖1

(506)

Sum of k largest absolute entries of an n-length vector x is expressible asa supremum of 2kn!/(k!(n − k)!) linear functions of x ; (Figure 73) hence,this norm is convex (nonmonotonic) in x . [59, exer.6.3e]

minimizex∈R

n‖x‖n

k

subject to x ∈ C≡

minimizez∈R

n , t∈R , x∈Rn

k t + 1Tz

subject to −t1 − z ¹ x ¹ t1 + zz º 0x ∈ C (507)

3.3.2.1.1 Exercise. Polyhedral epigraph of k-largest norm.Make those card I = 2kn!/(k!(n − k)!) linear functions explicit for ‖x‖2

2and

‖x‖21

on R2 and ‖x‖32

on R3. Plot ‖x‖22

and ‖x‖21

in three dimensions. H

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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 225

3.3.2.1.2 Example. Compressed sensing problem.Conventionally posed as convex problem (496), we showed: the compressedsensing problem can always be posed equivalently with a nonnegative variableas in convex statement (501). The 1-norm predominantly appears in theliterature because it is convex, it inherently minimizes cardinality under sometechnical conditions, [66] and because the desirable 0-norm is intractable.

Assuming a cardinality-k solution exists, the compressed sensing problemmay be written as a difference of two convex functions: for A∈ Rm×n

minimizex∈R

n‖x‖1 − ‖x‖n

k

subject to Ax = bx º 0

≡find x ∈ Rn

subject to Ax = bx º 0‖x‖0 ≤ k

(508)

which is a nonconvex statement, a minimization of smallest entries of variablevector x ; but a statement of compressed sensing more precise than (501)because of its equivalence to 0-norm. ‖x‖n

kis the convex k-largest norm of x

(monotonic on Rn+) while ‖x‖0 (quasiconcave on Rn

+) expresses its cardinality.Global optimality occurs at a zero objective of minimization; id est, when thesmallest n−k entries of variable vector x are zeroed. Under nonnegativityconstraint, this compressed sensing problem (508) becomes the same as

minimizez(x) , x∈R

n(1 − z)Tx

subject to Ax = bx º 0

(509)

where1 = ∇‖x‖1

z = ∇‖x‖nk

, x º 0 (510)

where gradient of k-largest norm is an optimal solution to a convex problem:

‖x‖nk

= maximizey∈R

nyTx

subject to 0 ¹ y ¹ 1yT1 = k

∇‖x‖nk

= arg maximizey∈R

nyTx

subject to 0 ¹ y ¹ 1yT1 = k

, x º 0 (511)

2

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226 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.3.2.1.3 Exercise. k-largest norm gradient.Prove (510). Find ∇‖x‖1 and ∇‖x‖n

kon Rn .3.11 H

3.3.3 clipping

Zeroing negative vector entries under 1-norm is accomplished:

‖x+‖1 = minimizet∈R

n1Tt

subject to x ¹ t

0 ¹ t

(512)

where, for x=[xi , i=1 . . . n]∈ Rn

x+ , t⋆ =

[

xi , xi ≥ 0

0 , xi < 0

, i=1 . . . n

]

=1

2(x + |x|) (513)

(clipping)

minimizex∈R

n‖x+‖1

subject to x ∈ C≡

minimizex∈R

n , t∈Rn

1Tt

subject to x ¹ t

0 ¹ t

x ∈ C

(514)

3.4 Inverted functions and roots

A given function f is convex iff −f is concave. Both functions are looselyreferred to as convex since −f is simply f inverted about the abscissa axis,and minimization of f is equivalent to maximization of −f .

A given positive function f is convex iff 1/f is concave; f inverted aboutordinate 1 is concave. Minimization of f is maximization of 1/f .

We wish to implement objectives of the form x−1. Suppose we have a2×2 matrix

T ,

[

x zz y

]

∈ R2 (515)

3.11Hint: §D.2.1.

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3.4. INVERTED FUNCTIONS AND ROOTS 227

which is positive semidefinite by (1487) when

T º 0 ⇔ x > 0 and xy ≥ z2 (516)

A polynomial constraint such as this is therefore called a conic constraint.3.12

This means we may formulate convex problems, having inverted variables,as semidefinite programs in Schur-form; e.g.,

minimizex∈R

x−1

subject to x > 0

x ∈ C≡

minimizex , y ∈ R

y

subject to

[

x 11 y

]

º 0

x ∈ C

(517)

rather

x > 0 , y ≥ 1

x⇔

[

x 11 y

]

º 0 (518)

(inverted) For vector x=[xi , i=1 . . . n]∈ Rn

minimizex∈R

n

n∑

i=1

x−1i

subject to x ≻ 0

x ∈ C

minimizex∈R

n , y∈R

y

subject to

[

xi

√n√

n y

]

º 0 , i=1 . . . n

x ∈ C (519)

rather

x ≻ 0 , y ≥ tr(

δ(x)−1)

⇔[

xi

√n√

n y

]

º 0 , i=1 . . . n (520)

3.4.1 fractional power

[145] To implement an objective of the form xα for positive α , we quantizeα and work instead with that approximation. Choose nonnegative integer qfor adequate quantization of α like so:

α ,k

2q (521)

where k∈0, 1, 2 . . . 2q−1. Any k from that set may be written

k=q

i=1

bi 2i−1 where bi∈0, 1. Define vector y=[yi , i=0 . . . q] with y0 =1 :

3.12In this dimension, the convex cone formed from the set of all values x , y , z satisfyingconstraint (516) is called a rotated quadratic or circular cone or positive semidefinite cone.

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228 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.4.1.1 positive

Then we have the equivalent semidefinite program for maximizing a concavefunction xα, for quantized 0≤α<1

maximizex∈R

subject to x > 0

x ∈ C≡

maximizex∈R , y∈R

q+1yq

subject to

[

yi−1 yi

yi xbi

]

º 0 , i=1 . . . q

x ∈ C (522)

where nonnegativity of yq is enforced by maximization; id est,

x > 0 , yq ≤ xα ⇔[

yi−1 yi

yi xbi

]

º 0 , i=1 . . . q (523)

3.4.1.2 negative

It is also desirable to implement an objective of the form x−α for positive α .The technique is nearly the same as before: for quantized 0≤α<1

minimizex∈R

x−α

subject to x > 0

x ∈ C≡

minimizex , z∈R , y∈R

q+1z

subject to

[

yi−1 yi

yi xbi

]

º 0 , i=1 . . . q

[

z 1

1 yq

]

º 0

x ∈ C (524)

rather

x > 0 , z ≥ x−α ⇔

[

yi−1 yi

yi xbi

]

º 0 , i=1 . . . q

[

z 1

1 yq

]

º 0

(525)

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3.5. AFFINE FUNCTION 229

3.4.1.3 positive inverted

Now define vector t=[ti , i=0 . . . q] with t0 =1. To implement an objectivex1/α for quantized 0≤α<1 as in (521)

minimizex∈R

x1/α

subject to x ≥ 0

x ∈ C≡

minimizey∈R , t∈R

q+1y

subject to

[

ti−1 ti

ti ybi

]

º 0 , i=1 . . . q

x = tq ≥ 0

x ∈ C (526)

rather

x ≥ 0 , y ≥ x1/α ⇔

[

ti−1 ti

ti ybi

]

º 0 , i=1 . . . q

x = tq ≥ 0

(527)

3.5 Affine function

A function f(X) is affine when it is continuous and has the dimensionallyextensible form (confer §2.9.1.0.2)

f(X) = AX + B (528)

All affine functions are simultaneously convex and concave. This means:both −AX+B and AX+B , for example, are convex functions of X .The linear functions constitute a proper subset of affine functions; e.g., whenB=0, function f(X) is linear.

Affine multidimensional functions are recognized by existence of nomultivariate terms (multiplicative in argument entries) and no polynomialterms of degree higher than 1 ; id est, entries of the function are characterizedonly by linear combinations of argument entries plus constants.

Trace is an example of affine function; actually, a real linear functionexpressible as inner product f(X) = 〈A , X〉 with matrix A being the identity.The real affine function in Figure 72 illustrates hyperplanes, in its domain,constituting contours of equal function-value (level sets (533)).

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230 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

C

a

H−

A

H+ z∈ R 2| a Tz = κ1 z∈ R 2| a Tz = κ2 z∈ R 2| a Tz = κ3

f(z)

z2

z1

Figure 72: Three hyperplanes intersecting convex set C⊂ R2 from Figure 28.Cartesian axes in R3 : Plotted is affine subset A= f(R2)⊂ R2× R ; a planewith third dimension. Sequence of hyperplanes, w.r.t domain R2 of affinefunction f(z)= aTz + b : R2→ R , is increasing in direction of gradient a(§3.7.0.0.3) because affine function increases in normal direction (Figure 26).Minimization of aTz + b over C is equivalent to minimization of aTz .

3.5.0.0.1 Example. Engineering control. [385, §2.2]3.13

For X∈ SM and matrices A ,B , Q , R of any compatible dimensions, forexample, the expression XAX is not affine in X whereas

g(X) =

[

R BTXXB Q + ATX + XA

]

(529)

is an affine multidimensional function. Such a function is typical inengineering control. [57] [147] 2

(confer Figure 16) Any single- or many-valued inverse of an affine functionis affine.

3.13The interpretation from this citation of X∈ SM | g(X)º 0 as “an intersectionbetween a linear subspace and the cone of positive semidefinite matrices” is incorrect.(See §2.9.1.0.2 for a similar example.) The conditions they state under which strongduality holds for semidefinite programming are conservative. (confer §4.2.3.0.1)

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3.5. AFFINE FUNCTION 231

3.5.0.0.2 Example. Linear objective.Consider minimization of a real affine function f(z)= aTz + b over theconvex feasible set C in its domain R2 illustrated in Figure 72. Since scalar bis fixed, the problem posed is the same as the convex optimization

minimizez

aTz

subject to z∈ C(530)

whose objective of minimization is a real linear function. Were convex set Cpolyhedral (§2.12), then this problem would be called a linear program. Wereconvex set C an intersection with a positive semidefinite cone, then thisproblem would be called a semidefinite program.

There are two distinct ways to visualize this problem: one in the objectivefunction’s domain R2, the other including the ambient space of the objective

function’s range as in

[

R2

R

]

. Both visualizations are illustrated in Figure 72.

Visualization in the function domain is easier because of lower dimension andbecause level sets (533) of any affine function are affine. (§2.1.9)

In this circumstance, the level sets are parallel hyperplanes with respectto R2. One solves optimization problem (530) graphically by finding thathyperplane intersecting feasible set C furthest right (in the direction ofnegative gradient −a (§3.7)). 2

When a differentiable convex objective function f is nonlinear, thenegative gradient −∇f is a viable search direction (replacing −a in (530)).(§2.13.10.1, Figure 66) [149] Then the nonlinear objective function can bereplaced with a dynamic linear objective; linear as in (530).

3.5.0.0.3 Example. Support function. [196, §C.2.1-§C.2.3.1]For arbitrary set Y ⊆ Rn, its support function σY(a) : Rn→ R is defined

σY(a) , supz∈Y

aTz (531)

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232 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

a

aTz1 + b1 | a∈R

aTz2 + b2 | a∈R

aTz3 + b3 | a∈R

aTz4 + b4 | a∈R

aTz5 + b5 | a∈R

supi

aTpzi + bi

Figure 73: Pointwise supremum of convex functions remains convex; byepigraph intersection. Supremum of affine functions in variable a evaluatedat argument ap is illustrated. Topmost affine function per a is supremum.

a positively homogeneous function of direction a whose range contains ±∞.[246, p.135] For each z ∈ Y , aTz is a linear function of vector a . BecauseσY(a) is a pointwise supremum of linear functions, it is convex in a(Figure 73). Application of the support function is illustrated in Figure 29afor one particular normal a . Given nonempty closed bounded convex sets Yand Z in Rn and nonnegative scalars β and γ [368, p.234]

σβY+γZ(a) = βσY(a) + γσZ(a) (532)

2

3.5.0.0.4 Exercise. Level sets.Given a function f and constant κ , its level sets are defined

Lκκf , z | f(z)=κ (533)

Give two distinct examples of convex function, that are not affine, havingconvex level sets. H

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3.6. EPIGRAPH, SUBLEVEL SET 233

quasiconvex convex

f(x)q(x)

xx

Figure 74: Quasiconvex function q epigraph is not necessarily convex, butconvex function f epigraph is convex in any dimension. Sublevel sets arenecessarily convex for either function, but sufficient only for quasiconvexity.

3.6 Epigraph, Sublevel set

It is well established that a continuous real function is convex if andonly if its epigraph makes a convex set. [196] [303] [355] [368] [246]Thereby, piecewise-continuous convex functions are admitted. Epigraph isthe connection between convex sets and convex functions. Its generalizationto a vector-valued function f(X) : Rp×k→RM is straightforward: [291]

epi f , (X , t)∈ Rp×k× RM | X∈ dom f , f(X) ¹RM

+

t (534)

id est,

f convex ⇔ epi f convex (535)

Necessity is proven: [59, exer.3.60] Given any (X, u) , (Y , v)∈ epi f , wemust show for all µ∈ [0, 1] that µ(X, u) + (1−µ)(Y , v)∈ epi f ; id est, wemust show

f(µX + (1−µ)Y ) ¹RM

+

µu + (1−µ)v (536)

Yet this holds by definition because f(µX+(1−µ)Y ) ¹ µf(X)+(1−µ)f(Y ).The converse also holds. ¨

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234 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.6.0.0.1 Exercise. Epigraph sufficiency.Prove that converse: Given any (X, u) , (Y , v)∈ epi f , if for all µ∈ [0, 1]µ(X, u) + (1−µ)(Y , v)∈ epi f holds, then f must be convex. H

Sublevel sets of a convex real function are convex. Likewise, correspondingto each and every ν ∈ RM

Lνf , X∈ dom f | f(X) ¹RM

+

ν ⊆ Rp×k (537)

sublevel sets of a convex vector-valued function are convex. As for convexreal functions, the converse does not hold. (Figure 74)

To prove necessity of convex sublevel sets: For any X,Y ∈Lνf we mustshow for each and every µ∈ [0, 1] that µX + (1−µ)Y ∈Lνf . By definition,

f(µX + (1−µ)Y ) ¹RM

+

µf(X) + (1−µ)f(Y ) ¹RM

+

ν (538)

¨

When an epigraph (534) is artificially bounded above, t ¹ ν , then thecorresponding sublevel set can be regarded as an orthogonal projection ofepigraph on the function domain.

Sense of the inequality is reversed in (534), for concave functions, and weuse instead the nomenclature hypograph. Sense of the inequality in (537) isreversed, similarly, with each convex set then called superlevel set.

3.6.0.0.2 Example. Matrix pseudofractional function.Consider a real function of two variables

f(A , x) : Sn× Rn→ R = xTA†x (539)

on dom f = Sn+×R(A). This function is convex simultaneously in both

variables when variable matrix A belongs to the entire positive semidefinitecone Sn

+ and variable vector x is confined to range R(A) of matrix A .

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3.6. EPIGRAPH, SUBLEVEL SET 235

To explain this, we need only demonstrate that the function epigraph isconvex. Consider Schur-form (1484) from §A.4: for t ∈ R

G(A , z , t) =

[

A zzT t

]

º 0

⇔zT(I − AA†) = 0

t − zTA†z ≥ 0

A º 0

(540)

Inverse image of the positive semidefinite cone Sn+1+ under affine mapping

G(A , z , t) is convex by Theorem 2.1.9.0.1. Of the equivalent conditions forpositive semidefiniteness of G , the first is an equality demanding vector zbelong to R(A). Function f(A , z)= zTA†z is convex on convex domainSn

+×R(A) because the Cartesian product constituting its epigraph

epi f(A , z) =

(A , z , t) | A º 0 , z∈R(A) , zTA†z ≤ t

= G−1(

Sn+1+

)

(541)

is convex. 2

3.6.0.0.3 Exercise. Matrix product function.Continue Example 3.6.0.0.2 by introducing vector variable x and makingthe substitution z←Ax . Because of matrix symmetry (§E), for all x∈Rn

f(A , z(x)) = zTA†z = xTATA†Ax = xTAx = f(A , x) (542)

whose epigraph is

epi f(A , x) =

(A , x , t) | A º 0 , xTAx ≤ t

(543)

Provide two simple explanations why f(A , x) = xTAx is not a functionconvex simultaneously in positive semidefinite matrix A and vector x ondom f = Sn

+× Rn. H

3.6.0.0.4 Example. Matrix fractional function. (confer §3.8.2.0.1)Continuing Example 3.6.0.0.2, now consider a real function of two variableson dom f = Sn

+×Rn for small positive constant ǫ (confer (1833))

f(A , x) = ǫ xT(A + ǫ I )−1x (544)

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236 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

where the inverse always exists by (1427). This function is convexsimultaneously in both variables over the entire positive semidefinite cone Sn

+

and all x∈Rn : Consider Schur-form (1487) from §A.4: for t ∈ R

G(A , z , t) =

[

A + ǫ I zzT ǫ−1 t

]

º 0

⇔t − ǫ zT(A + ǫ I )−1z ≥ 0

A + ǫ I ≻ 0

(545)

Inverse image of the positive semidefinite cone Sn+1+ under affine mapping

G(A , z , t) is convex by Theorem 2.1.9.0.1. Function f(A , z) is convex onSn

+×Rn because its epigraph is that inverse image:

epi f(A , z) =

(A , z , t) | A + ǫ I ≻ 0 , ǫ zT(A + ǫ I )−1z ≤ t

= G−1(

Sn+1+

)

(546)2

3.6.1 matrix fractional projector function

Consider nonlinear function f having orthogonal projector W as argument:

f(W , x) = ǫ xT(W + ǫ I )−1x (547)

Projection matrix W has property W † = WT = W º 0 (1878). Anyorthogonal projector can be decomposed into an outer product oforthonormal matrices W = UUT where UTU = I as explained in§E.3.2. From (1833) for any ǫ > 0 and idempotent symmetric W ,ǫ(W + ǫ I )−1 = I − (1 + ǫ)−1W from which

f(W , x) = ǫ xT(W + ǫ I )−1x = xT(

I − (1 + ǫ)−1W)

x (548)

Therefore

limǫ→0+

f(W , x) = limǫ→0+

ǫ xT(W + ǫ I )−1x = xT(I − W )x (549)

where I − W is also an orthogonal projector (§E.2).We learned from Example 3.6.0.0.4 that f(W , x)= ǫ xT(W + ǫ I )−1x is

convex simultaneously in both variables over all x ∈ Rn when W ∈ Sn+ is

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3.6. EPIGRAPH, SUBLEVEL SET 237

confined to the entire positive semidefinite cone (including its boundary). Itis now our goal to incorporate f into an optimization problem such thatan optimal solution returned always comprises a projection matrix W . Theset of orthogonal projection matrices is a nonconvex subset of the positivesemidefinite cone. So f cannot be convex on the projection matrices, andits equivalent (for idempotent W )

f(W , x) = xT(

I − (1 + ǫ)−1W)

x (550)

cannot be convex simultaneously in both variables on either the positivesemidefinite or symmetric projection matrices.

Suppose we allow dom f to constitute the entire positive semidefinitecone but constrain W to a Fantope (90); e.g., for convex set C and 0 < k < nas in

minimizex∈R

n, W∈S

nǫ xT(W + ǫ I )−1x

subject to 0 ¹ W ¹ I

tr W = k

x ∈ C

(551)

Although this is a convex problem, there is no guarantee that optimal W isa projection matrix because only extreme points of a Fantope are orthogonalprojection matrices UUT.

Let’s try partitioning the problem into two convex parts (one for x andone for W ), substitute equivalence (548), and then iterate solution of convexproblem

minimizex∈R

nxT(I − (1 + ǫ)−1W )x

subject to x ∈ C(552)

with convex problem

(a)

minimizeW∈S

nx⋆T(I − (1 + ǫ)−1W )x⋆

subject to 0 ¹ W ¹ I

tr W = k

≡maximize

W∈Sn

x⋆TWx⋆

subject to 0 ¹ W ¹ I

tr W = k

(553)

until convergence, where x⋆ represents an optimal solution of (552) fromany particular iteration. The idea is to optimally solve for the partitionedvariables which are later combined to solve the original problem (551).What makes this approach sound is that the constraints are separable, the

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238 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

partitioned feasible sets are not interdependent, and the fact that the originalproblem (though nonlinear) is convex simultaneously in both variables.3.14

But partitioning alone does not guarantee a projector. To makeorthogonal projector W a certainty, we must invoke a known analyticaloptimal solution to problem (553): Diagonalize optimal solution fromproblem (552) x⋆x⋆T , QΛQT (§A.5.1) and set U⋆ = Q(: , 1:k)∈ Rn×k

per (1663c);

W = U⋆U⋆T =x⋆x⋆T

‖x⋆‖2+ Q(: , 2:k)Q(: , 2:k)T (554)

Then optimal solution (x⋆, U⋆) to problem (551) is found, for small ǫ , byiterating solution to problem (552) with optimal (projector) solution (554)to convex problem (553).

Proof. Optimal vector x⋆ is orthogonal to the last n−1 columns oforthogonal matrix Q , so

f ⋆(552) = ‖x⋆‖2(1 − (1 + ǫ)−1) (555)

after each iteration. Convergence of f ⋆(552) is proven with the observation that

iteration (552) (553a) is a nonincreasing sequence that is bounded below by 0.Any bounded monotonic sequence in R is convergent. [254, §1.2] [41, §1.1]Expression (554) for optimal projector W holds at each iteration, therefore‖x⋆‖2(1 − (1 + ǫ)−1) must also represent the optimal objective value f ⋆

(552)

at convergence.Because the objective f(551) from problem (551) is also bounded below

by 0 on the same domain, this convergent optimal objective value f ⋆(552) (for

positive ǫ arbitrarily close to 0) is necessarily optimal for (551); id est,

f ⋆(552) ≥ f ⋆

(551) ≥ 0 (556)

by (1646), and

limǫ→0+

f ⋆(552) = 0 (557)

Since optimal (x⋆, U⋆) from problem (552) is feasible to problem (551), andbecause their objectives are equivalent for projectors by (548), then converged(x⋆, U⋆) must also be optimal to (551) in the limit. Because problem (551)is convex, this represents a globally optimal solution. ¨

3.14A convex problem has convex feasible set, and the objective surface has one and onlyone global minimum. [298, p.123]

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3.6. EPIGRAPH, SUBLEVEL SET 239

3.6.2 semidefinite program via Schur

Schur complement (1484) can be used to convert a projection problemto an optimization problem in epigraph form. Suppose, for example,we are presented with the constrained projection problem studied byHayden & Wells in [180] (who provide analytical solution): Given A∈RM×M

and some full-rank matrix S∈ RM×L with L < M

minimizeX∈ SM

‖A − X‖2F

subject to STXS º 0(558)

Variable X is constrained to be positive semidefinite, but only on a subspacedetermined by S . First we write the epigraph form:

minimizeX∈ SM , t∈R

t

subject to ‖A − X‖2F ≤ t

STXS º 0

(559)

Next we use Schur complement [273, §6.4.3] [244] and matrix vectorization(§2.2):

minimizeX∈ SM , t∈R

t

subject to

[

tI vec(A − X)vec(A − X)T 1

]

º 0

STXS º 0

(560)

This semidefinite program (§4) is an epigraph form in disguise, equivalentto (558); it demonstrates how a quadratic objective or constraint can beconverted to a semidefinite constraint.

Were problem (558) instead equivalently expressed without the square

minimizeX∈ SM

‖A − X‖F

subject to STXS º 0(561)

then we get a subtle variation:

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240 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

minimizeX∈ SM , t∈R

t

subject to ‖A − X‖F ≤ t

STXS º 0

(562)

that leads to an equivalent for (561) (and for (558) by (491))

minimizeX∈ SM , t∈R

t

subject to

[

tI vec(A − X)vec(A − X)T t

]

º 0

STXS º 0

(563)

3.6.2.0.1 Example. Schur anomaly.Consider a problem abstract in the convex constraint, given symmetricmatrix A

minimizeX∈ SM

‖X‖2F − ‖A − X‖2

F

subject to X∈ C(564)

the minimization of a difference of two quadratic functions each convex inmatrix X . Observe equality

‖X‖2F − ‖A − X‖2

F = 2 tr(XA) − ‖A‖2F (565)

So problem (564) is equivalent to the convex optimization

minimizeX∈ SM

tr(XA)

subject to X∈ C(566)

But this problem (564) does not have Schur-form

minimizeX∈ SM , α , t

t − α

subject to X∈ C‖X‖2

F ≤ t

‖A − X‖2F ≥ α

(567)

because the constraint in α is nonconvex. (§2.9.1.0.1) 2

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3.7. GRADIENT 241

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Y1

Y2

Figure 75: Gradient in R2 evaluated on grid over some open disc in domainof convex quadratic bowl f(Y )= Y TY : R2→ R illustrated in Figure 76.Circular contours are level sets; each defined by a constant function-value.

3.7 Gradient

Gradient ∇f of any differentiable multidimensional function f (formallydefined in §D.1) maps each entry fi to a space having the same dimensionas the ambient space of its domain. Notation ∇f is shorthand for gradient∇xf(x) of f with respect to x . ∇f(y) can mean ∇yf(y) or gradient∇xf(y) of f(x) with respect to x evaluated at y ; a distinction that shouldbecome clear from context.

Gradient of a differentiable real function f(x) : RK→R with respect toits vector argument is defined

∇f(x) =

∂f(x)∂x1

∂f(x)∂x2...

∂f(x)∂xK

∈ RK (1722)

while the second-order gradient of the twice differentiable real function with

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242 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

respect to its vector argument is traditionally called the Hessian ;3.15

∇2f(x) =

∂2f(x)

∂x21

∂2f(x)∂x1∂x2

· · · ∂2f(x)∂x1∂xK

∂2f(x)∂x2∂x1

∂2f(x)

∂x22

· · · ∂2f(x)∂x2∂xK

......

. . ....

∂2f(x)∂xK∂x1

∂2f(x)∂xK∂x2

· · · ∂2f(x)

∂x2K

∈ SK (1723)

The gradient can be interpreted as a vector pointing in the direction ofgreatest change, [215, §15.6] or polar to direction of steepest descent.3.16 [372]The gradient can also be interpreted as that vector normal to a sublevel set;e.g., Figure 77, Figure 66.

For the quadratic bowl in Figure 76, the gradient maps to R2 ; illustratedin Figure 75. For a one-dimensional function of real variable, for example,the gradient evaluated at any point in the function domain is just the slope(or derivative) of that function there. (confer §D.1.4.1) For any differentiable multidimensional function, zero gradient ∇f = 0

is a condition necessary for its unconstrained minimization [149, §3.2]:

3.7.0.0.1 Example. Projection on a rank-1 subset.For A∈SN having eigenvalues λ(A)= [λi]∈ RN , consider the unconstrainednonconvex optimization that is a projection on the rank-1 subset (§2.9.2.1)of positive semidefinite cone SN

+ : Defining λ1 , maxiλ(A)i andcorresponding eigenvector v1

minimizex

‖xxT− A‖2F = minimize

xtr(xxT(xTx) − 2AxxT + ATA)

=

‖λ(A)‖2 , λ1 ≤ 0

‖λ(A)‖2 − λ21 , λ1 > 0

(1658)

arg minimizex

‖xxT− A‖2F =

0 , λ1 ≤ 0

v1

√λ1 , λ1 > 0

(1659)

3.15Jacobian is the Hessian transpose, so commonly confused in matrix calculus.3.16Newton’s direction −∇2f(x)−1∇f(x) is better for optimization of nonlinear functionswell approximated locally by a quadratic function. [149, p.105]

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3.7. GRADIENT 243

From (1752) and §D.2.1, the gradient of ‖xxT− A‖2F is

∇x

(

(xTx)2 − 2xTAx)

= 4(xTx)x − 4Ax (568)

Setting the gradient to 0

Ax = x(xTx) (569)

is necessary for optimal solution. Replace vector x with a normalizedeigenvector vi of A∈SN , corresponding to a positive eigenvalue λi , scaledby square root of that eigenvalue. Then (569) is satisfied

x ← vi

λi ⇒ Avi = viλi (570)

xxT = λi vivTi is a rank-1 matrix on the positive semidefinite cone boundary,

and the minimum is achieved (§7.1.2) when λi =λ1 is the largest positiveeigenvalue of A . If A has no positive eigenvalue, then x=0 yields theminimum. 2

Differentiability is a prerequisite neither to convexity or to numericalsolution of a convex optimization problem. The gradient provides a necessaryand sufficient condition (330) (430) for optimality in the constrained case,nevertheless, as it does in the unconstrained case: For any differentiable multidimensional convex function, zero gradient

∇f = 0 is a necessary and sufficient condition for its unconstrainedminimization [59, §5.5.3]:

3.7.0.0.2 Example. Pseudoinverse.The pseudoinverse matrix is the unique solution to an unconstrained convexoptimization problem [155, §5.5.4]: given A∈Rm×n

minimizeX∈R

n×m‖XA − I‖2

F (571)

where

‖XA − I‖2F = tr

(

ATXTXA − XA − ATXT + I)

(572)

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244 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

whose gradient (§D.2.3)

∇X‖XA − I‖2F = 2

(

XAAT − AT)

= 0 (573)

vanishes when

XAAT = AT (574)

When A is fat full-rank, then AAT is invertible, X⋆ = AT(AAT)−1 is thepseudoinverse A† , and AA†= I . Otherwise, we can make AAT invertibleby adding a positively scaled identity, for any A∈Rm×n

X = AT(AAT + t I )−1 (575)

Invertibility is guaranteed for any finite positive value of t by (1427). Thenmatrix X becomes the pseudoinverse X → A† , X⋆ in the limit t→ 0+.Minimizing instead ‖AX − I‖2

F yields the second flavor in (1832). 2

3.7.0.0.3 Example. Hyperplane, line, described by affine function.Consider the real affine function of vector variable, (confer Figure 72)

f(x) : Rp→ R = aTx + b (576)

whose domain is Rp and whose gradient ∇f(x)=a is a constant vector(independent of x). This function describes the real line R (its range), andit describes a nonvertical [196, §B.1.2] hyperplane ∂H in the space Rp×Rfor any particular vector a (confer §2.4.2);

∂H =

[

xaTx + b

]

| x∈Rp

⊂ Rp×R (577)

having nonzero normal

η =

[

a−1

]

∈ Rp×R (578)

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3.7. GRADIENT 245

This equivalence to a hyperplane holds only for real functions.3.17 Epigraph

of real affine function f(x) is therefore a halfspace in

[

Rp

R

]

, so we have:

The real affine function is to convex functionsas

the hyperplane is to convex sets.

Similarly, the matrix-valued affine function of real variable x , for anyparticular matrix A∈RM×N ,

h(x) : R→RM×N = Ax + B (579)

describes a line in RM×N in direction A

Ax + B | x∈R ⊆ RM×N (580)

and describes a line in R×RM×N

[

xAx + B

]

| x∈R

⊂ R×RM×N (581)

whose slope with respect to x is A . 2

3.17To prove that, consider a vector-valued affine function

f(x) : Rp→RM = Ax + b

having gradient ∇f(x)=AT∈ Rp×M : The affine set

[

xAx + b

]

| x∈Rp

⊂ Rp×RM

is perpendicular to

η ,

[

∇f(x)−I

]

∈ Rp×M×RM×M

because

ηT

([

xAx + b

]

−[

0b

])

= 0 ∀x ∈ Rp

Yet η is a vector (in Rp×RM ) only when M = 1. ¨

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246 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.7.1 monotonic function

A real function of real argument is called monotonic when it is exclusivelynonincreasing or nondecreasing over the whole of its domain. A realdifferentiable function of real argument is monotonic when its first derivative(not necessarily continuous) maintains sign over the function domain.

3.7.1.0.1 Definition. Monotonicity.Multidimensional function f(X) is

nondecreasing monotonic when Y º X ⇒ f(Y ) º f(X)nonincreasing monotonic when Y º X ⇒ f(Y ) ¹ f(X)

(582)

∀X,Y ∈ dom f .

For monotonicity of vector-valued functions, compared with respect tothe nonnegative orthant, it is necessary and sufficient for each entry fi to bemonotonic in the same sense.

Any affine function is monotonic. tr(Y TX) is a nondecreasing monotonicfunction of matrix X∈ SM when constant matrix Y is positive semidefinite,for example; which follows from a result (356) of Fejer.

A convex function can be characterized by another kind of nondecreasingmonotonicity of its gradient:

3.7.1.0.2 Theorem. Gradient monotonicity. [196, §B.4.1.4][53, §3.1 exer.20] Given f(X) : Rp×k→R a real differentiable function withmatrix argument on open convex domain, the condition

〈∇f(Y ) −∇f(X) , Y − X〉 ≥ 0 for each and every X,Y ∈ dom f (583)

is necessary and sufficient for convexity of f . Strict inequality and caveatdistinct Y ,X provide necessary and sufficient conditions for strict convexity.

3.7.1.0.3 Example. Composition of functions. [59, §3.2.4] [196, §B.2.1]Monotonic functions play a vital role determining convexity of functionsconstructed by transformation. Given functions g : Rk→ R andh : Rn→ Rk, their composition f = g(h) : Rn→ R defined by

f(x) = g(h(x)) , dom f = x∈ dom h | h(x)∈ dom g (584)

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3.7. GRADIENT 247

is convex if g is convex nondecreasing monotonic and h is convex g is convex nonincreasing monotonic and h is concave

and composite function f is concave if g is concave nondecreasing monotonic and h is concave g is concave nonincreasing monotonic and h is convex

where ∞ (−∞) is assigned to convex (concave) g when evaluated outside itsdomain. For differentiable functions, these rules are consequent to (1753). Convexity (concavity) of any g is preserved when h is affine. 2

If f and g are nonnegative convex real functions, then (f(x)k + g(x)k)1/k

is also convex for integer k≥1. [245, p.44] A squared norm is convex havingthe same minimum because a squaring operation is convex nondecreasingmonotonic on the nonnegative real line.

3.7.1.0.4 Exercise. Order of composition.Real function f =x−2 is convex on R+ but not predicted so by results inExample 3.7.1.0.3 when g=h(x)−1 and h=x2. Explain this anomaly. H

The following result for product of real functions is extensible to innerproduct of multidimensional functions on real domain:

3.7.1.0.5 Exercise. Product and ratio of convex functions. [59, exer.3.32]In general the product or ratio of two convex functions is not convex. [226]However, there are some results that apply to functions on R [real domain].Prove the following.3.18

(a) If f and g are convex, both nondecreasing (or nonincreasing), andpositive functions on an interval, then fg is convex.

(b) If f , g are concave, positive, with one nondecreasing and the othernonincreasing, then fg is concave.

(c) If f is convex, nondecreasing, and positive, and g is concave,nonincreasing, and positive, then f/g is convex. H

3.18Hint: Prove §3.7.1.0.5a by verifying Jensen’s inequality ((475) at µ= 12).

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248 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.7.2 first-order convexity condition, real function

Discretization of wº 0 in (476) invites refocus to the real-valued function:

3.7.2.0.1 Theorem. Necessary and sufficient convexity condition.[59, §3.1.3] [129, §I.5.2] [388, §1.2.3] [41, §1.2] [327, §4.2] [302, §3] For realdifferentiable function f(X) : Rp×k→R with matrix argument on openconvex domain, the condition (confer §D.1.7)

f(Y ) ≥ f(X) + 〈∇f(X) , Y − X〉 for each and every X,Y ∈ dom f (585)

is necessary and sufficient for convexity of f . ⋄

When f(X) : Rp→R is a real differentiable convex function with vectorargument on open convex domain, there is simplification of the first-ordercondition (585); for each and every X,Y ∈ dom f

f(Y ) ≥ f(X) + ∇f(X)T(Y − X) (586)

From this we can find ∂H− a unique [368, p.220-229] nonvertical [196, §B.1.2]hyperplane (§2.4), expressed in terms of function gradient, supporting epi f

at

[

Xf(X)

]

: videlicet, defining f(Y /∈ dom f ) , ∞ [59, §3.1.7]

[

Yt

]

∈ epi f ⇔ t ≥ f(Y ) ⇒[

∇f(X)T −1]

([

Yt

]

−[

Xf(X)

])

≤ 0

(587)

This means, for each and every point X in the domain of a convex realfunction f(X) , there exists a hyperplane ∂H− in Rp× R having normal[

∇f(X)−1

]

supporting the function epigraph at

[

Xf(X)

]

∈ ∂H−

∂H− =

[

Yt

]

∈[

Rp

R

]

[

∇f(X)T −1]

([

Yt

]

−[

Xf(X)

])

= 0

(588)

One such supporting hyperplane (confer Figure 29a) is illustrated inFigure 76 for a convex quadratic.

From (586) we deduce, for each and every X,Y ∈ dom f

∇f(X)T(Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (589)

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3.7. GRADIENT 249

∂H−

f(Y )

[

∇f(X)−1

]

Figure 76: When a real function f is differentiable at each point in its opendomain, there is an intuitive geometric interpretation of function convexity interms of its gradient ∇f and its epigraph: Drawn is a convex quadratic bowlin R2×R (confer Figure 158, p.668); f(Y )= Y TY : R2→ R versus Y onsome open disc in R2. Unique strictly supporting hyperplane ∂H−∈ R2× R(only partially drawn) and its normal vector [∇f(X)T −1 ]T at theparticular point of support [XT f(X) ]T are illustrated. The interpretation:At each and every coordinate Y , there is a unique hyperplane containing[ Y T f(Y ) ]T and supporting the epigraph of convex differentiable f .

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250 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

α

β

γ∇f(X)

Y−X

Z | f(Z) = α

Y | ∇f(X)T(Y − X) = 0 , f(X)=α (590)

α ≥ β ≥ γ

Figure 77:(confer Figure 66) Shown is a plausible contour plot in R2 of somearbitrary real differentiable convex function f(Z) at selected levels α , β ,and γ ; contours of equal level f (level sets) drawn in the function’s domain.A convex function has convex sublevel sets Lf(X)f (591). [303, §4.6] Thesublevel set whose boundary is the level set at α , for instance, comprisesall the shaded regions. For any particular convex function, the familycomprising all its sublevel sets is nested. [196, p.75] Were sublevel sets notconvex, we may certainly conclude the corresponding function is neitherconvex. Contour plots of real affine functions are illustrated in Figure 26and Figure 72.

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3.7. GRADIENT 251

meaning, the gradient at X identifies a supporting hyperplane there in Rp

Y ∈ Rp | ∇f(X)T(Y − X) = 0 (590)

to the convex sublevel sets of convex function f (confer (537))

Lf(X)f , Z ∈ dom f | f(Z) ≤ f(X) ⊆ Rp (591)

illustrated for an arbitrary convex real function in Figure 77 and Figure 66.That supporting hyperplane is unique for twice differentiable f . [215, p.501]

3.7.3 first-order convexity condition, vector-valued f

Now consider the first-order necessary and sufficient condition for convexity ofa vector-valued function: Differentiable function f(X) : Rp×k→RM is convexif and only if dom f is open, convex, and for each and every X,Y ∈ dom f

f(Y ) ºRM

+

f(X) +→Y −X

df(X) = f(X) +d

dt

t=0

f(X+ t (Y − X)) (592)

where→Y −X

df(X) is the directional derivative3.19 [215] [329] of f at X in directionY −X . This, of course, follows from the real-valued function case: by dualgeneralized inequalities (§2.13.2.0.1),

f(Y ) − f(X) −→Y −X

df(X) ºRM

+

0 ⇔⟨

f(Y ) − f(X) −→Y −X

df(X) , w

≥ 0 ∀w ºRM∗

+

0

(593)where

→Y −X

df(X) =

tr(

∇f1(X)T(Y − X))

tr(

∇f2(X)T(Y − X))

...

tr(

∇fM(X)T(Y − X))

∈ RM (594)

3.19We extend the traditional definition of directional derivative in §D.1.4 so that directionmay be indicated by a vector or a matrix, thereby broadening the scope of the Taylorseries (§D.1.7). The right-hand side of the inequality (592) is the first-order Taylor seriesexpansion of f about X .

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252 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Necessary and sufficient discretization (476) allows relaxation of thesemiinfinite number of conditions wº 0 instead to w ∈ ei , i=1 . . . Mthe extreme directions of the selfdual nonnegative orthant. Each extreme

direction picks out a real entry fi and→Y −X

df(X)i from the vector-valuedfunction and its directional derivative, then Theorem 3.7.2.0.1 applies.

The vector-valued function case (592) is therefore a straightforwardapplication of the first-order convexity condition for real functions to eachentry of the vector-valued function.

3.7.4 second-order convexity condition

Again, by discretization (476), we are obliged only to consider each individualentry fi of a vector-valued function f ; id est, the real functions fi.

For f(X) : Rp→RM , a twice differentiable vector-valued function withvector argument on open convex domain,

∇2fi(X) ºS

p+

0 ∀X∈ dom f , i=1 . . . M (595)

is a necessary and sufficient condition for convexity of f . Obviously,when M = 1, this convexity condition also serves for a real function.Condition (595) demands nonnegative curvature, intuitively, henceprecluding points of inflection as in Figure 78 (p.260).

Strict inequality in (595) provides only a sufficient condition for strictconvexity, but that is nothing new; videlicet, the strictly convex real functionfi(x)=x4 does not have positive second derivative at each and every x∈R .Quadratic forms constitute a notable exception where the strict-case converseholds reliably.

3.7.4.0.1 Exercise. Real fractional function. (confer §3.4, §3.6.0.0.4)Prove that real function f(x, y) = x/y is not convex on the first quadrant.Also exhibit this in a plot of the function. (f is quasilinear (p.262) ony > 0 , in fact, and nonmonotonic even on the first quadrant.) H

3.7.4.0.2 Exercise. Stress function.Define |x− y|,

(x− y)2 and

X = [ x1 · · · xN ] ∈ R1×N (76)

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3.8. CONVEX MATRIX-VALUED FUNCTION 253

Given symmetric nonnegative data [hij] ∈ SN∩ RN×N+ , consider function

f(vec X) =N−1∑

i=1

N∑

j=i+1

(|xi − xj| − hij)2 ∈ R (1284)

Take the gradient and Hessian of f . Then explain why f is not a convexfunction; id est, why doesn’t second-order condition (595) apply to theconstant positive semidefinite Hessian matrix you found. For N = 6 and hij

data from (1357), apply line theorem 3.8.3.0.1 to plot f along some arbitrarylines through its domain. H

3.7.4.1 second-order ⇒ first-order condition

For a twice-differentiable real function fi(X) : Rp→R having open domain,a consequence of the mean value theorem from calculus allows compressionof its complete Taylor series expansion about X∈ dom fi (§D.1.7) to threeterms: On some open interval of ‖Y ‖2 , so that each and every line segment[X,Y ] belongs to dom fi , there exists an α∈ [0 , 1] such that [388, §1.2.3][41, §1.1.4]

fi(Y ) = fi(X) + ∇fi(X)T(Y −X) +1

2(Y −X)T∇2fi(αX + (1−α)Y )(Y −X)

(596)

The first-order condition for convexity (586) follows directly from this andthe second-order condition (595).

3.8 Convex matrix-valued function

We need different tools for matrix argument: We are primarily interested incontinuous matrix-valued functions g(X). We choose symmetric g(X)∈ SM

because matrix-valued functions are most often compared (597) with respectto the positive semidefinite cone SM

+ in the ambient space of symmetricmatrices.3.20

3.20Function symmetry is not a necessary requirement for convexity; indeed, for A∈Rm×p

and B∈Rm×k, g(X) = AX + B is a convex (affine) function in X on domain Rp×k withrespect to the nonnegative orthant Rm×k

+ . Symmetric convex functions share the samebenefits as symmetric matrices. Horn & Johnson [199, §7.7] liken symmetric matrices toreal numbers, and (symmetric) positive definite matrices to positive real numbers.

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254 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.8.0.0.1 Definition. Convex matrix-valued function:1) Matrix-definition.A function g(X) : Rp×k→SM is convex in X iff dom g is a convex set and,for each and every Y, Z∈dom g and all 0≤µ≤1 [211, §2.3.7]

g(µY + (1 − µ)Z) ¹SM+

µ g(Y ) + (1 − µ)g(Z) (597)

Reversing sense of the inequality flips this definition to concavity. Strictconvexity is defined less a stroke of the pen in (597) similarly to (478).2) Scalar-definition.It follows that g(X) : Rp×k→SM is convex in X iff wTg(X)w : Rp×k→R isconvex in X for each and every ‖w‖= 1 ; shown by substituting the defininginequality (597). By dual generalized inequalities we have the equivalent butmore broad criterion, (§2.13.5)

g convex w.r.t SM+ ⇔ 〈W , g〉 convex for each and every W º

SM∗

+

0 (598)

Strict convexity on both sides requires caveat W 6= 0. Because theset of all extreme directions for the selfdual positive semidefinite cone(§2.9.2.4) comprises a minimal set of generators for that cone, discretization(§2.13.4.2.1) allows replacement of matrix W with symmetric dyad wwT asproposed.

3.8.0.0.2 Example. Taxicab distance matrix.Consider an n-dimensional vector space Rn with metric induced by the1-norm. Then distance between points x1 and x2 is the norm of theirdifference: ‖x1−x2‖1 . Given a list of points arranged columnar in a matrix

X = [ x1 · · · xN ] ∈ Rn×N (76)

then we could define a taxicab distance matrix

D1(X) , (I ⊗ 1Tn ) | vec(X)1T − 1⊗X | ∈ SN

h ∩ RN×N+

=

0 ‖x1 − x2‖1 ‖x1 − x3‖1 · · · ‖x1 − xN‖1

‖x1 − x2‖1 0 ‖x2 − x3‖1 · · · ‖x2 − xN‖1

‖x1 − x3‖1 ‖x2 − x3‖1 0 ‖x3 − xN‖1...

.... . .

...‖x1 − xN‖1 ‖x2 − xN‖1 ‖x3 − xN‖1 · · · 0

(599)

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3.8. CONVEX MATRIX-VALUED FUNCTION 255

where 1n is a vector of ones having dim1n = n and where ⊗ representsKronecker product. This matrix-valued function is convex with respect to thenonnegative orthant since, for each and every Y, Z∈Rn×N and all 0≤µ≤1

D1(µY + (1 − µ)Z) ¹R

N×N+

µD1(Y ) + (1 − µ)D1(Z) (600)

2

3.8.0.0.3 Exercise. 1-norm distance matrix.The 1-norm is called taxicab distance because to go from one point to anotherin a city by car, road distance is a sum of grid lengths. Prove (600). H

3.8.1 first-order convexity condition, matrix-valued f

From the scalar-definition (§3.8.0.0.1) of a convex matrix-valued function,for differentiable function g and for each and every real vector w of unitnorm ‖w‖= 1, we have

wTg(Y )w ≥ wTg(X)w + wT→Y −X

dg(X) w (601)

that follows immediately from the first-order condition (585) for convexity ofa real function because

wT→Y −X

dg(X) w =⟨

∇X wTg(X)w , Y − X⟩

(602)

where→Y −X

dg(X) is the directional derivative (§D.1.4) of function g at X indirection Y −X . By discretized dual generalized inequalities, (§2.13.5)

g(Y ) − g(X) −→Y −X

dg(X) ºSM+

0 ⇔⟨

g(Y ) − g(X) −→Y −X

dg(X) , wwT

≥ 0 ∀wwT(ºSM∗

+

0)

(603)For each and every X,Y ∈ dom g (confer (592))

g(Y ) ºSM+

g(X) +→Y −X

dg(X) (604)

must therefore be necessary and sufficient for convexity of a matrix-valuedfunction of matrix variable on open convex domain.

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256 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.8.2 epigraph of matrix-valued function, sublevel sets

We generalize epigraph to a continuous matrix-valued function [33, p.155]g(X) : Rp×k→SM :

epi g , (X , T )∈ Rp×k× SM | X∈ dom g , g(X) ¹SM+

T (605)

from which it follows

g convex ⇔ epi g convex (606)

Proof of necessity is similar to that in §3.6 on page 233.Sublevel sets of a convex matrix-valued function corresponding to each

and every S∈ SM (confer (537))

LSg , X∈ dom g | g(X) ¹

SM+

S ⊆ Rp×k (607)

are convex. There is no converse.

3.8.2.0.1 Example. Matrix fractional function redux. [33, p.155]Generalizing Example 3.6.0.0.4 consider a matrix-valued function of twovariables on dom g = SN

+×Rn×N for small positive constant ǫ (confer (1833))

g(A , X) = ǫX(A + ǫ I )−1XT (608)

where the inverse always exists by (1427). This function is convexsimultaneously in both variables over the entire positive semidefinite cone SN

+

and all X∈ Rn×N : Consider Schur-form (1487) from §A.4: for T ∈ Sn

G(A , X , T ) =

[

A + ǫ I XT

X ǫ−1 T

]

º 0

⇔T − ǫX(A + ǫ I )−1XT º 0

A + ǫ I ≻ 0

(609)

By Theorem 2.1.9.0.1, inverse image of the positive semidefinite cone SN+n+

under affine mapping G(A , X , T ) is convex. Function g(A , X ) is convexon SN

+×Rn×N because its epigraph is that inverse image:

epi g(A , X ) =

(A , X , T ) | A + ǫ I ≻ 0 , ǫX(A + ǫ I )−1XT ¹ T

= G−1(

SN+n+

)

(610)2

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3.8. CONVEX MATRIX-VALUED FUNCTION 257

3.8.3 second-order convexity condition, matrix-valued f

The following line theorem is a potent tool for establishing convexity of amultidimensional function. To understand it, what is meant by line must firstbe solidified. Given a function g(X) : Rp×k→SM and particular X, Y ∈ Rp×k

not necessarily in that function’s domain, then we say a line X+ t Y | t ∈ Rpasses through dom g when X+ t Y ∈ dom g over some interval of t ∈ R .

3.8.3.0.1 Theorem. Line theorem. [59, §3.1.1]Matrix-valued function g(X) : Rp×k→SM is convex in X if and only if itremains convex on the intersection of any line with its domain. ⋄

Now we assume a twice differentiable function.

3.8.3.0.2 Definition. Differentiable convex matrix-valued function.Matrix-valued function g(X) : Rp×k→SM is convex in X iff dom g is anopen convex set, and its second derivative g′′(X+ t Y ) : R→SM is positivesemidefinite on each point of intersection along every line X+ t Y | t ∈ Rthat intersects dom g ; id est, iff for each and every X, Y ∈ Rp×k such thatX+ t Y ∈ dom g over some open interval of t ∈ R

d2

dt2g(X+ t Y ) º

SM+

0 (611)

Similarly, ifd2

dt2g(X+ t Y ) ≻

SM+

0 (612)

then g is strictly convex; the converse is generally false. [59, §3.1.4]3.21

3.8.3.0.3 Example. Matrix inverse. (confer §3.4.1)The matrix-valued function Xµ is convex on int SM

+ for −1≤µ≤0or 1≤µ≤2 and concave for 0≤µ≤1. [59, §3.6.2] In particular, thefunction g(X) = X−1 is convex on int SM

+ . For each and every Y ∈ SM

(§D.2.1, §A.3.1.0.5)

d2

dt2g(X+ t Y ) = 2(X+ t Y )−1 Y (X+ t Y )−1 Y (X+ t Y )−1 º

SM+

0 (613)

3.21The strict-case converse is reliably true for quadratic forms.

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258 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

on some open interval of t ∈ R such that X + t Y ≻ 0. Hence, g(X) isconvex in X . This result is extensible;3.22 tr X−1 is convex on that samedomain. [199, §7.6, prob.2] [53, §3.1, exer.25] 2

3.8.3.0.4 Example. Matrix squared.Iconic real function f(x)= x2 is strictly convex on R . The matrix-valuedfunction g(X)=X2 is convex on the domain of symmetric matrices; forX, Y ∈ SM and any open interval of t ∈ R (§D.2.1)

d2

dt2g(X+ t Y ) =

d2

dt2(X+ t Y )2 = 2Y 2 (614)

which is positive semidefinite when Y is symmetric because then Y 2 = Y TY(1433).3.23

A more appropriate matrix-valued counterpart for f is g(X)=XTXwhich is a convex function on domain X∈ Rm×n , and strictly convexwhenever X is skinny-or-square full-rank. This matrix-valued function canbe generalized to g(X)=XTAX which is convex whenever matrix A ispositive semidefinite (p.680), and strictly convex when A is positive definiteand X is skinny-or-square full-rank (Corollary A.3.1.0.5). 2

‖X‖2 , sup‖a‖=1

‖Xa‖2 = σ(X)1 =√

λ(XTX)1 = minimizet∈R

t

subject to

[

tI XXT tI

]

º 0

(615)

The matrix 2-norm (spectral norm) coincides with largest singular value.This supremum of a family of convex functions in X must be convex becauseit constitutes an intersection of epigraphs of convex functions.

3.8.3.0.5 Exercise. Squared maps.Give seven examples of distinct polyhedra P for which the set

XTX | X∈ P ⊆ Sn+ (616)

were convex. Is this set convex, in general, for any polyhedron P ?(confer (1212) (1219)) Is the epigraph of function g(X)=XTX convex forany polyhedral domain? H

3.22 d/dt tr g(X+ t Y ) = tr d/dt g(X+ t Y ). [200, p.491]3.23By (1451) in §A.3.1, changing the domain instead to all symmetric and nonsymmetricpositive semidefinite matrices, for example, will not produce a convex function.

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3.8. CONVEX MATRIX-VALUED FUNCTION 259

3.8.3.0.6 Exercise. Squared inverse. (confer §3.8.2.0.1)For positive scalar a , real function f(x)= ax−2 is convex on the nonnegativereal line. Given positive definite matrix constant A , prove via line theoremthat g(X)= tr

(

(XTA−1X)−1)

is generally not convex unless X≻ 0 .3.24

From this result, show how it follows via Definition 3.8.0.0.1-2 thath(X) = (XTA−1X)−1 is generally neither convex. H

3.8.3.0.7 Example. Matrix exponential.The matrix-valued function g(X) = eX : SM → SM is convex on the subspaceof circulant [164] symmetric matrices. Applying the line theorem, for all t∈Rand circulant X, Y ∈ SM , from Table D.2.7 we have

d2

dt2eX+ t Y = Y eX+ t Y Y º

SM+

0 , (XY )T = XY (617)

because all circulant matrices are commutative and, for symmetric matrices,XY = Y X ⇔ (XY )T = XY (1450). Given symmetric argument, the matrixexponential always resides interior to the cone of positive semidefinitematrices in the symmetric matrix subspace; eA≻ 0 ∀A∈SM (1830). Thenfor any matrix Y of compatible dimension, Y TeAY is positive semidefinite.(§A.3.1.0.5)

The subspace of circulant symmetric matrices contains all diagonalmatrices. The matrix exponential of any diagonal matrix eΛ exponentiateseach individual entry on the main diagonal. [247, §5.3] So, changingthe function domain to the subspace of real diagonal matrices reduces thematrix exponential to a vector-valued function in an isometrically isomorphicsubspace RM ; known convex (§3.1.1) from the real-valued function case[59, §3.1.5]. 2

There are, of course, multifarious methods to determine functionconvexity, [59] [41] [129] each of them efficient when appropriate.

3.8.3.0.8 Exercise. log det function.Matrix determinant is neither a convex or concave function, in general,but its inverse is convex when domain is restricted to interior of a positivesemidefinite cone. [33, p.149] Show by three different methods: On interior ofthe positive semidefinite cone, log det X = − log det X−1 is concave. H

3.24Hint: §D.2.3.

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260 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

Figure 78: Iconic unimodal differentiable quasiconvex function of twovariables graphed in R2× R on some open disc in R2. Note reversal ofcurvature in direction of gradient.

3.9 Quasiconvex

Quasiconvex functions [59, §3.4] [196] [327] [368] [240, §2] are useful inpractical problem solving because they are unimodal (by definition whennonmonotonic); a global minimum is guaranteed to exist over any convex setin the function domain; e.g., Figure 78.

3.9.0.0.1 Definition. Quasiconvex function.f(X) : Rp×k→R is a quasiconvex function of matrix X iff dom f is a convexset and for each and every Y, Z∈dom f , 0≤µ≤1

f(µY + (1 − µ)Z) ≤ maxf(Y ) , f(Z) (618)

A quasiconcave function is determined:

f(µY + (1 − µ)Z) ≥ minf(Y ) , f(Z) (619)

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3.9. QUASICONVEX 261

Unlike convex functions, quasiconvex functions are not necessarilycontinuous; e.g., quasiconcave rank(X) on SM

+ (§2.9.2.6.2) and card(x)on RM

+ . Although insufficient for convex functions, convexity of each andevery sublevel set serves as a definition of quasiconvexity:

3.9.0.0.2 Definition. Quasiconvex multidimensional function.Scalar-, vector-, or matrix-valued function g(X) : Rp×k→SM is a quasiconvexfunction of matrix X iff dom g is a convex set and the sublevel setcorresponding to each and every S∈ SM

LSg = X∈ dom g | g(X) ¹ S ⊆ Rp×k (607)

is convex. Vectors are compared with respect to the nonnegative orthant RM+

while matrices are with respect to the positive semidefinite cone SM+ .

Convexity of the superlevel set corresponding to each and every S∈ SM ,likewise

LSg = X∈ dom g | g(X) º S ⊆ Rp×k (620)

is necessary and sufficient for quasiconcavity.

3.9.0.0.3 Exercise. Nonconvexity of matrix product.Consider real function f on a positive definite domain

f(X) = tr(X1X2) , X ,

[

X1

X2

]

∈ dom f ,

[

rel int SN+

rel int SN+

]

(621)

with superlevel sets

Lsf = X ∈ dom f | 〈X1 , X2〉 ≥ s (622)

Prove: f(X) is not quasiconcave except when N = 1, nor is it quasiconvexunless X1 = X2 . H

3.9.1 bilinear

Bilinear function3.25 xTy of vectors x and y is quasiconcave (monotonic) onthe entirety of the nonnegative orthants only when vectors are of dimension 1.

3.25Convex envelope of bilinear functions is well known. [3]

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262 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

3.9.2 quasilinear

When a function is simultaneously quasiconvex and quasiconcave, it is calledquasilinear. Quasilinear functions are completely determined by convexlevel sets. One-dimensional function f(x) = x3 and vector-valued signumfunction sgn(x) , for example, are quasilinear. Any monotonic function isquasilinear3.26 (but not vice versa, Exercise 3.7.4.0.1).

3.10 Salient properties

of convex and quasiconvex functions

1. A convex function is assumed continuous but not necessarilydifferentiable on the relative interior of its domain. [303, §10] A quasiconvex function is not necessarily a continuous function.

2. convex epigraph ⇔ convexity ⇒ quasiconvexity ⇔ convex sublevel sets.convex hypograph ⇔ concavity ⇒ quasiconcavity ⇔ convex superlevel.

monotonicity ⇒ quasilinearity ⇔ convex level sets.

3.log-convex ⇒ convex ⇒ quasiconvex.

concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave.3.27

4. (homogeneity) Convexity, concavity, quasiconvexity, andquasiconcavity are invariant to nonnegative scaling of function. g convex ⇔ −g concaveg quasiconvex ⇔ −g quasiconcaveg log-convex ⇔ 1/g log-concave

5. The line theorem (§3.8.3.0.1) translates identically to quasiconvexity(quasiconcavity). [59, §3.4.2]

6. (affine transformation of argument) Composition g(h(X)) of convex(concave) function g with any affine function h : Rm×n → Rp×k

remains convex (concave) in X∈ Rm×n, where h(Rm×n) ∩ dom g 6= ∅ .[196, §B.2.1] Likewise for the quasiconvex (quasiconcave) functions g .

3.26 e.g., a monotonic concave function is quasiconvex, but dare not confuse these terms.3.27Log-convex means: logarithm of function f is convex on dom f .

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3.10. SALIENT PROPERTIES 263

7. – Nonnegatively weighted sum of convex (concave) functionsremains convex (concave). (§3.1.2.1.1)

– Nonnegatively weighted nonzero sum of strictly convex(concave) functions remains strictly convex (concave).

– Nonnegatively weighted maximum (minimum) of convex3.28

(concave) functions remains convex (concave).

– Pointwise supremum (infimum) of convex (concave) functionsremains convex (concave). (Figure 73) [303, §5] – Sum of quasiconvex functions not necessarily quasiconvex.

– Nonnegatively weighted maximum (minimum) of quasiconvex(quasiconcave) functions remains quasiconvex (quasiconcave).

– Pointwise supremum (infimum) of quasiconvex (quasiconcave)functions remains quasiconvex (quasiconcave).

3.28Supremum and maximum of convex functions are proven convex by intersection ofepigraphs.

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264 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS

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Chapter 4

Semidefinite programming

Prior to 1984, linear and nonlinear programming,4.1 one a subsetof the other, had evolved for the most part along unconnectedpaths, without even a common terminology. (The use of“programming” to mean “optimization” serves as a persistentreminder of these differences.)

−Forsgren, Gill, & Wright (2002) [141]

Given some practical application of convex analysis, it may at first seempuzzling why a search for its solution ends abruptly with a formalizedstatement of the problem itself as a constrained optimization. Theexplanation is: typically we do not seek analytical solution because there arerelatively few. (§3.6.2, §C) If a problem can be expressed in convex form,rather, then there exist computer programs providing efficient numericalglobal solution. [163] [381] [382] [380] [345] [333] The goal, then, becomesconversion of a given problem (perhaps a nonconvex or combinatorialproblem statement) to an equivalent convex form or to an alternation ofconvex subproblems convergent to a solution of the original problem:

4.1 nascence of polynomial-time interior-point methods of solution [360] [378].Linear programming ⊂ (convex ∩ nonlinear) programming.© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

265

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266 CHAPTER 4. SEMIDEFINITE PROGRAMMING

By the fundamental theorem of Convex Optimization, any locally optimalpoint of a convex problem is globally optimal. [59, §4.2.2] [302, §1] Givenconvex real objective function g and convex feasible set D⊆dom g , whichis the set of all variable values satisfying the problem constraints, we pose ageneric convex optimization problem

minimizeX

g(X)

subject to X∈ D(623)

where constraints are abstract here in membership of variable X to convexfeasible set D . Inequality constraint functions of a convex optimizationproblem are convex while equality constraint functions are conventionallyaffine, but not necessarily so. Affine equality constraint functions, as opposedto the superset of all convex equality constraint functions having convex levelsets (§3.5.0.0.4), make convex optimization tractable.

Similarly, the problem

maximizeX

g(X)

subject to X∈ D(624)

is called convex were g a real concave function and feasible set D convex.As conversion to convex form is not always possible, there is much ongoingresearch to determine which problem classes have convex expression orrelaxation. [33] [57] [147] [273] [342] [145]

4.1 Conic problem

Still, we are surprised to see the relatively small number ofsubmissions to semidefinite programming (SDP) solvers, as thisis an area of significant current interest to the optimizationcommunity. We speculate that semidefinite programming issimply experiencing the fate of most new areas: Users have yet tounderstand how to pose their problems as semidefinite programs,and the lack of support for SDP solvers in popular modellinglanguages likely discourages submissions.

−SIAM News, 2002. [112, p.9]

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4.1. CONIC PROBLEM 267

(confer p.154) Consider a conic problem (p) and its dual (d): [290, §3.3.1][235, §2.1] [236]

(p)

minimizex

cTx

subject to x ∈ KAx = b

maximizey,s

bTy

subject to s ∈ K∗

ATy + s = c

(d) (281)

where K is a closed convex cone, K∗is its dual, matrix A is fixed, and the

remaining quantities are vectors.When K is a polyhedral cone (§2.12.1), then each conic problem becomes

a linear program [92]; the selfdual nonnegative orthant providing theprototypical primal linear program and its dual. More generally, eachoptimization problem is convex when K is a closed convex cone. Unlikethe optimal objective value, a solution to each problem is not necessarilyunique; in other words, the optimal solution set x⋆ or y⋆, s⋆ is convex andmay comprise more than a single point although the corresponding optimalobjective value is unique when the feasible set is nonempty.

4.1.1 a Semidefinite program

When K is the selfdual cone of positive semidefinite matrices Sn+ in the

subspace of symmetric matrices Sn, then each conic problem is called asemidefinite program (SDP); [273, §6.4] primal problem (P) having matrixvariable X∈ Sn while corresponding dual (D) has slack variable S∈ Sn andvector variable y = [yi]∈ Rm : [10] [11, §2] [388, §1.3.8]

(P)

minimizeX∈ S

n〈C , X 〉

subject to X º 0

A svec X = b

maximizey∈R

m, S∈S

n〈b , y〉

subject to S º 0

svec−1(ATy) + S = C

(D)

(625)

This is the prototypical semidefinite program and its dual, where matrixC ∈ Sn and vector b∈Rm are fixed, as is

A ,

svec(A1)T

...svec(Am)T

∈ Rm×n(n+1)/2 (626)

where Ai∈ Sn, i=1 . . . m , are given. Thus

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268 CHAPTER 4. SEMIDEFINITE PROGRAMMING

A svec X =

〈A1 , X 〉...

〈Am , X 〉

svec−1(ATy) =m∑

i=1

yiAi

(627)

The vector inner-product for matrices is defined in the Euclidean/Frobeniussense in the isomorphic vector space Rn(n+1)/2 ; id est,

〈C , X 〉 , tr(CTX) = svec(C)Tsvec X (38)

where svec X defined by (56) denotes symmetric vectorization.

In a national planning problem of some size, one may easily run into several

hundred variables and perhaps a hundred or more degrees of freedom. . . . It

should always be remembered that any mathematical method and particularly

methods in linear programming must be judged with reference to the type of

computing machinery available. Our outlook may perhaps be changed when

we get used to the super modern, high capacity electronic computor that will

be available here from the middle of next year. −Ragnar Frisch [143]

Linear programming, invented by Dantzig in 1947 [92], is now integralto modern technology. The same cannot yet be said of semidefiniteprogramming whose roots trace back to systems of positive semidefinitelinear inequalities studied by Bellman & Fan in 1963. [31] [99] Interior-pointmethods for numerical solution can be traced back to the logarithmic barrierof Frisch in 1954 and Fiacco & McCormick in 1968 [138]. Karmarkar’spolynomial-time interior-point method sparked a log-barrier renaissancein 1984, [271, §11] [378] [360] [273, p.3] but numerical performanceof contemporary general-purpose semidefinite program solvers remainslimited: Computational intensity for dense systems varies as O(m2n)(m constraints ≪ n variables) based on interior-point methods that producesolutions no more relatively accurate than 1E-8. There are no solverscapable of handling in excess of n=100,000 variables without significant,sometimes crippling, loss of precision or time.4.2 [34] [272, p.258] [65, p.3]

4.2Heuristics are not ruled out by SIOPT; indeed I would suspect that most successfulmethods have (appropriately described) heuristics under the hood - my codes certainly do.. . .Of course, there are still questions relating to high-accuracy and speed, but for manyapplications a few digits of accuracy suffices and overnight runs for non-real-time deliveryis acceptable. −Nicholas I. M. Gould, Editor-in-Chief, SIOPT

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4.1. CONIC PROBLEM 269

linear program

second-order cone program

semidefinite program

quadratic program

PC

Figure 79: Venn diagram of programming hierarchy. Semidefinite program isa subset of convex program PC . Semidefinite program subsumes other convexprogram classes excepting geometric program. Second-order cone programand quadratic program each subsume linear program. Nonconvex program\PC comprises those for which convex equivalents have not yet been found.

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270 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Nevertheless, semidefinite programming has recently emerged toprominence because it admits a new class of problem previously unsolvable byconvex optimization techniques, [57] and because it theoretically subsumesother convex techniques: (Figure 79) linear programming and quadraticprogramming and second-order cone programming.4.3 Determination of theRiemann mapping function from complex analysis [283] [29, §8, §13], forexample, can be posed as a semidefinite program.

4.1.1.1 Maximal complementarity

It has been shown [388, §2.5.3] that contemporary interior-point methods[379] [286] [273] [11] [59, §11] [141] (developed circa 1990 [147] for numericalsolution of semidefinite programs) can converge to a solution of maximalcomplementarity ; [172, §5] [387] [249] [154] not a vertex-solution but asolution of highest cardinality or rank among all optimal solutions.4.4

This phenomenon can be explained by recognizing that interior-pointmethods generally find solutions relatively interior to a feasible set bydesign.4.5 [6, p.3] Log barriers are designed to fail numerically at the feasibleset boundary. So low-rank solutions, all on the boundary, are rendered moredifficult to find as numerical error becomes more prevalent there.

4.1.1.2 Reduced-rank solution

A simple rank reduction algorithm, for construction of a primal optimalsolution X⋆ to (625P) satisfying an upper bound on rank governed byProposition 2.9.3.0.1, is presented in §4.3. That proposition asserts existenceof feasible solutions with an upper bound on their rank; [26, §II.13.1]specifically, it asserts an extreme point (§2.6.0.0.1) of primal feasible setA ∩ Sn

+ satisfies upper bound

rank X ≤⌊√

8m + 1 − 1

2

(252)

where, given A∈Rm×n(n+1)/2 and b∈Rm,

A , X∈ Sn | A svec X = b (628)

4.3SOCP came into being in the 1990s; it is not posable as a quadratic program. [244]4.4This characteristic might be regarded as a disadvantage to this method of numerical

solution, but this behavior is not certain and depends on solver implementation.4.5Simplex methods, in contrast, generally find vertex solutions.

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4.1. CONIC PROBLEM 271

is the affine subset from primal problem (625P).

4.1.1.3 Coexistence of low- and high-rank solutions; analogy

That low-rank and high-rank optimal solutions X⋆ of (625P) coexist maybe grasped with the following analogy: We compare a proper polyhedral coneS3

+ in R3 (illustrated in Figure 80) to the positive semidefinite cone S3

+ inisometrically isomorphic R6, difficult to visualize. The analogy is good: int S3

+ is constituted by rank-3 matrices.

intS3

+ has three dimensions. boundary ∂S3

+ contains rank-0, rank-1, and rank-2 matrices.

boundary ∂S3

+ contains 0-, 1-, and 2-dimensional faces. the only rank-0 matrix resides in the vertex at the origin. Rank-1 matrices are in one-to-one correspondence with extremedirections of S3

+ and S3

+ . The set of all rank-1 symmetric matrices inthis dimension

G ∈ S3

+ | rank G=1

(629)

is not a connected set. Rank of a sum of members F +G in Lemma 2.9.2.6.1 and location ofa difference F−G in §2.9.2.9.1 similarly hold for S3

+ and S3

+ . Euclidean distance from any particular rank-3 positive semidefinitematrix (in the cone interior) to the closest rank-2 positive semidefinitematrix (on the boundary) is generally less than the distance to theclosest rank-1 positive semidefinite matrix. (§7.1.2) distance from any point in ∂S3

+ to int S3

+ is infinitesimal (§2.1.7.1.1).

distance from any point in ∂S3

+ to intS3

+ is infinitesimal.

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272 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Γ2

Γ1

S3

+

0

A=∂H

C

P

Figure 80: Visualizing positive semidefinite cone in high dimension: Properpolyhedral cone S3

+⊂ R3 representing positive semidefinite cone S3

+⊂ S3 ;analogizing its intersection with hyperplane S3

+ ∩ ∂H . Number of facets isarbitrary (analogy is not inspired by eigenvalue decomposition). The rank-0positive semidefinite matrix corresponds to the origin in R3, rank-1 positivesemidefinite matrices correspond to the edges of the polyhedral cone, rank-2to the facet relative interiors, and rank-3 to the polyhedral cone interior.Vertices Γ1 and Γ2 are extreme points of polyhedron P=∂H ∩ S3

+ , andextreme directions of S3

+ . A given vector C is normal to another hyperplane(not illustrated but independent w.r.t ∂H) containing line segment Γ1Γ2

minimizing real linear function 〈C, X 〉 on P . (confer Figure 26, Figure 30)

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4.1. CONIC PROBLEM 273 faces of S3

+ correspond to faces of S3

+ (confer Table 2.9.2.3.1):

k dimF(S3

+) dimF(S3

+) dimF(S3

+∋ rank-k matrix)0 0 0 0

boundary 1 1 1 12 2 3 3

interior 3 3 6 6

Integer k indexes k-dimensional faces F of S3

+ . Positive semidefinitecone S3

+ has four kinds of faces, including cone itself (k = 3,boundary + interior), whose dimensions in isometrically isomorphicR6 are listed under dimF(S3

+). Smallest face F(S3

+∋ rank-k matrix)that contains a rank-k positive semidefinite matrix has dimensionk(k + 1)/2 by (213). For A equal to intersection of m hyperplanes having independentnormals, and for X∈ S3

+ ∩ A , we have rank X ≤ m ; the analogueto (252).

Proof. With reference to Figure 80: Assume one (m = 1) hyperplaneA= ∂H intersects the polyhedral cone. Every intersecting planecontains at least one matrix having rank less than or equal to 1 ; id est,from all X∈ ∂H ∩ S3

+ there exists an X such that rankX≤ 1. Rank 1is therefore an upper bound in this case.

Now visualize intersection of the polyhedral cone with two (m = 2)hyperplanes having linearly independent normals. The hyperplaneintersection A makes a line. Every intersecting line contains at least onematrix having rank less than or equal to 2, providing an upper bound.In other words, there exists a positive semidefinite matrix X belongingto any line intersecting the polyhedral cone such that rankX ≤ 2.

In the case of three independent intersecting hyperplanes (m = 3), thehyperplane intersection A makes a point that can reside anywhere inthe polyhedral cone. The upper bound on a point in S3

+ is also thegreatest upper bound: rankX ≤ 3. ¨

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274 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.1.1.3.1 Example. Optimization over A ∩ S3

+ .Consider minimization of the real linear function 〈C , X 〉 over

P , A ∩ S3

+ (630)

a polyhedral feasible set;

f ⋆0 , minimize

X〈C , X 〉

subject to X ∈ A ∩ S3

+

(631)

As illustrated for particular vector C and hyperplane A= ∂H in Figure 80,this linear function is minimized on any X belonging to the face of Pcontaining extreme points Γ1 , Γ2 and all the rank-2 matrices in between;id est, on any X belonging to the face of P

F(P) = X | 〈C , X 〉 = f ⋆0 ∩ A ∩ S3

+ (632)

exposed by the hyperplane X | 〈C , X 〉=f ⋆0. In other words, the set of all

optimal points X⋆ is a face of P

X⋆ = F(P) = Γ1Γ2 (633)

comprising rank-1 and rank-2 positive semidefinite matrices. Rank 1 isthe upper bound on existence in the feasible set P for this case m = 1hyperplane constituting A . The rank-1 matrices Γ1 and Γ2 in face F(P)are extreme points of that face and (by transitivity (§2.6.1.2)) extremepoints of the intersection P as well. As predicted by analogy to Barvinok’sProposition 2.9.3.0.1, the upper bound on rank of X existent in the feasibleset P is satisfied by an extreme point. The upper bound on rank of anoptimal solution X⋆ existent in F(P) is thereby also satisfied by an extremepoint of P precisely because X⋆ constitutes F(P) ;4.6 in particular,

X⋆∈ P | rank X⋆≤ 1 = Γ1 , Γ2 ⊆ F(P) (634)

As all linear functions on a polyhedron are minimized on a face, [92] [248][270] [276] by analogy we so demonstrate coexistence of optimal solutions X⋆

of (625P) having assorted rank. 2

4.6 and every face contains a subset of the extreme points of P by the extremeexistence theorem (§2.6.0.0.2). This means: because the affine subset A and hyperplaneX | 〈C , X 〉 = f⋆

0 must intersect a whole face of P , calculation of an upper bound onrank of X⋆ ignores counting the hyperplane when determining m in (252).

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4.1. CONIC PROBLEM 275

4.1.1.4 Previous work

Barvinok showed, [24, §2.2] when given a positive definite matrix C and anarbitrarily small neighborhood of C comprising positive definite matrices,there exists a matrix C from that neighborhood such that optimal solutionX⋆ to (625P) (substituting C ) is an extreme point of A ∩ Sn

+ and satisfiesupper bound (252).4.7 Given arbitrary positive definite C , this meansnothing inherently guarantees that an optimal solution X⋆ to problem (625P)satisfies (252); certainly nothing given any symmetric matrix C , as theproblem is posed. This can be proved by example:

4.1.1.4.1 Example. (Ye) Maximal Complementarity.Assume dimension n to be an even positive number. Then the particularinstance of problem (625P),

minimizeX∈ S

n

⟨[

I 00 2I

]

, X

subject to X º 0

〈I , X 〉 = n

(635)

has optimal solution

X⋆ =

[

2I 00 0

]

∈ Sn (636)

with an equal number of twos and zeros along the main diagonal. Indeed,optimal solution (636) is a terminal solution along the central path taken bythe interior-point method as implemented in [388, §2.5.3]; it is also a solutionof highest rank among all optimal solutions to (635). Clearly, rank of thisprimal optimal solution exceeds by far a rank-1 solution predicted by upperbound (252). 2

4.1.1.5 Later developments

This rational example (635) indicates the need for a more generally applicableand simple algorithm to identify an optimal solution X⋆ satisfying Barvinok’sProposition 2.9.3.0.1. We will review such an algorithm in §4.3, but first weprovide more background.

4.7Further, the set of all such C in that neighborhood is open and dense.

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276 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.2 Framework

4.2.1 Feasible sets

Denote by D and D∗ the convex sets of primal and dual points respectivelysatisfying the primal and dual constraints in (625), each assumed nonempty;

D =

X∈ Sn+ |

〈A1 , X 〉...

〈Am , X 〉

= b

= A ∩ Sn+

D∗ =

S∈ Sn+ , y = [yi]∈ Rm |

m∑

i=1

yiAi + S = C

(637)

These are the primal feasible set and dual feasible set. Geometrically,primal feasible A ∩ Sn

+ represents an intersection of the positive semidefinitecone Sn

+ with an affine subset A of the subspace of symmetric matrices Sn

in isometrically isomorphic Rn(n+1)/2. A has dimension n(n+1)/2−mwhen the vectorized Ai are linearly independent. Dual feasible set D∗ isa Cartesian product of the positive semidefinite cone with its inverse image(§2.1.9.0.1) under affine transformation4.8 C−∑

yiAi . Both feasible sets areconvex, and the objective functions are linear on a Euclidean vector space.Hence, (625P) and (625D) are convex optimization problems.

4.2.1.1 A ∩ Sn+ emptiness determination via Farkas’ lemma

4.2.1.1.1 Lemma. Semidefinite Farkas’ lemma.Given set Ai∈ Sn, i=1 . . . m , vector b = [bi]∈ Rm, and affine subset

(628) A = X∈ Sn |〈Ai , X 〉= bi , i=1 . . . m Ä A svec X |Xº 0 (358) is closed,

then primal feasible set A ∩ Sn+ is nonempty if and only if yTb≥ 0 holds for

each and every vector y = [yi]∈ Rm such thatm∑

i=1

yiAi º 0.

Equivalently, primal feasible set A ∩ Sn+ is nonempty if and only if

yTb≥ 0 holds for each and every vector ‖y‖= 1 such thatm∑

i=1

yiAi º 0. ⋄

4.8Inequality C−∑

yiAiº 0 follows directly from (625D) (§2.9.0.1.1) and is known as alinear matrix inequality. (§2.13.5.1.1) Because

yiAi¹C , matrix S is known as a slackvariable (a term borrowed from linear programming [92]) since its inclusion raises thisinequality to equality.

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4.2. FRAMEWORK 277

Semidefinite Farkas’ lemma provides necessary and sufficient conditionsfor a set of hyperplanes to have nonempty intersection A ∩ Sn

+ with thepositive semidefinite cone. Given

A =

svec(A1)T

...svec(Am)T

∈ Rm×n(n+1)/2 (626)

semidefinite Farkas’ lemma assumes that a convex cone

K = A svec X | Xº 0 (358)

is closed per membership relation (298) from which the lemma springs:[228, §I] K closure is attained when matrix A satisfies the cone closednessinvariance corollary (p.175). Given closed convex cone K and its dual fromExample 2.13.5.1.1

K∗= y |

m∑

j=1

yjAj º 0 (365)

then we can apply membership relation

b ∈ K ⇔ 〈y , b〉 ≥ 0 ∀ y ∈ K∗(298)

to obtain the lemma

b ∈ K ⇔ ∃X º 0 Ä A svec X = b ⇔ A ∩ Sn+ 6= ∅ (638)

b ∈ K ⇔ 〈y , b〉 ≥ 0 ∀ y ∈ K∗ ⇔ A ∩ Sn+ 6= ∅ (639)

The final equivalence synopsizes semidefinite Farkas’ lemma.While the lemma is correct as stated, a positive definite version is required

for semidefinite programming [388, §1.3.8] because existence of a feasiblesolution in the cone interior A ∩ int Sn

+ is required by Slater’s condition4.9

to achieve 0 duality gap (optimal primal−dual objective difference §4.2.3,Figure 57). Geometrically, a positive definite lemma is required to insurethat a point of intersection closest to the origin is not at infinity; e.g.,

4.9Slater’s sufficient constraint qualification is satisfied whenever any primal or dualstrictly feasible solution exists; id est, any point satisfying the respective affine constraintsand relatively interior to the convex cone. [327, §6.6] [40, p.325] If the cone were polyhedral,then Slater’s constraint qualification is satisfied when any feasible solution exists (relativelyinterior to the cone or on its relative boundary). [59, §5.2.3]

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278 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Figure 44. Then given A∈Rm×n(n+1)/2 having rank m , we wish to detectexistence of nonempty primal feasible set interior to the PSD cone;4.10 (361)

b ∈ intK ⇔ 〈y , b〉 > 0 ∀ y ∈ K∗, y 6= 0 ⇔ A∩ int Sn

+ 6= ∅ (640)

Positive definite Farkas’ lemma is made from proper cones, K (358) and K∗

(365), and membership relation (304) for which K closedness is unnecessary:

4.2.1.1.2 Lemma. Positive definite Farkas’ lemma.Given l.i. set Ai∈ Sn, i=1 . . . m and vector b = [bi]∈ Rm, make affine set

A = X∈ Sn |〈Ai , X 〉= bi , i=1 . . . m (628)

Primal feasible cone interior A ∩ int Sn+ is nonempty if and only if yTb > 0

holds for each and every vector y = [yi] 6= 0 such thatm∑

i=1

yiAi º 0.

Equivalently, primal feasible cone interior A ∩ int Sn+ is nonempty if and

only if yTb > 0 holds for each and every vector ‖y‖= 1 Äm∑

i=1

yiAi º 0. ⋄

4.2.1.1.3 Example. “New” Farkas’ lemma.Lasserre [228, §III] presented an example in 1995, originally offered byBen-Israel in 1969 [32, p.378], to support closedness in semidefinite Farkas’Lemma 4.2.1.1.1:

A ,

[

svec(A1)T

svec(A2)T

]

=

[

0 1 00 0 1

]

, b =

[

10

]

(641)

Intersection A ∩ Sn+ is practically empty because the solution set

Xº 0 | A svec X = b =

[

α 1√2

1√2

0

]

º 0 | α∈R

(642)

is positive semidefinite only asymptotically (α→∞). Yet the dual systemm∑

i=1

yiAiº 0 ⇒ yTb≥0 erroneously indicates nonempty intersection because

K (358) violates a closedness condition of the lemma; videlicet, for ‖y‖= 1

y1

[

0 1√2

1√2

0

]

+ y2

[

0 00 1

]

º 0 ⇔ y =

[

01

]

⇒ yTb = 0 (643)

4.10Detection of A ∩ int Sn+ 6= ∅ by examining intK instead is a trick need not be lost.

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4.2. FRAMEWORK 279

On the other hand, positive definite Farkas’ Lemma 4.2.1.1.2 certifies thatA ∩ int Sn

+ is empty; what we need to know for semidefinite programming.Lasserre suggested addition of another condition to semidefinite Farkas’

lemma (§4.2.1.1.1) to make a new lemma having no closedness condition.But positive definite Farkas’ lemma (§4.2.1.1.2) is simpler and obviates theadditional condition proposed. 2

4.2.1.2 Theorem of the alternative for semidefinite programming

Because these Farkas’ lemmas follow from membership relations, we mayconstruct alternative systems from them. Applying the method of §2.13.2.1.1,then from positive definite Farkas’ lemma we get

A ∩ int Sn+ 6= ∅

or in the alternative

yTb≤ 0 ,m∑

i=1

yiAi º 0 , y 6= 0

(644)

Any single vector y satisfying the alternative certifies A ∩ int Sn+ is empty.

Such a vector can be found as a solution to another semidefinite program:for linearly independent (vectorized) set Ai∈ Sn, i=1 . . . m

minimizey

yTb

subject tom∑

i=1

yiAi º 0

‖y‖2 ≤ 1

(645)

If an optimal vector y⋆ 6= 0 can be found such that y⋆Tb≤ 0, then primalfeasible cone interior A ∩ int Sn

+ is empty.

4.2.1.3 Boundary-membership criterion

(confer (639) (640)) From boundary-membership relation (308) for linearmatrix inequality proper cones K (358) and K∗

(365)

b ∈ ∂K ⇔ ∃ y 6= 0 Ä 〈y , b〉= 0 , y ∈ K∗, b ∈ K ⇔ ∂Sn

+ ⊃ A∩ Sn+ 6= ∅

(646)

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280 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Whether vector b ∈ ∂K belongs to cone K boundary, that is a determinationwe can indeed make; one that is certainly expressible as a feasibility problem:Given linearly independent set4.11 Ai∈ Sn, i=1 . . . m , for b ∈ K (638)

find y 6= 0

subject to yTb = 0m∑

i=1

yiAi º 0

(647)

Any such nonzero solution y certifies that affine subset A (628) intersects thepositive semidefinite cone Sn

+ only on its boundary; in other words, nonemptyfeasible set A ∩ Sn

+ belongs to the positive semidefinite cone boundary ∂Sn+ .

4.2.2 Duals

The dual objective function from (625D) evaluated at any feasible solutionrepresents a lower bound on the primal optimal objective value from (625P).We can see this by direct substitution: Assume the feasible sets A ∩ Sn

+ andD∗ are nonempty. Then it is always true:

〈C , X 〉 ≥ 〈b , y〉⟨

i

yiAi + S , X

≥ [ 〈A1 , X 〉 · · · 〈Am , X 〉 ] y

〈S , X 〉 ≥ 0

(648)

The converse also follows because

X º 0 , S º 0 ⇒ 〈S , X 〉 ≥ 0 (1458)

Optimal value of the dual objective thus represents the greatest lower boundon the primal. This fact is known as the weak duality theorem for semidefiniteprogramming, [388, §1.3.8] and can be used to detect convergence in anyprimal/dual numerical method of solution.

4.11From the results of Example 2.13.5.1.1, vector b on the boundary of K cannotbe detected simply by looking for 0 eigenvalues in matrix X . We do not consider askinny-or-square matrix A because then feasible set A ∩ Sn

+ is at most a single point.

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4.2. FRAMEWORK 281

4.2.2.1 Dual problem statement is not unique

Even subtle but equivalent restatements of a primal convex problem canlead to vastly different statements of a corresponding dual problem. Thisphenomenon is of interest because a particular instantiation of dual problemmight be easier to solve numerically or it might take one of few forms forwhich analytical solution is known.

Here is a canonical restatement of prototypical dual semidefinite program(625D), for example, equivalent by (185):

(D)

maximizey∈R

m, S∈Sn

〈b , y〉subject to S º 0

svec−1(ATy) + S = C

≡maximize

y∈Rm

〈b , y〉subject to svec−1(ATy) ¹ C

(625D)

Dual feasible cone interior in int Sn+ (637) (627) thereby corresponds with

canonical dual (D) feasible interior

rel int D∗ ,

y∈ Rm |m

i=1

yiAi ≺ C

(649)

4.2.2.1.1 Exercise. Primal prototypical semidefinite program.Derive prototypical primal (625P) from its canonical dual (625D); id est,demonstrate that particular connectivity in Figure 81. H

4.2.3 Optimality conditions

When primal feasible cone interior A ∩ int Sn+ exists in Sn or when canonical

dual feasible interior rel int D∗ exists in Rm, then these two problems (625P)(625D) become strong duals by Slater’s sufficient condition (p.277). Inother words, the primal optimal objective value becomes equal to the dualoptimal objective value: there is no duality gap (Figure 57); id est, if∃X∈ A ∩ int Sn

+ or ∃ y∈ rel int D∗ then

〈C , X⋆〉 = 〈b , y⋆〉⟨

i

y⋆i Ai + S⋆ , X⋆

= [ 〈A1 , X⋆〉 · · · 〈Am , X⋆〉 ] y⋆

〈S⋆ , X⋆〉 = 0

(650)

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282 CHAPTER 4. SEMIDEFINITE PROGRAMMING

P

D

P

D

dualityduality

transformation

Figure 81: Connectivity indicates paths between particular primal and dualproblems from Exercise 4.2.2.1.1. More generally, any path between primalproblems P (and equivalent P) and dual D (and equivalent D) is possible:implying, any given path is not necessarily circuital; dual of a dual problemis not necessarily stated in precisely same manner as corresponding primalconvex problem, in other words, although its solution set is equivalent towithin some transformation.

where S⋆, y⋆ denote a dual optimal solution.4.12 We summarize this:

4.2.3.0.1 Corollary. Optimality and strong duality. [356, §3.1][388, §1.3.8] For semidefinite programs (625P) and (625D), assume primaland dual feasible sets A ∩ Sn

+⊂ Sn and D∗⊂ Sn× Rm (637) are nonempty.Then X⋆ is optimal for (625P) S⋆, y⋆ are optimal for (625D) duality gap 〈C,X⋆〉−〈b , y⋆〉 is 0

if and only if

i) ∃X∈ A ∩ int Sn+ or ∃ y ∈ rel int D∗

and

ii) 〈S⋆ , X⋆〉 = 0 ⋄

4.12Optimality condition 〈S⋆ , X⋆〉=0 is called a complementary slackness condition, inkeeping with linear programming tradition [92], that forbids dual inequalities in (625) tosimultaneously hold strictly. [302, §4]

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4.2. FRAMEWORK 283

For symmetric positive semidefinite matrices, requirement ii is equivalentto the complementarity (§A.7.4)

〈S⋆ , X⋆〉 = 0 ⇔ S⋆X⋆ = X⋆S⋆ = 0 (651)

Commutativity of diagonalizable matrices is a necessary and sufficientcondition [199, §1.3.12] for these two optimal symmetric matrices to besimultaneously diagonalizable. Therefore

rank X⋆ + rankS⋆ ≤ n (652)

Proof. To see that, the product of symmetric optimal matricesX⋆, S⋆∈ Sn must itself be symmetric because of commutativity. (1450) Thesymmetric product has diagonalization [11, cor.2.11]

S⋆X⋆ = X⋆S⋆ = QΛS⋆ΛX⋆QT = 0 ⇔ ΛX⋆ΛS⋆ = 0 (653)

where Q is an orthogonal matrix. The product of the nonnegative diagonal Λmatrices can be 0 if their main diagonal zeros are complementary or coincide.Due only to symmetry, rankX⋆ = rank ΛX⋆ and rank S⋆ = rank ΛS⋆ forthese optimal primal and dual solutions. (1436) So, because of thecomplementarity, the total number of nonzero diagonal entries from both Λcannot exceed n . ¨

When equality is attained in (652)

rank X⋆ + rank S⋆ = n (654)

there are no coinciding main diagonal zeros in ΛX⋆ΛS⋆ , and so we have whatis called strict complementarity.4.13 Logically it follows that a necessary andsufficient condition for strict complementarity of an optimal primal and dualsolution is

X⋆ + S⋆ ≻ 0 (655)

4.2.3.1 solving primal problem via dual

The beauty of Corollary 4.2.3.0.1 is its conjugacy; id est, one can solveeither the primal or dual problem in (625) and then find a solution tothe other via the optimality conditions. When a dual optimal solution isknown, for example, a primal optimal solution belongs to the hyperplaneX | 〈S⋆ , X 〉=0.

4.13 distinct from maximal complementarity (§4.1.1.1).

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284 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.2.3.1.1 Example. Minimum cardinality Boolean.[91] [33, §4.3.4] [342](confer Example 4.5.1.4.1) Consider finding a minimum cardinality Booleansolution x to the classic linear algebra problem Ax = b given noiseless dataA∈Rm×n and b∈Rm ;

minimizex

‖x‖0

subject to Ax = b

xi ∈ 0, 1 , i=1 . . . n

(656)

where ‖x‖0 denotes cardinality of vector x (a.k.a, 0-norm; not a convexfunction).

A minimum cardinality solution answers the question: “Which fewestlinear combination of columns in A constructs vector b ?” Cardinalityproblems have extraordinarily wide appeal, arising in many fields of scienceand across many disciplines. [314] [210] [167] [166] Yet designing an efficientalgorithm to optimize cardinality has proved difficult. In this example, wealso constrain the variable to be Boolean. The Boolean constraint forcesan identical solution were the norm in problem (656) instead the 1-norm or2-norm; id est, the two problems

(656)

minimizex

‖x‖0

subject to Ax = b

xi ∈ 0, 1 , i=1 . . . n

=

minimizex

‖x‖1

subject to Ax = b

xi ∈ 0, 1 , i=1 . . . n

(657)

are the same. The Boolean constraint makes the 1-norm problem nonconvex.

Given data

A =

−1 1 8 1 1 0

−3 2 8 12

13

12− 1

3

−9 4 8 14

19

14− 1

9

, b =

11214

(658)

the obvious and desired solution to the problem posed,

x⋆ = e4 ∈ R6 (659)

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4.2. FRAMEWORK 285

has norm ‖x⋆‖2 =1 and minimum cardinality; the minimum number ofnonzero entries in vector x . The Matlab backslash command x=A\b ,for example, finds

xM

=

2128

05

128

090128

0

(660)

having norm ‖xM‖2 = 0.7044 . Coincidentally, x

Mis a 1-norm solution;

id est, an optimal solution to

minimizex

‖x‖1

subject to Ax = b(496)

The pseudoinverse solution (rounded)

xP

= A†b =

−0.0456−0.1881

0.06230.26680.3770

−0.1102

(661)

has least norm ‖xP‖2 =0.5165 ; id est, the optimal solution to (§E.0.1.0.1)

minimizex

‖x‖2

subject to Ax = b(662)

Certainly none of the traditional methods provide x⋆ = e4 (659) because, andin general, for Ax = b

∥arg inf ‖x‖2

2≤

∥arg inf ‖x‖1

2≤

∥arg inf ‖x‖0

2(663)

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286 CHAPTER 4. SEMIDEFINITE PROGRAMMING

We can reformulate this minimum cardinality Boolean problem (656) asa semidefinite program: First transform the variable

x , (x + 1)12

(664)

so xi∈−1, 1 ; equivalently,

minimizex

‖(x + 1)12‖0

subject to A(x + 1)12

= b

δ(xxT) = 1

(665)

where δ is the main-diagonal linear operator (§A.1). By assigning (§B.1)

G =

[

x1

]

[ xT 1 ]=

[

X xxT 1

]

,

[

xxT xxT 1

]

∈ Sn+1 (666)

problem (665) becomes equivalent to: (Theorem A.3.1.0.7)

minimizeX∈ S

n , x∈Rn

1Tx

subject to A(x + 1)12

= b

G =

[

X xxT 1

]

(º 0)

δ(X) = 1rank G = 1

(667)

where solution is confined to rank-1 vertices of the elliptope in Sn+1

(§5.9.1.0.1) by the rank constraint, the positive semidefiniteness, and theequality constraints δ(X)=1. The rank constraint makes this problemnonconvex; by removing it4.14 we get the semidefinite program

4.14Relaxed problem (668) can also be derived via Lagrange duality; it is a dual of adual program [sic ] to (667). [300] [59, §5, exer.5.39] [374, §IV] [146, §11.3.4] The relaxedproblem must therefore be convex having a larger feasible set; its optimal objective valuerepresents a generally loose lower bound (1646) on the optimal objective of problem (667).

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4.2. FRAMEWORK 287

minimizeX∈ S

n, x∈R

n1Tx

subject to A(x + 1)12

= b

G =

[

X xxT 1

]

º 0

δ(X) = 1

(668)

whose optimal solution x⋆ (664) is identical to that of minimum cardinalityBoolean problem (656) if and only if rank G⋆ =1. Hope4.15 of acquiring arank-1 solution is not ill-founded because 2n elliptope vertices have rank 1,and we are minimizing an affine function on a subset of the elliptope(Figure 128) containing rank-1 vertices; id est, by assumption that thefeasible set of minimum cardinality Boolean problem (656) is nonempty,a desired solution resides on the elliptope relative boundary at a rank-1vertex.4.16

For that data given in (658), our semidefinite program solver [381] [382](accurate in solution to approximately 1E-8)4.17 finds optimal solution to(668)

round(G⋆) =

1 1 1 −1 1 1 −11 1 1 −1 1 1 −11 1 1 −1 1 1 −1

−1 −1 −1 1 −1 −1 11 1 1 −1 1 1 −11 1 1 −1 1 1 −1

−1 −1 −1 1 −1 −1 1

(669)

near a rank-1 vertex of the elliptope in Sn+1 (§5.9.1.0.2); its sorted

4.15A more deterministic approach to constraining rank and cardinality is developed in§4.6.0.0.11.4.16Confinement to the elliptope can be regarded as a kind of normalization akin tomatrix A column normalization suggested in [117] and explored in Example 4.2.3.1.2.4.17A typically ignored limitation of interior-point solution methods is their relativeaccuracy of only about 1E-8 on a machine using 64-bit (double precision) floating-pointarithmetic; id est, optimal solution x⋆ cannot be more accurate than square root ofmachine epsilon (ǫ=2.2204E-16). Nonzero primal−dual objective difference is not a goodmeasure of solution accuracy.

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288 CHAPTER 4. SEMIDEFINITE PROGRAMMING

eigenvalues,

λ(G⋆) =

6.999999777990990.000000226872410.000000022502960.00000000262974

−0.00000000999738−0.00000000999875−0.00000001000000

(670)

Negative eigenvalues are undoubtedly finite-precision effects. Because thelargest eigenvalue predominates by many orders of magnitude, we can expectto find a good approximation to a minimum cardinality Boolean solution bytruncating all smaller eigenvalues. We find, indeed, the desired result (659)

x⋆ = round

0.000000001279470.000000005273690.000000001810010.999999974690440.000000014089500.00000000482903

= e4 (671)

2

4.2.3.1.2 Example. Optimization over elliptope versus 1-norm polyhedronfor minimum cardinality Boolean Example 4.2.3.1.1.

A minimum cardinality problem is typically formulated via, what is by now,the standard practice [117] [66, §3.2, §3.4] of column normalization applied toa 1-norm problem surrogate like (496). Suppose we define a diagonal matrix

Λ ,

‖A(: , 1)‖2 0‖A(: , 2)‖2

. . .

0 ‖A(: , 6)‖2

∈ S6 (672)

used to normalize the columns (assumed nonzero) of given noiseless datamatrix A . Then approximate the minimum cardinality Boolean problem

minimizex

‖x‖0

subject to Ax = b

xi ∈ 0, 1 , i=1 . . . n

(656)

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4.2. FRAMEWORK 289

as

minimizey

‖y‖1

subject to AΛ−1y = b1 º Λ−1y º 0

(673)

where optimal solution

y⋆ = round(Λ−1y⋆) (674)

The inequality in (673) relaxes Boolean constraint yi∈0, 1 from (656);bounding any solution y⋆ to a nonnegative unit hypercube whose vertices arebinary numbers. Convex problem (673) is justified by the convex envelope

cenv ‖x‖0 on x∈Rn | ‖x‖∞≤κ =1

κ‖x‖1 (1342)

Donoho concurs with this particular formulation, equivalently expressible asa linear program via (492).

Approximation (673) is therefore equivalent to minimization of an affinefunction (§3.3) on a bounded polyhedron, whereas semidefinite program

minimizeX∈ S

n, x∈R

n1Tx

subject to A(x + 1)12

= b

G =

[

X xxT 1

]

º 0

δ(X) = 1

(668)

minimizes an affine function on an intersection of the elliptope withhyperplanes. Although the same Boolean solution is obtained fromthis approximation (673) as compared with semidefinite program (668),when given that particular data from Example 4.2.3.1.1, Singer confides acounter-example: Instead, given data

A =

[

1 0 1√2

0 1 1√2

]

, b =

[

1

1

]

(675)

then solving approximation (673) yields

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290 CHAPTER 4. SEMIDEFINITE PROGRAMMING

y⋆ = round

1 − 1√2

1 − 1√2

1

=

0

0

1

(676)

(infeasible, with or without rounding, with respect to original problem (656))whereas solving semidefinite program (668) produces

round(G⋆) =

1 1 −1 11 1 −1 1

−1 −1 1 −11 1 −1 1

(677)

with sorted eigenvalues

λ(G⋆) =

3.999999650572640.00000035942736

−0.00000000000000−0.00000001000000

(678)

Truncating all but the largest eigenvalue, from (664) we obtain (confer y⋆)

x⋆ = round

0.999999996252990.999999996252990.00000001434518

=

110

(679)

the desired minimum cardinality Boolean result. 2

4.2.3.1.3 Exercise. Minimum cardinality Boolean art.Assess general performance of standard-practice approximation (673) ascompared with the proposed semidefinite program (668). H

4.2.3.1.4 Exercise. Conic independence.Matrix A from (658) is full-rank having three-dimensional nullspace. Findits four conically independent columns. (§2.10) To what part of proper coneK= Ax | xº 0 does vector b belong? H

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4.3. RANK REDUCTION 291

4.3 Rank reduction

. . . it is not clear generally how to predict rank X⋆ or rank S⋆

before solving the SDP problem.

−Farid Alizadeh (1995) [11, p.22]

The premise of rank reduction in semidefinite programming is: an optimalsolution found does not satisfy Barvinok’s upper bound (252) on rank. Theparticular numerical algorithm solving a semidefinite program may haveinstead returned a high-rank optimal solution (§4.1.1.1; e.g., (636)) whena lower-rank optimal solution was expected.

4.3.1 Posit a perturbation of X⋆

Recall from §4.1.1.2, there is an extreme point of A ∩ Sn+ (628) satisfying

upper bound (252) on rank. [24, §2.2] It is therefore sufficient to locatean extreme point of the intersection whose primal objective value (625P) isoptimal:4.18 [110, §31.5.3] [235, §2.4] [236] [7, §3] [288]

Consider again affine subset

A = X∈ Sn | A svec X = b (628)

where for Ai∈ Sn

A ,

svec(A1)T

...svec(Am)T

∈ Rm×n(n+1)/2 (626)

Given any optimal solution X⋆ to

minimizeX∈ S

n〈C , X 〉

subject to X ∈ A ∩ Sn+

(625P)

whose rank does not satisfy upper bound (252), we posit existence of a setof perturbations

4.18There is no known construction for Barvinok’s tighter result (257).−Monique Laurent (2004)

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292 CHAPTER 4. SEMIDEFINITE PROGRAMMING

tjBj | tj ∈ R , Bj ∈ Sn, j =1 . . . n (680)

such that, for some 0≤ i≤n and scalars tj , j =1 . . . i ,

X⋆ +i

j=1

tjBj (681)

becomes an extreme point of A ∩ Sn+ and remains an optimal solution of

(625P). Membership of (681) to affine subset A is secured for the ith

perturbation by demanding

〈Bi , Aj〉 = 0 , j =1 . . . m (682)

while membership to the positive semidefinite cone Sn+ is insured by small

perturbation (691). Feasibility of (681) is insured in this manner, optimalityis proved in §4.3.3.

The following simple algorithm has very low computational intensity andlocates an optimal extreme point, assuming a nontrivial solution:

4.3.1.0.1 Procedure. Rank reduction. [Wıκımization]initialize: Bi = 0 ∀ ifor iteration i=1...n

1. compute a nonzero perturbation matrix Bi of X⋆ +i−1∑

j=1

tjBj

2. maximize ti

subject to X⋆ +i

j=1

tjBj ∈ Sn+

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4.3. RANK REDUCTION 293

A rank-reduced optimal solution is then

X⋆ ← X⋆ +i

j=1

tjBj (683)

4.3.2 Perturbation form

Perturbations of X⋆ are independent of constants C ∈ Sn and b∈Rm inprimal and dual problems (625). Numerical accuracy of any rank-reducedresult, found by perturbation of an initial optimal solution X⋆, is thereforequite dependent upon initial accuracy of X⋆.

4.3.2.0.1 Definition. Matrix step function. (confer §A.6.5.0.1)Define the signum-like quasiconcave real function ψ : Sn→ R

ψ(Z ) ,

1 , Z º 0−1 , otherwise

(684)

The value −1 is taken for indefinite or nonzero negative semidefiniteargument.

Deza & Laurent [110, §31.5.3] prove: every perturbation matrix Bi ,i=1 . . . n , is of the form

Bi = −ψ(Zi)RiZiRTi ∈ Sn (685)

where

X⋆ , R1RT1 , X⋆ +

i−1∑

j=1

tjBj , RiRTi ∈ Sn (686)

where the tj are scalars and Ri∈ Rn×ρ is full-rank and skinny where

ρ , rank

(

X⋆ +i−1∑

j=1

tjBj

)

(687)

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294 CHAPTER 4. SEMIDEFINITE PROGRAMMING

and where matrix Zi∈ Sρ is found at each iteration i by solving a verysimple feasibility problem: 4.19

find Zi∈ Sρ

subject to 〈Zi , RTi AjRi〉 = 0 , j =1 . . . m

(688)

Were there a sparsity pattern common to each member of the setRT

i AjRi∈ Sρ, j =1 . . . m , then a good choice for Zi has 1 in each entrycorresponding to a 0 in the pattern; id est, a sparsity pattern complement.At iteration i

X⋆ +i−1∑

j=1

tjBj + tiBi = Ri(I − tiψ(Zi)Zi)RTi (689)

By fact (1426), therefore

X⋆ +i−1∑

j=1

tjBj + tiBi º 0 ⇔ 1 − tiψ(Zi)λ(Zi) º 0 (690)

where λ(Zi)∈ Rρ denotes the eigenvalues of Zi .

Maximization of each ti in step 2 of the Procedure reduces rank of (689)and locates a new point on the boundary ∂(A ∩ Sn

+) .4.20 Maximization ofti thereby has closed form;

4.19A simple method of solution is closed-form projection of a random nonzero point onthat proper subspace of isometrically isomorphic Rρ(ρ+1)/2 specified by the constraints.(§E.5.0.0.6) Such a solution is nontrivial assuming the specified intersection of hyperplanesis not the origin; guaranteed by ρ(ρ + 1)/2 > m . Indeed, this geometric intuition aboutforming the perturbation is what bounds any solution’s rank from below; m is fixed bythe number of equality constraints in (625P) while rank ρ decreases with each iteration i .Otherwise, we might iterate indefinitely.4.20This holds because rank of a positive semidefinite matrix in Sn is diminished belown by the number of its 0 eigenvalues (1436), and because a positive semidefinite matrixhaving one or more 0 eigenvalues corresponds to a point on the PSD cone boundary (184).Necessity and sufficiency are due to the facts: Ri can be completed to a nonsingular matrix(§A.3.1.0.5), and I − ti ψ(Zi)Zi can be padded with zeros while maintaining equivalencein (689).

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4.3. RANK REDUCTION 295

(t⋆i )−1 = max ψ(Zi)λ(Zi)j , j =1 . . . ρ (691)

When Zi is indefinite, direction of perturbation (determined by ψ(Zi)) isarbitrary. We may take an early exit from the Procedure were Zi to become0 or were

rank[

svec RTi A1Ri svec RT

i A2Ri · · · svec RTi AmRi

]

= ρ(ρ + 1)/2 (692)

which characterizes rank ρ of any [sic] extreme point in A ∩ Sn+ . [235, §2.4]

[236]

Proof. Assuming the form of every perturbation matrix is indeed (685),then by (688)

svec Zi ⊥[

svec(RTi A1Ri) svec(RT

i A2Ri) · · · svec(RTi AmRi)

]

(693)

By orthogonal complement we have

rank[

svec(RTi A1Ri) · · · svec(RT

i AmRi)]⊥

+ rank[

svec(RTi A1Ri) · · · svec(RT

i AmRi)]

= ρ(ρ + 1)/2(694)

When Zi can only be 0, then the perturbation is null because an extremepoint has been found; thus

[

svec(RTi A1Ri) · · · svec(RT

i AmRi)]⊥

= 0 (695)

from which the stated result (692) directly follows. ¨

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296 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.3.3 Optimality of perturbed X⋆

We show that the optimal objective value is unaltered by perturbation (685);id est,

〈C , X⋆ +i

j=1

tjBj〉 = 〈C , X⋆〉 (696)

Proof. From Corollary 4.2.3.0.1 we have the necessary and sufficientrelationship between optimal primal and dual solutions under assumption ofnonempty primal feasible cone interior A ∩ int Sn

+ :

S⋆X⋆ = S⋆R1RT1 = X⋆S⋆ = R1R

T1 S⋆ = 0 (697)

This means R(R1) ⊆ N (S⋆) and R(S⋆) ⊆ N (RT1 ). From (686) and (689)

we get the sequence:

X⋆ = R1RT1

X⋆ + t1B1 = R2RT2 = R1(I − t1ψ(Z1)Z1)R

T1

X⋆ + t1B1 + t2B2 = R3RT3 = R2(I − t2ψ(Z2)Z2)R

T2 = R1(I − t1ψ(Z1)Z1)(I − t2ψ(Z2)Z2)R

T1

...

X⋆ +i

j=1

tjBj = R1

(

i∏

j=1

(I − tj ψ(Zj)Zj)

)

RT1 (698)

Substituting C = svec−1(ATy⋆) + S⋆ from (625),

〈C , X⋆ +i

j=1

tjBj〉 =

svec−1(ATy⋆) + S⋆ , R1

(

i∏

j=1

(I − tj ψ(Zj)Zj)

)

RT1

=

m∑

k=1

y⋆kAk , X⋆ +

i∑

j=1

tjBj

=

m∑

k=1

y⋆kAk + S⋆ , X⋆

= 〈C , X⋆〉 (699)

because 〈Bi , Aj〉=0 ∀ i , j by design (682). ¨

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4.3. RANK REDUCTION 297

4.3.3.0.1 Example. Aδ(X) = b .This academic example demonstrates that a solution found by rank reductioncan certainly have rank less than Barvinok’s upper bound (252): Assume agiven vector b∈Rm belongs to the conic hull of columns of a given matrixA∈Rm×n ;

A =

−1 1 8 1 1

−3 2 8 12

13

−9 4 8 14

19

, b =

11214

(700)

Consider the convex optimization problem

minimizeX∈ S5

tr X

subject to X º 0

Aδ(X) = b

(701)

that minimizes the 1-norm of the main diagonal; id est, problem (701) is thesame as

minimizeX∈ S5

‖δ(X)‖1

subject to X º 0

Aδ(X) = b

(702)

that finds a solution to Aδ(X)= b . Rank-3 solution X⋆ = δ(xM

) is optimal,where (confer (660))

xM

=

2128

05

128

090128

(703)

Yet upper bound (252) predicts existence of at most a

rank-

(⌊√

8m + 1 − 1

2

= 2

)

(704)

feasible solution from m = 3 equality constraints. To find a lowerrank ρ optimal solution to (701) (barring combinatorics), we invokeProcedure 4.3.1.0.1:

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298 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Initialize:

C = I , ρ=3, Aj , δ(A(j, :)) , j =1, 2, 3, X⋆ = δ(xM

) , m=3, n=5.

Iteration i=1:

Step 1: R1 =

2128

0 0

0 0 0

0√

5128

0

0 0 0

0 0√

90128

.

find Z1∈ S3

subject to 〈Z1 , RT1AjR1〉 = 0 , j =1, 2, 3

(705)

A nonzero randomly selected matrix Z1 , having 0 main diagonal,is a solution yielding nonzero perturbation matrix B1 . Choosearbitrarily

Z1 = 11T− I ∈ S3 (706)

then (rounding)

B1 =

0 0 0.0247 0 0.10480 0 0 0 0

0.0247 0 0 0 0.16570 0 0 0 0

0.1048 0 0.1657 0 0

(707)

Step 2: t⋆1 = 1 because λ(Z1)= [−1 −1 2 ]T. So,

X⋆ ← δ(xM

) + B1 =

2128

0 0.0247 0 0.10480 0 0 0 0

0.0247 0 5128

0 0.16570 0 0 0 0

0.1048 0 0.1657 0 90128

(708)

has rank ρ←1 and produces the same optimal objective value.

2

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4.3. RANK REDUCTION 299

4.3.3.0.2 Exercise. Rank reduction of maximal complementarity.Apply rank reduction Procedure 4.3.1.0.1 to the maximal complementarityexample (§4.1.1.4.1). Demonstrate a rank-1 solution; which can certainly befound (by Barvinok’s Proposition 2.9.3.0.1) because there is only one equalityconstraint. H

4.3.4 thoughts regarding rank reduction

Because rank reduction Procedure 4.3.1.0.1 is guaranteed only to produceanother optimal solution conforming to Barvinok’s upper bound (252), theProcedure will not necessarily produce solutions of arbitrarily low rank; but ifthey exist, the Procedure can. Arbitrariness of search direction when matrixZi becomes indefinite, mentioned on page 295, and the enormity of choicesfor Zi (688) are liabilities for this algorithm.

4.3.4.1 inequality constraints

The question naturally arises: what to do when a semidefinite program (notin prototypical form (625))4.21 has linear inequality constraints of the form

αTi svec X¹ βi , i = 1 . . . k (709)

where βi are given scalars and αi are given vectors. One expedient wayto handle this circumstance is to convert the inequality constraints to equalityconstraints by introducing a slack variable γ ; id est,

αTi svec X + γi = βi , i = 1 . . . k , γ º 0 (710)

thereby converting the problem to prototypical form.Alternatively, we say the ith inequality constraint is active when it is

met with equality; id est, when for particular i in (709), αTi svec X⋆ = βi .

An optimal high-rank solution X⋆ is, of course, feasible (satisfying all theconstraints). But for the purpose of rank reduction, inactive inequalityconstraints are ignored while active inequality constraints are interpreted as

4.21Contemporary numerical packages for solving semidefinite programs can solve a rangeof problems wider than prototype (625). Generally, they do so by transforming a givenproblem into prototypical form by introducing new constraints and variables. [11] [382]We are momentarily considering a departure from the primal prototype that augments theconstraint set with linear inequalities.

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300 CHAPTER 4. SEMIDEFINITE PROGRAMMING

equality constraints. In other words, we take the union of active inequalityconstraints (as equalities) with equality constraints A svec X = b to forma composite affine subset A substituting for (628). Then we proceed withrank reduction of X⋆ as though the semidefinite program were in prototypicalform (625P).

4.4 Rank-constrained semidefinite program

We generalize the trace heuristic (§7.2.2.1), for finding low-rank optimalsolutions to SDPs of a more general form:

4.4.1 rank constraint by convex iteration

Consider a semidefinite feasibility problem of the form

findG∈SN

G

subject to G ∈ CG º 0rank G ≤ n

(711)

where C is a convex set presumed to contain positive semidefinite matricesof rank n or less; id est, C intersects the positive semidefinite cone boundary.We propose that this rank-constrained feasibility problem can be equivalentlyexpressed as the problem sequence (712) (1663a) having convex constraints:

minimizeG∈SN

〈G , W 〉subject to G ∈ C

G º 0

(712)

where direction vector 4.22 W is an optimal solution to semidefinite program,for 0≤n≤N−1

N∑

i=n+1

λ(G⋆)i = minimizeW∈ SN

〈G⋆, W 〉subject to 0 ¹ W ¹ I

tr W = N − n

(1663a)

4.22Search direction W is a hyperplane-normal pointing opposite to direction of movementdescribing minimization of a real linear function 〈G , W 〉 (p.75).

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 301

whose feasible set is a Fantope (§2.3.2.0.1), and where G⋆ is an optimalsolution to problem (712) given some iterate W . The idea is to iteratesolution of (712) and (1663a) until convergence as defined in §4.4.1.2: 4.23

N∑

i=n+1

λ(G⋆)i = 〈G⋆, W ⋆〉 = λ(G⋆)Tλ(W ⋆) , 0 (713)

defines global convergence of the iteration; a vanishing objective that is acertificate of global optimality but cannot be guaranteed. Optimal directionvector W ⋆ is defined as any positive semidefinite matrix yielding optimalsolution G⋆ of rank n or less to then convex equivalent (712) of feasibilityproblem (711):

(711)

findG∈SN

G

subject to G ∈ CG º 0rank G ≤ n

≡minimize

G∈SN〈G , W ⋆〉

subject to G ∈ CG º 0

(712)

id est, any direction vector for which the last N− n nonincreasingly orderedeigenvalues λ of G⋆ are zero. In any semidefinite feasibility problem, asolution of lowest rank must be an extreme point of the feasible set. Thenthere must exist a linear objective function such that this lowest-rank feasiblesolution optimizes the resultant SDP. [235, §2.4] [236]

We emphasize that convex problem (712) is not a relaxation ofrank-constrained feasibility problem (711); at global convergence, convexiteration (712) (1663a) makes it instead an equivalent problem.4.24

4.4.1.1 direction matrix interpretation

(confer §4.5.1.1) The feasible set of direction matrices in (1663a) is the convexhull of outer product of all rank-(N− n) orthonormal matrices; videlicet,

conv

UUT | U ∈ RN×N−n, UTU = I

=

A∈ SN | I º A º 0 , 〈I , A 〉= N− n

(90)

4.23Proposed iteration is neither dual projection (Figure 162) or alternating projection(Figure 164). Sum of eigenvalues follows from results of Ky Fan (page 646). Innerproduct of eigenvalues follows from (1556) and properties of commutative matrix products(page 595).4.24Terminology equivalent problem meaning, optimal solution to one problem can bederived from optimal solution to another. Terminology same problem means: optimalsolution set for one problem is identical to the optimal solution set of another (withouttransformation).

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302 CHAPTER 4. SEMIDEFINITE PROGRAMMING

This set (92), argument to conv , comprises the extreme points of thisFantope (90). An optimal solution W to (1663a), that is an extreme point,is known in closed form (p.646): Given ordered diagonalizationG⋆ = QΛQT∈ SN (§A.5.1), then direction matrix W = U⋆U⋆T is optimal andextreme where U⋆ = Q(: , n+1:N)∈RN×N−n. Eigenvalue vector λ(W ) has 1in each entry corresponding to the N− n smallest entries of δ(Λ) and has 0elsewhere. By (212) (214), polar direction −W can be regarded as pointingtoward the set of all rank-n (or less) positive semidefinite matrices whosenullspace contains that of G⋆. For that particular closed-form solution W ,consequently

N∑

i=n+1

λ(G⋆)i = 〈G⋆, W 〉 = λ(G⋆)Tλ(W ) (714)

This is the connection to cardinality minimization of vectors;4.25 id est,eigenvalue λ cardinality (rank) is analogous to vector x cardinality via (736).

So as not to be misconstrued under closed-form solution W to (1663a):Define (confer (212))

Sn , (I−W )G(I−W ) |G∈ SN = X∈ SN | N (X) ⊇ N (G⋆) (715)

as the symmetric subspace of rank≤n matrices whose nullspace containsN (G⋆). Then projection of G⋆ on Sn is (I−W )G⋆(I−W ). (§E.7)Direction of projection is −WG⋆W . (Figure 82) tr(WG⋆W ) is ameasure of proximity to Sn because its orthogonal complement isS⊥

n = WGW |G∈ SN ; the point being, convex iteration incorporatingconstrained tr(WGW ) = 〈G , W 〉 minimization is not a projection method:certainly, not on these two subspaces.

Closed-form solution W to problem (1663a), though efficient, comes witha caveat : there exist cases where this projection matrix solution W does notprovide the most efficient route to an optimal rank-n solution G⋆ ; id est,direction W is not unique. So we sometimes choose to solve (1663a) insteadof employing a known closed-form solution.

4.25 not trace minimization of a nonnegative diagonal matrix δ(x) as in [132, §1] [296, §2].To make rank-constrained problem (711) resemble cardinality problem (508), we couldmake C an affine subset:

find X∈ SN

subject to A svec X = bX º 0rankX ≤ n

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 303

Sn

S⊥n

(I−W )G⋆(I−W )

WG⋆W

G⋆

Figure 82: Projection on subspace Sn of rank≤n matrices whose nullspacecontains N (G⋆). This direction W is closed-form solution to (1663a).

When direction matrix W = I , as in the trace heuristic for example,then −W points directly at the origin (the rank-0 PSD matrix, Figure 83).Vector inner-product of an optimization variable with direction matrix Wis therefore a generalization of the trace heuristic (§7.2.2.1) for rankminimization; −W is instead trained toward the boundary of the positivesemidefinite cone.

4.4.1.2 convergence

We study convergence to ascertain conditions under which a direction matrixwill reveal a feasible solution G , of rank n or less, to semidefinite program(712). Denote by W ⋆ a particular optimal direction matrix from semidefiniteprogram (1663a) such that (713) holds (feasible rank G≤n found). Then wedefine global convergence of the iteration (712) (1663a) to correspond withthis vanishing vector inner-product (713) of optimal solutions.

Because this iterative technique for constraining rank is not a projectionmethod, it can find a rank-n solution G⋆ ((713) will be satisfied) only if atleast one exists in the feasible set of program (712).

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304 CHAPTER 4. SEMIDEFINITE PROGRAMMING

I

0S2

+

∂H = G | 〈G , I 〉 = κ

Figure 83: Trace heuristic can be interpreted as minimization of a hyperplane,with normal I , over positive semidefinite cone drawn here in isometricallyisomorphic R3. Polar of direction vector W = I points toward origin.

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 305

4.4.1.2.1 Proof. Suppose 〈G⋆, W 〉= τ is satisfied for some nonnegativeconstant τ after any particular iteration (712) (1663a) of the twominimization problems. Once a particular value of τ is achieved, it can neverbe exceeded by subsequent iterations because existence of feasible G and Whaving that vector inner-product τ has been established simultaneously ineach problem. Because the infimum of vector inner-product of two positivesemidefinite matrix variables is zero, the nonincreasing sequence of iterationsis thus bounded below hence convergent because any bounded monotonicsequence in R is convergent. [254, §1.2] [41, §1.1] Local convergence to somenonnegative objective value τ is thereby established. ¨

Local convergence, in this context, means convergence to a fixed point ofpossibly infeasible rank. Only local convergence can be established becauseobjective 〈G , W 〉 , when instead regarded simultaneously in two variables(G , W ) , is generally multimodal. (§3.9.0.0.3)

Local convergence, convergence to τ 6= 0 and definition of a stall, neverimplies nonexistence of a rank-n feasible solution to (712). A nonexistentrank-n feasible solution would mean certain failure to converge globally bydefinition (713) (convergence to τ 6= 0) but, as proved, convex iteration alwaysconverges locally if not globally.

When a rank-n feasible solution to (712) exists, it remains an openproblem to state conditions under which 〈G⋆, W ⋆〉= τ =0 (713) is achievedby iterative solution of semidefinite programs (712) and (1663a). Thenrank G⋆≤ n and pair (G⋆, W ⋆) becomes a globally optimal fixed point ofiteration. There can be no proof of global convergence because of theimplicit high-dimensional multimodal manifold in variables (G , W ). Whenstall occurs, direction vector W can be manipulated to steer out; e.g.,reversal of search direction as in Example 4.6.0.0.1, or reinitialization to arandom rank-(N− n) matrix in the same positive semidefinite cone face(§2.9.2.3) demanded by the current iterate: given ordered diagonalizationG⋆ = QΛQT∈ SN , then W = U⋆ΦU⋆T where U⋆ = Q(: , n+1:N)∈RN×N−n

and where eigenvalue vector λ(W )1:N−n = λ(Φ) has nonnegative uniformlydistributed random entries by selection of Φ∈SN−n

+ while λ(W )N−n+1:N = 0.When this direction works, rank w.r.t iteration tends to be noisily monotonic.Zero eigenvalues act as memory while randomness largely reduces likelihoodof stall.

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306 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.4.1.2.2 Exercise. Completely positive semidefinite matrix. [39]

Given rank-2 positive semidefinite matrix G =

0.50 0.55 0.200.55 0.61 0.220.20 0.22 0.08

, find a

positive factorization G = XTX (876) by solving

findX∈R2×3

X ≥ 0

subject to Z =

[

I XXT G

]

º 0

rank Z ≤ 2

(716)

via convex iteration. H

4.4.1.2.3 Exercise. Nonnegative matrix factorization.

Given rank-2 nonnegative matrix X =

17 28 4216 47 5117 82 72

, find a nonnegative

factorizationX = WH (717)

by solving

findA∈S3, B∈S3, W∈R3×2, H∈R2×3

W , H

subject to Z =

I WT HW A XHT XT B

º 0

W ≥ 0H ≥ 0rank Z ≤ 2

(718)

which follows from the fact, at optimality,

Z⋆ =

IWHT

[ I WT H ](719)

Use the known closed-form solution for a direction vector Y to regulate rankby convex iteration; set Z⋆ = QΛQT∈ S8 to an ordered diagonalization andU⋆ = Q(: , 3:8)∈ R8×6, then Y = U⋆U⋆T (§4.4.1.1).

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 307

In summary, initialize Y then iterate numerical solution of (convex)semidefinite program

minimizeA∈S3, B∈S3, W∈R3×2, H∈R2×3

〈Z , Y 〉

subject to Z =

I WT HW A XHT XT B

º 0

W ≥ 0H ≥ 0

(720)

with Y = U⋆U⋆T until convergence (which is global and occurs in very fewiterations for this instance). H

Now, an application to optimal regulation of affine dimension:

4.4.1.2.4 Example. Sensor-Network Localization or Wireless Location.Heuristic solution to a sensor-network localization problem, proposedby Carter, Jin, Saunders, & Ye in [70],4.26 is limited to two Euclideandimensions and applies semidefinite programming (SDP) to littlesubproblems. There, a large network is partitioned into smaller subnetworks(as small as one sensor − a mobile point, whereabouts unknown) and thensemidefinite programming and heuristics called spaseloc are applied tolocalize each and every partition by two-dimensional distance geometry.Their partitioning procedure is one-pass, yet termed iterative; a termapplicable only in so far as adjoining partitions can share localized sensorsand anchors (absolute sensor positions known a priori). But there isno iteration on the entire network, hence the term “iterative” is perhapsinappropriate. As partitions are selected based on “rule sets” (heuristics, notgeographics), they also term the partitioning adaptive. But no adaptation ofa partition actually occurs once it has been determined.

One can reasonably argue that semidefinite programming methods areunnecessary for localization of small partitions of large sensor networks.[274] [84] In the past, these nonlinear localization problems were solvedalgebraically and computed by least squares solution to hyperbolic equations;

4.26The paper constitutes Jin’s dissertation for University of Toronto [212] although hername appears as second author. Ye’s authorship is honorary.

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308 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Figure 84: Sensor-network localization in R2, illustrating connectivity andcircular radio-range per sensor. Smaller dark grey regions each hold ananchor at their center; known fixed sensor positions. Sensor/anchor distanceis measurable with negligible uncertainty for sensor within those grey regions.Graphic by Geoff Merrett.

called multilateration.4.27 [224] [262] Indeed, practical contemporarynumerical methods for global positioning (GPS) by satellite do not rely onconvex optimization. [287]

Modern distance geometry is inextricably melded with semidefiniteprogramming. The beauty of semidefinite programming, as relates tolocalization, lies in convex expression of classical multilateration: So & Yeshowed [316] that the problem of finding unique solution, to a noiselessnonlinear system describing the common point of intersection of hyperspheresin real Euclidean vector space, can be expressed as a semidefinite program

4.27Multilateration − literally, having many sides; shape of a geometric figure formed bynearly intersecting lines of position. In navigation systems, therefore: Obtaining a fix frommultiple lines of position. Multilateration can be regarded as noisy trilateration.

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 309

via distance geometry.

But the need for SDP methods in Carter & Jin et alii is enigmatic fortwo more reasons: 1) guessing solution to a partition whose intersensormeasurement data or connectivity is inadequate for localization by distancegeometry, 2) reliance on complicated and extensive heuristics for partitioninga large network that could instead be efficiently solved whole by onesemidefinite program [220, §3]. While partitions range in size between 2 and10 sensors, 5 sensors optimally, heuristics provided are only for two spatialdimensions (no higher-dimensional heuristics are proposed). For these smallnumbers it remains unclarified as to precisely what advantage is gained overtraditional least squares: it is difficult to determine what part of their noiseperformance is attributable to SDP and what part is attributable to theirheuristic geometry.

Partitioning of large sensor networks is a compromise to rapid growthof SDP computational intensity with problem size. But when impact ofnoise on distance measurement is of most concern, one is averse to apartitioning scheme because noise-effects vary inversely with problem size.[51, §2.2] (§5.13.2) Since an individual partition’s solution is not iteratedin Carter & Jin and is interdependent with adjoining partitions, we expecterrors to propagate from one partition to the next; the ultimate partitionsolved, expected to suffer most.

Heuristics often fail on real-world data because of unanticipatedcircumstances. When heuristics fail, generally they are repaired byadding more heuristics. Tenuous is any presumption, for example, thatdistance measurement errors have distribution characterized by circularcontours of equal probability about an unknown sensor-location. (Figure 84)That presumption effectively appears within Carter & Jin’s optimizationproblem statement as affine equality constraints relating unknowns todistance measurements that are corrupted by noise. Yet in most allurban environments, this measurement noise is more aptly characterized byellipsoids of varying orientation and eccentricity as one recedes from a sensor.(Figure 124) Each unknown sensor must therefore instead be bound to itsown particular range of distance, primarily determined by the terrain.4.28

The nonconvex problem we must instead solve is:

4.28A distinct contour map corresponding to each anchor is required in practice.

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310 CHAPTER 4. SEMIDEFINITE PROGRAMMING

findi , j ∈ I

xi , xjsubject to dij ≤ ‖xi − xj‖2 ≤ dij

(721)

where xi represents sensor location, and where dij and dij respectively

represent lower and upper bounds on measured distance-square from ith toj th sensor (or from sensor to anchor). Figure 89 illustrates contours of equalsensor-location uncertainty. By establishing these individual upper and lowerbounds, orientation and eccentricity can effectively be incorporated into theproblem statement.

Generally speaking, there can be no unique solution to the sensor-networklocalization problem because there is no unique formulation; that is the art ofOptimization. Any optimal solution obtained depends on whether or how anetwork is partitioned, whether distance data is complete, presence of noise,and how the problem is formulated. When a particular formulation is aconvex optimization problem, then the set of all optimal solutions forms aconvex set containing the actual or true localization. Measurement noiseprecludes equality constraints representing distance. The optimal solutionset is consequently expanded; necessitated by introduction of distanceinequalities admitting more and higher-rank solutions. Even were the optimalsolution set a single point, it is not necessarily the true localization becausethere is little hope of exact localization by any algorithm once significantnoise is introduced.

Carter & Jin gauge performance of their heuristics to the SDP formulationof author Biswas whom they regard as vanguard to the art. [14, §1] Biswasposed localization as an optimization problem minimizing a distance measure.[46] [44] Intuitively, minimization of any distance measure yields compactedsolutions; (confer §6.4.0.0.1) precisely the anomaly motivating Carter & Jin.Their two-dimensional heuristics outperformed Biswas’ localizations bothin execution-time and proximity to the desired result. Perhaps, instead ofheuristics, Biswas’ approach to localization can be improved: [43] [45].

The sensor-network localization problem is considered difficult. [14, §2]Rank constraints in optimization are considered more difficult. Control ofaffine dimension in Carter & Jin is suboptimal because of implicit projectionon R2. In what follows, we present the localization problem as a semidefiniteprogram (equivalent to (721)) having an explicit rank constraint whichcontrols affine dimension of an optimal solution. We show how to achievethat rank constraint only if the feasible set contains a matrix of desired rank.

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 311

23

41

Figure 85: 2-lattice in R2, hand-drawn. Nodes 3 and 4 are anchors;remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

Our problem formulation is extensible to any spatial dimension.

proposed standardized test

Jin proposes an academic test in two-dimensional real Euclidean space R2

that we adopt. In essence, this test is a localization of sensors andanchors arranged in a regular triangular lattice. Lattice connectivity issolely determined by sensor radio range; a connectivity graph is assumedincomplete. In the interest of test standardization, we propose adoptionof a few small examples: Figure 85 through Figure 88 and their particularconnectivity represented by matrices (722) through (725) respectively.

0 • ? •• 0 • •? • 0 • • 0

(722)

Matrix entries dot • indicate measurable distance between nodes whileunknown distance is denoted by ? (question mark). Matrix entries hollowdot represent known distance between anchors (to high accuracy) whilezero distance is denoted 0. Because measured distances are quite unreliablein practice, our solution to the localization problem substitutes a distinct

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312 CHAPTER 4. SEMIDEFINITE PROGRAMMING

7

8

91 2

3 4

5 6

Figure 86: 3-lattice in R2, hand-drawn. Nodes 7, 8, and 9 are anchors;remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

range of possible distance for each measurable distance; equality constraintsexist only for anchors.

Anchors are chosen so as to increase difficulty for algorithms dependenton existence of sensors in their convex hull. The challenge is to find a solutionin two dimensions close to the true sensor positions given incomplete noisyintersensor distance information.

0 • • ? • ? ? • •• 0 • • ? • ? • •• • 0 • • • • • •? • • 0 ? • • • •• ? • ? 0 • • • •? • • • • 0 • • •? ? • • • • 0 • • • • • • 0 • • • • • • 0

(723)

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 313

2

9

4

8

10

147

3161

15 5 6

13 11 12

Figure 87: 4-lattice in R2, hand-drawn. Nodes 13, 14, 15, and 16 are anchors;remaining nodes are sensors. Radio range of sensor 1 indicated by arc.

0 ? ? • ? ? • ? ? ? ? ? ? ? • •? 0 • • • • ? • ? ? ? ? ? • • •? • 0 ? • • ? ? • ? ? ? ? ? • •• • ? 0 • ? • • ? • ? ? • • • •? • • • 0 • ? • • ? • • • • • •? • • ? • 0 ? • • ? • • ? ? ? ?• ? ? • ? ? 0 ? ? • ? ? • • • •? • ? • • • ? 0 • • • • • • • •? ? • ? • • ? • 0 ? • • • ? • ?? ? ? • ? ? • • ? 0 • ? • • • ?? ? ? ? • • ? • • • 0 • • • • ?? ? ? ? • • ? • • ? • 0 ? ? ? ?? ? ? • • ? • • • • • ? 0 ? • ? • • ? • • ? • • ? 0 • • • • • ? • • • • • ? 0 • • • • • ? • • ? ? ? ? 0

(724)

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314 CHAPTER 4. SEMIDEFINITE PROGRAMMING

3

16

25

15

17

22

9

421

13 14

18 21 19 20

5 6 24 7 8

10 23 11 12

Figure 88: 5-lattice in R2. Nodes 21 through 25 are anchors.

0 • ? ? • • ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?• 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? • •? ? 0 • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? ? • • •? ? • 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? • ?• • ? ? 0 • ? ? • • ? ? • • ? ? • ? ? ? ? ? • ? •• • • ? • 0 • ? • • • ? ? • ? ? ? ? ? ? ? ? • • •? ? • • ? • 0 • ? ? • • ? ? • • ? ? ? ? ? ? • • •? ? • • ? ? • 0 ? ? • • ? ? • • ? ? ? ? ? ? ? • ?• ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ? ? ? ? ?? • ? ? • • ? ? • 0 • ? • • ? ? ? • ? ? • • • • •? ? • ? ? • • • ? • 0 • ? • • • ? ? • ? ? • • • •? ? • • ? ? • • ? ? • 0 ? ? • • ? ? • • ? • • • ?? ? ? ? • ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ?? ? ? ? • • ? ? • • • ? • 0 • ? • • • ? • • • • ?? ? ? ? ? ? • • ? ? • • ? • 0 • ? ? • • • • • • ?? ? ? ? ? ? • • ? ? • • ? ? • 0 ? ? • • ? • ? ? ?? ? ? ? • ? ? ? • ? ? ? • • ? ? 0 • ? ? • ? ? ? ?? ? ? ? ? ? ? ? • • ? ? • • ? ? • 0 • ? • • • ? ?? ? ? ? ? ? ? ? ? ? • • ? • • • ? • 0 • • • • ? ?? ? ? ? ? ? ? ? ? ? ? • ? ? • • ? ? • 0 • • ? ? ?? ? ? ? ? ? ? ? ? • ? ? • • • ? • • • • 0 ? ? ? ? ? ? ? ? ? • • • • • • • ? • • • 0 ? ? • ? • • • ? ? • • • ? • • ? ? • • ? 0 ? • • • ? • • • ? • • • ? • • ? ? ? ? ? 0 ? • • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? 0

(725)

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 315

MAR

KET S

t.

Figure 89: Location uncertainty ellipsoid in R2 for each of 15 sensors • withinthree city blocks in downtown San Francisco. Data by Polaris Wireless. [319]

problem statement

Ascribe points in a list xℓ ∈ Rn, ℓ=1 . . . N to the columns of a matrix X ;

X = [ x1 · · · xN ] ∈ Rn×N (76)

where N is regarded as cardinality of list X . Positive semidefinite matrixXTX , formed from inner product of the list, is a Gram matrix ; [246, §3.6]

G = XTX =

‖x1‖2 xT1x2 xT

1x3 · · · xT1xN

xT2x1 ‖x2‖2 xT

2x3 · · · xT2xN

xT3x1 xT

3x2 ‖x3‖2 . . . xT3xN

......

. . . . . ....

xTNx1 xT

Nx2 xTNx3 · · · ‖xN‖2

∈ SN+ (876)

where SN+ is the convex cone of N ×N positive semidefinite matrices in the

symmetric matrix subspace SN .Existence of noise precludes measured distance from the input data. We

instead assign measured distance to a range estimate specified by individualupper and lower bounds: dij is an upper bound on distance-square from ith

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316 CHAPTER 4. SEMIDEFINITE PROGRAMMING

to j th sensor, while dij is a lower bound. These bounds become the inputdata. Each measurement range is presumed different from the others becauseof measurement uncertainty; e.g., Figure 89.

Our mathematical treatment of anchors and sensors is notdichotomized.4.29 A sensor position that is known a priori to highaccuracy (with absolute certainty) xi is called an anchor. Then thesensor-network localization problem (721) can be expressed equivalently:Given a number of anchors m , and I a set of indices (corresponding to allmeasurable distances • ), for 0 < n < N

findG∈SN , X∈Rn×N

X

subject to dij ≤ 〈G , (ei − ej)(ei − ej)T〉 ≤ dij ∀(i, j)∈ I

〈G , eieTi 〉 = ‖xi‖2 , i = N−m + 1 . . . N

〈G , (eieTj + eje

Ti )/2〉 = xT

i xj , i < j , ∀ i, j∈N−m + 1 . . . NX(: , N−m + 1:N) = [ xN−m+1 · · · xN ]

Z =

[

I XXT G

]

º 0

rank Z = n(726)

where ei is the ith member of the standard basis for RN . Distance-square

dij = ‖xi − xj‖22 = 〈xi − xj , xi − xj〉 (863)

is related to Gram matrix entries G, [gij] by vector inner-product

dij = gii + gjj − 2gij

= 〈G , (ei − ej)(ei − ej)T〉 = tr(GT(ei − ej)(ei − ej)

T)(878)

hence the scalar inequalities. Each linear equality constraint in G∈ SN

represents a hyperplane in isometrically isomorphic Euclidean vector spaceRN(N+1)/2, while each linear inequality pair represents a convex Euclideanbody known as slab (an intersection of two parallel but opposing halfspaces,Figure 11). By Schur complement (§A.4), any solution G , X providescomparison with respect to the positive semidefinite cone

G º XTX (913)

4.29Wireless location problem thus stated identically; difference being: fewer sensors.

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 317

which is a convex relaxation of the desired equality constraint

[

I XXT G

]

=

[

IXT

]

[ I X ](914)

The rank constraint insures this equality holds, by Theorem A.4.0.0.5, thusrestricting solution to Rn. Assuming full-rank solution (list) X

rank Z = rank G = rank X (727)

convex equivalent problem statement

Problem statement (726) is nonconvex because of the rank constraint. Wedo not eliminate or ignore the rank constraint; rather, we find a convex wayto enforce it: for 0 < n < N

minimizeG∈SN , X∈Rn×N

〈Z , W 〉subject to dij ≤ 〈G , (ei − ej)(ei − ej)

T〉 ≤ dij ∀(i, j)∈ I〈G , eie

Ti 〉 = ‖xi‖2 , i = N−m + 1 . . . N

〈G , (eieTj + eje

Ti )/2〉 = xT

i xj , i < j , ∀ i, j∈N−m + 1 . . . NX(: , N−m + 1:N) = [ xN−m+1 · · · xN ]

Z =

[

I XXT G

]

º 0 (728)

Convex function tr Z is a well-known heuristic whose sole purpose is torepresent convex envelope of rankZ . (§7.2.2.1) In this convex optimizationproblem (728), a semidefinite program, we substitute a vector inner-productobjective function for trace;

tr Z = 〈Z , I 〉 ← 〈Z , W 〉 (729)

a generalization of the trace heuristic for minimizing convex envelope of rank,where W ∈ SN+n

+ is constant with respect to (728). Matrix W is normal toa hyperplane in SN+n minimized over a convex feasible set specified by theconstraints in (728). Matrix W is chosen so −W points in direction of rank-nfeasible solutions G . Thus the purpose of vector inner-product objective(729) is to locate a rank-n feasible Gram matrix assumed existent on theboundary of positive semidefinite cone SN

+ , as explained beginning in §4.4.1.

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318 CHAPTER 4. SEMIDEFINITE PROGRAMMING

−0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 90: Typical solution for 2-lattice in Figure 85 with noise factorη = 0.1 . Two red rightmost nodes are anchors; two remaining nodes aresensors. Radio range of sensor 1 indicated by arc; radius = 1.14 . Actualsensor indicated by target # while its localization is indicated by bullet • .Rank-2 solution found in 1 iteration (728) (1663a) subject to reflection error.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 91: Typical solution for 3-lattice in Figure 86 with noise factorη = 0.1 . Three red vertical middle nodes are anchors; remaining nodes aresensors. Radio range of sensor 1 indicated by arc; radius = 1.12 . Actualsensor indicated by target # while its localization is indicated by bullet • .Rank-2 solution found in 2 iterations (728) (1663a).

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 319

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 92: Typical solution for 4-lattice in Figure 87 with noise factorη = 0.1 . Four red vertical middle-left nodes are anchors; remaining nodesare sensors. Radio range of sensor 1 indicated by arc; radius = 0.75 . Actualsensor indicated by target # while its localization is indicated by bullet • .Rank-2 solution found in 7 iterations (728) (1663a).

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 93: Typical solution for 5-lattice in Figure 88 with noise factorη = 0.1 . Five red vertical middle nodes are anchors; remaining nodes aresensors. Radio range of sensor 1 indicated by arc; radius = 0.56 . Actualsensor indicated by target # while its localization is indicated by bullet • .Rank-2 solution found in 3 iterations (728) (1663a).

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320 CHAPTER 4. SEMIDEFINITE PROGRAMMING

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 94: Typical solution for 10-lattice with noise factor η = 0.1 comparesbetter than Carter & Jin [70, fig.4.2]. Ten red vertical middle nodes areanchors; the rest are sensors. Radio range of sensor 1 indicated by arc;radius = 0.25 . Actual sensor indicated by target # while its localization isindicated by bullet • . Rank-2 solution found in 5 iterations (728) (1663a).

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 95: Typical localization of 100 randomized noiseless sensors (η = 0) isexact despite incomplete EDM. Ten red vertical middle nodes are anchors;remaining nodes are sensors. Radio range of sensor at origin indicated by arc;radius = 0.25 . Actual sensor indicated by target # while its localization isindicated by bullet • . Rank-2 solution found in 3 iterations (728) (1663a).

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 321

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 96: Typical solution for 100 randomized sensors with noise factorη = 0.1 ; worst measured average sensor error ≈ 0.0044 compares better thanCarter & Jin’s 0.0154 computed in 0.71s [70, p.19]. Ten red vertical middlenodes are anchors; same as before. Remaining nodes are sensors. Interioranchor placement makes localization difficult. Radio range of sensor at originindicated by arc; radius = 0.25 . Actual sensor indicated by target # whileits localization is indicated by bullet • . After 1 iteration rankG=92,after 2 iterations rankG=4. Rank-2 solution found in 3 iterations (728)(1663a). (Regular lattice in Figure 94 is actually harder to solve, requiringmore iterations.) Runtime for SDPT3 [345] under cvx [163] is a few minuteson 2009 vintage laptop Core 2 Duo CPU (Intel T6400@2GHz, 800MHz FSB).

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322 CHAPTER 4. SEMIDEFINITE PROGRAMMING

direction matrix W

Denote by Z⋆ an optimal composite matrix from semidefinite program (728).Then for Z⋆∈ SN+n whose eigenvalues λ(Z⋆)∈ RN+n are arranged innonincreasing order, (Ky Fan)

N+n∑

i=n+1

λ(Z⋆)i = minimizeW∈ SN+n

〈Z⋆, W 〉subject to 0 ¹ W ¹ I

tr W = N

(1663a)

which has an optimal solution that is known in closed form (p.646, §4.4.1.1).This eigenvalue sum is zero when Z⋆ has rank n or less.

Foreknowledge of optimal Z⋆, to make possible this search for W ,implies iteration; id est, semidefinite program (728) is solved for Z⋆

initializing W = I or W = 0. Once found, Z⋆ becomes constant insemidefinite program (1663a) where a new normal direction W is found asits optimal solution. Then this cycle (728) (1663a) iterates until convergence.When rank Z⋆ = n , solution via this convex iteration solves sensor-networklocalization problem (721) and its equivalent (726).

numerical solution

In all examples to follow, number of anchors

m =√

N (730)

equals square root of cardinality N of list X . Indices set I identifying allmeasurable distances • is ascertained from connectivity matrix (722), (723),(724), or (725). We solve iteration (728) (1663a) in dimension n = 2 for eachrespective example illustrated in Figure 85 through Figure 88.

In presence of negligible noise, true position is reliably localized for everystandardized example; noteworthy in so far as each example represents anincomplete graph. This implies that the set of all optimal solutions havinglowest rank must be small.

To make the examples interesting and consistent with previous work, werandomize each range of distance-square that bounds 〈G , (ei− ej)(ei− ej)

T〉in (728); id est, for each and every (i, j)∈ I

dij = dij(1 +√

3 η χl)2

dij = dij(1 −√

3 η χl+1

)2 (731)

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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 323

where η = 0.1 is a constant noise factor, χl

is the lth sample of a noiseprocess realization uniformly distributed in the interval [0, 1] like rand(1)from Matlab, and dij is actual distance-square from ith to j th sensor.Because of distinct function calls to rand() , each range of distance-square[ dij , dij ] is not necessarily centered on actual distance-square dij . Unit

stochastic variance is provided by factor√

3.Figure 90 through Figure 93 each illustrate one realization of numerical

solution to the standardized lattice problems posed by Figure 85 throughFigure 88 respectively. Exact localization, by any method, is impossiblebecause of measurement noise. Certainly, by inspection of their publishedgraphical data, our results are better than those of Carter & Jin.(Figure 94, 95, 96) Obviously our solutions do not suffer from thosecompaction-type errors (clustering of localized sensors) exhibited by Biswas’graphical results for the same noise factor η .

localization example conclusion

Solution to this sensor-network localization problem became apparent byunderstanding geometry of optimization. Trace of a matrix, to a student oflinear algebra, is perhaps a sum of eigenvalues. But to us, trace representsthe normal I to some hyperplane in Euclidean vector space. (Figure 83)

Our solutions are globally optimal, requiring: 1) no centralized-gradientpostprocessing heuristic refinement as in [43] because there is effectivelyno relaxation of (726) at optimality, 2) no implicit postprojection onrank-2 positive semidefinite matrices induced by nonzero G−XTX denotingsuboptimality as occurs in [44] [45] [46] [70] [212] [220]; indeed, G⋆ = X⋆TX⋆

by convex iteration.Numerical solution to noisy problems, containing sensor variables well

in excess of 100, becomes difficult via the holistic semidefinite programwe proposed. When problem size is within reach of contemporarygeneral-purpose semidefinite program solvers, then the convex iteration wepresented inherently overcomes limitations of Carter & Jin with respect toboth noise performance and ability to localize in any desired affine dimension.

The legacy of Carter, Jin, Saunders, & Ye [70] is a sobering demonstrationof the need for more efficient methods for solution of semidefinite programs,while that of So & Ye [316] forever bonds distance geometry to semidefiniteprogramming. Elegance of our semidefinite problem statement (728),for constraining affine dimension of sensor-network localization, should

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324 CHAPTER 4. SEMIDEFINITE PROGRAMMING

provide some impetus to focus more research on computational intensity ofgeneral-purpose semidefinite program solvers. An approach different frominterior-point methods is required; higher speed and greater accuracy froma simplex-like solver is what is needed. 2

We numerically tested the foregoing technique for constraining rank ona wide range of problems including localization of randomized positions(Figure 96), stress (§7.2.2.7.1), ball packing (§5.4.2.3.4), and cardinalityproblems (§4.6). We have had some success introducing the direction matrixinner-product (729) as a regularization term4.30

minimizeZ∈SN

f(Z) + w〈Z , W 〉subject to Z ∈ C

Z º 0

(732)

minimizeW∈ SN

f(Z⋆) + w〈Z⋆, W 〉subject to 0 ¹ W ¹ I

tr W = N − n

(733)

whose purpose is to constrain rank, affine dimension, or cardinality:

4.5 Constraining cardinalityThe convex iteration technique for constraining rank can be applied tocardinality problems. There are parallels in its development analogous tohow prototypical semidefinite program (625) resembles linear program (281)on page 267 [379]:

4.5.1 nonnegative variable

Our goal has been to reliably constrain rank in a semidefinite program. Thereis a direct analogy to linear programming that is simpler to present but,perhaps, more difficult to solve. In Optimization, that analogy is known asthe cardinality problem.

4.30 called multiobjective- or vector optimization. Proof of convergence for this convexiteration is identical to that in §4.4.1.2.1 because f is a convex real function, hence boundedbelow, and f(Z⋆) is constant in (733). Positive scalar w is chosen by cut-and-try.

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4.5. CONSTRAINING CARDINALITY 325

Consider a feasibility problem Ax = b , but with an upper bound k oncardinality ‖x‖0 of a nonnegative solution x : for A∈Rm×n and vectorb∈R(A)

find x ∈ Rn

subject to Ax = b

x º 0

‖x‖0 ≤ k

(508)

where ‖x‖0 ≤ k means4.31 vector x has at most k nonzero entries; sucha vector is presumed existent in the feasible set. Nonnegativity constraintxº 0 is analogous to positive semidefiniteness; the notation means vector xbelongs to the nonnegative orthant Rn

+ . Cardinality is quasiconcave on Rn+

just as rank is quasiconcave on Sn+ . [59, §3.4.2]

We propose that cardinality-constrained feasibility problem (508) can beequivalently expressed as a sequence of convex problems: for 0≤k≤n−1

minimizex∈R

n〈x , y〉

subject to Ax = b

x º 0

(154)

n∑

i=k+1

π(x⋆)i = minimizey∈R

n〈x⋆, y〉

subject to 0 ¹ y ¹ 1

yT1 = n − k

(503)

where π is the (nonincreasing) presorting function. This sequence is iterateduntil x⋆Ty⋆ vanishes; id est, until desired cardinality is achieved. But thisglobal convergence cannot always be guaranteed.

Problem (503) is analogous to the rank constraint problem; (confer p.300)

N∑

i=k+1

λ(G⋆)i = minimizeW∈ SN

〈G⋆, W 〉subject to 0 ¹ W ¹ I

tr W = N − k

(1663a)

Linear program (503) sums the n−k smallest entries from vector x . Incontext of problem (508), we want n−k entries of x to sum to zero; id est,

4.31Although it is a metric (§5.2), cardinality ‖x‖0 cannot be a norm (§3.3) because it isnot positively homogeneous.

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326 CHAPTER 4. SEMIDEFINITE PROGRAMMING

1

0

R3

+

∂H = x | 〈x , 1〉 = κ

∂H

Figure 97: 1-norm heuristic for cardinality minimization can be interpreted asminimization of a hyperplane, ∂H with normal 1, over nonnegative orthantdrawn here in R3. Polar of direction vector y = 1 points toward origin.

we want a globally optimal objective x⋆Ty⋆ to vanish: more generally,

n∑

i=k+1

π(|x⋆|)i = 〈|x⋆| , y⋆〉 , 0 (734)

defines global convergence for the iteration. Then n−k entries of x⋆ arethemselves zero whenever their absolute sum is, and cardinality of x⋆∈ Rn isat most k . Optimal direction vector y⋆ is defined as any nonnegative vectorfor which

(508)

find x ∈ Rn

subject to Ax = b

x º 0

‖x‖0 ≤ k

≡minimize

x∈Rn

〈x , y⋆〉subject to Ax = b

x º 0

(154)

Existence of such a y⋆, whose nonzero entries are complementary to thoseof x⋆, is obvious assuming existence of a cardinality-k solution x⋆.

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4.5. CONSTRAINING CARDINALITY 327

4.5.1.1 direction vector interpretation

(confer §4.4.1.1) Vector y may be interpreted as a negative search direction;it points opposite to direction of movement of hyperplane x | 〈x , y〉= τin a minimization of real linear function 〈x , y〉 over the feasible set inlinear program (154). (p.75) Direction vector y is not unique. The feasibleset of direction vectors in (503) is the convex hull of all cardinality-(n−k)one-vectors; videlicet,

convu∈Rn | card u = n − k , ui∈0 , 1 = a∈Rn | 1 º a º 0 , 〈1 , a〉= n − k(735)

This set, argument to conv , comprises the extreme points of set (735)which is a nonnegative hypercube slice. An optimal solution y to (503),that is an extreme point of its feasible set, is known in closed form: ithas 1 in each entry corresponding to the n−k smallest entries of x⋆ andhas 0 elsewhere. That particular polar direction −y can be interpreted4.32

by Proposition 7.1.3.0.3 as pointing toward the nonnegative orthant inthe Cartesian subspace, whose basis is a subset of the Cartesian axes,containing all cardinality k (or less) vectors having the same ordering as x⋆.Consequently, for that closed-form solution,

n∑

i=k+1

π(|x⋆|)i = 〈|x⋆| , y〉 (736)

When y = 1, as in 1-norm minimization for example, then polar direction−y points directly at the origin (the cardinality-0 nonnegative vector) as inFigure 97. We sometimes solve (503) instead of employing a known closedform because a direction vector is not unique. Setting direction vector yinstead in accordance with an iterative inverse weighting scheme, calledreweighting [157], was described for the 1-norm by Huo [203, §4.11.3] in 1999.

4.5.1.2 convergence can mean stalling

Convex iteration (154) (503) always converges to a locally optimal solution,a fixed point of possibly infeasible cardinality, by virtue of a monotonicallynonincreasing real objective sequence. [254, §1.2] [41, §1.1] There can beno proof of global convergence, defined by (734). Constraining cardinality,

4.32Convex iteration (154) (503) is not a projection method because there is nothresholding or discard of variable-vector x entries.

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328 CHAPTER 4. SEMIDEFINITE PROGRAMMING

solution to problem (508), can often be achieved but simple examples canbe contrived that stall at a fixed point of infeasible cardinality; at a positiveobjective value 〈x⋆, y〉= τ >0. Direction vector y is then manipulated, ascountermeasure, to steer out of local minima; e.g., complete randomizationas in Example 4.5.1.4.1, or reinitialization to a random cardinality-(n−k)vector in the same nonnegative orthant face demanded by the current iterate:y has nonnegative uniformly distributed random entries corresponding tothe n−k smallest entries of x⋆ and has 0 elsewhere. When this particularheuristic is successful, cardinality versus iteration is characterized by noisymonotonicity. Zero entries behave like memory while randomness greatlydiminishes likelihood of a stall.

4.5.1.3 algebraic derivation of direction vector for convex iteration

In §3.3.2.1.2, the compressed sensing problem was precisely represented as anonconvex difference of convex functions bounded below by 0

find x ∈ Rn

subject to Ax = bx º 0‖x‖0 ≤ k

≡minimize

x∈Rn

‖x‖1 − ‖x‖nk

subject to Ax = bx º 0

(508)

where convex k-largest norm ‖x‖nk

is monotonic on Rn+ . There we showed

how (508) is equivalently stated in terms of gradients

minimizex∈R

n

x , ∇‖x‖1 − ∇‖x‖nk

subject to Ax = bx º 0

(737)

because

‖x‖1 = xT∇‖x‖1 , ‖x‖nk

= xT∇‖x‖nk

, x º 0 (738)

The objective function from (737) is a directional derivative (at x in directionx , §D.1.6, confer §D.1.4.1.1) of the objective function from (508) while thedirection vector of convex iteration

y = ∇‖x‖1 − ∇‖x‖nk

(739)

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4.5. CONSTRAINING CARDINALITY 329

is an objective gradient where ∇‖x‖1 = 1 under nonnegativity and

∇‖x‖nk

= arg maximizez∈R

nzTx

subject to 0 ¹ z ¹ 1zT1 = k

, x º 0 (511)

Substituting 1 − z ← z the direction vector becomes

y = 1 − arg maximizez∈R

nzTx = arg minimize

z∈Rn

zTx

subject to 0 ¹ z ¹ 1 subject to 0 ¹ z ¹ 1zT1 = k zT1 = n − k

(503)

4.5.1.4 optimality conditions for compressed sensing

Now we see how global optimality conditions can be stated without referenceto a dual problem: From conditions (447) for optimality of (508), it isnecessary [59, §5.5.3] that

x⋆ º 0 (1)

Ax⋆ = b (2)

∇‖x⋆‖1 − ∇‖x⋆‖nk

+ ATν⋆ º 0 (3)

〈∇‖x⋆‖1 − ∇‖x⋆‖nk

+ ATν⋆, x⋆〉 = 0 (4ℓ)

(740)

These conditions must hold at any optimal solution (locally or globally). By(738), the fourth condition is identical to

‖x⋆‖1 − ‖x⋆‖nk

+ ν⋆TAx⋆ = 0 (4ℓ) (741)

Because a 1-norm

‖x‖1 = ‖x‖nk

+ ‖π(|x|)k+1:n‖1 (742)

is separable into k largest and n−k smallest absolute entries,

‖π(|x|)k+1:n‖1 = 0 ⇔ ‖x‖0 ≤ k (4g) (743)

is a necessary condition for global optimality. By assumption, matrix Ais fat and b 6= 0 ⇒ Ax⋆ 6= 0. This means ν⋆∈ N (AT)⊂ Rm, and ν⋆ = 0when A is full-rank. By definition, ∇‖x‖1 º ∇‖x‖n

kalways holds. Assuming

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330 CHAPTER 4. SEMIDEFINITE PROGRAMMING

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

Donoho bound

approximation

x > 0 constraint

m/k

k/n

m > k log2(1+n/k)

minimizex

‖x‖1

subject to Ax = b(496)

minimizex

‖x‖1

subject to Ax = bx º 0

(501)

Figure 98: For Gaussian random matrix A∈Rm×n, graph illustratesDonoho/Tanner least lower bound on number of measurements m belowwhich recovery of k-sparse n-length signal x by linear programming failswith overwhelming probability. Hard problems are below curve, but not thereverse; id est, failure above depends on proximity. Inequality demarcatesapproximation (dashed curve) empirically observed in [23]. Problems havingnonnegativity constraint (dotted) are easier to solve. [119] [120]

existence of a cardinality-k solution, then only three of the four conditionsare necessary and sufficient for global optimality of (508):

x⋆ º 0 (1)

Ax⋆ = b (2)

‖x⋆‖1 − ‖x⋆‖nk

= 0 (4g)

(744)

meaning, global optimality of a feasible solution to (508) is identified by azero objective.

4.5.1.4.1 Example. Sparsest solution to Ax = b . [68] [115](confer Example 4.5.1.7.1) Problem (658) has sparsest solution not easilyrecoverable by least 1-norm; id est, not by compressed sensing because

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4.5. CONSTRAINING CARDINALITY 331

of proximity to a theoretical lower bound on number of measurements mdepicted in Figure 98: for A∈Rm×n Given data from Example 4.2.3.1.1, for m=3, n=6, k=1

A =

−1 1 8 1 1 0

−3 2 8 12

13

12− 1

3

−9 4 8 14

19

14− 1

9

, b =

11214

(658)

the sparsest solution to classical linear equation Ax = b is x = e4∈ R6

(confer (671)).

Although the sparsest solution is recoverable by inspection, we discernit instead by convex iteration; namely, by iterating problem sequence(154) (503) on page 325. From the numerical data given, cardinality ‖x‖0 = 1is expected. Iteration continues until xTy vanishes (to within some numericalprecision); id est, until desired cardinality is achieved. But this comes notwithout a stall.

Stalling, whose occurrence is sensitive to initial conditions of convexiteration, is a consequence of finding a local minimum of a multimodalobjective 〈x , y〉 when regarded as simultaneously variable in x and y .(§3.9.0.0.3) Stalls are simply detected as fixed points x of infeasiblecardinality, sometimes remedied by reinitializing direction vector y to arandom positive state.

Bolstered by success in breaking out of a stall, we then apply convexiteration to 22,000 randomized problems: Given random data for m=3, n=6, k=1, in Matlab notation

A=randn(3 , 6), index=round(5∗rand(1)) + 1, b=rand(1)∗A(: , index)(745)

the sparsest solution x∝eindex is a scaled standard basis vector.

Without convex iteration or a nonnegativity constraint xº 0, rate of failurefor this minimum cardinality problem Ax=b by 1-norm minimization of xis 22%. That failure rate drops to 6% with a nonnegativity constraint. Ifwe then engage convex iteration, detect stalls, and randomly reinitialize thedirection vector, failure rate drops to 0% but the amount of computation isapproximately doubled. 2

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332 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Stalling is not an inevitable behavior. For some problem types (beyondmere Ax = b), convex iteration succeeds nearly all the time. Here is acardinality problem, with noise, whose statement is just a bit more intricatebut easy to solve in a few convex iterations:

4.5.1.4.2 Example. Signal dropout. [118, §6.2]Signal dropout is an old problem; well studied from both an industrial andacademic perspective. Essentially dropout means momentary loss or gap ina signal, while passing through some channel, caused by some man-madeor natural phenomenon. The signal lost is assumed completely destroyedsomehow. What remains within the time-gap is system or idle channel noise.The signal could be voice over Internet protocol (VoIP), for example, audiodata from a compact disc (CD) or video data from a digital video disc (DVD),a television transmission over cable or the airwaves, or a typically ravagedcell phone communication, etcetera.

Here we consider signal dropout in a discrete-time signal corrupted byadditive white noise assumed uncorrelated to the signal. The linear channelis assumed to introduce no filtering. We create a discretized windowedsignal for this example by positively combining k randomly chosen vectorsfrom a discrete cosine transform (DCT) basis denoted Ψ∈ Rn×n. Frequencyincreases, in the Fourier sense, from DC toward Nyquist as column index ofbasis Ψ increases. Otherwise, details of the basis are unimportant exceptfor its orthogonality ΨT = Ψ−1. Transmitted signal is denoted

s = Ψz ∈ Rn (746)

whose upper bound on DCT basis coefficient cardinality card z≤ k isassumed known;4.33 hence a critical assumption: transmitted signal s issparsely supported (k < n) on the DCT basis. It is further assumed thatnonzero signal coefficients in vector z place each chosen basis vector abovethe noise floor.

We also assume that the gap’s beginning and ending in time are preciselylocalized to within a sample; id est, index ℓ locates the last sample prior tothe gap’s onset, while index n−ℓ+1 locates the first sample subsequent tothe gap: for rectangularly windowed received signal g possessing a time-gap

4.33This simplifies exposition, although it may be an unrealistic assumption in manyapplications.

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4.5. CONSTRAINING CARDINALITY 333

loss and additive noise η ∈ Rn

g =

s1:ℓ + η1:ℓ

ηℓ+1:n−ℓ

sn−ℓ+1:n + ηn−ℓ+1:n

∈ Rn (747)

The window is thereby centered on the gap and short enough so that the DCTspectrum of signal s can be assumed static over the window’s duration n .Signal to noise ratio within this window is defined

SNR , 20 log

[

s1:ℓ

sn−ℓ+1:n

]∥

‖η‖ (748)

In absence of noise, knowing the signal DCT basis and having agood estimate of basis coefficient cardinality makes perfectly reconstructinggap-loss easy: it amounts to solving a linear system of equations andrequires little or no optimization; with caveat, number of equations exceedscardinality of signal representation (roughly ℓ≥ k) with respect to DCTbasis.

But addition of a significant amount of noise η increases level ofdifficulty dramatically; a 1-norm based method of reducing cardinality, forexample, almost always returns DCT basis coefficients numbering in excessof minimum cardinality. We speculate that is because signal cardinality 2ℓbecomes the predominant cardinality. DCT basis coefficient cardinality is anexplicit constraint to the optimization problem we shall pose: In presence ofnoise, constraints equating reconstructed signal f to received signal g arenot possible. We can instead formulate the dropout recovery problem as abest approximation:

minimizex∈R

n

[

f1:ℓ − g1:ℓ

fn−ℓ+1:n − gn−ℓ+1:n

]∥

subject to f = Ψ xx º 0card x ≤ k

(749)

We propose solving this nonconvex problem (749) by moving the cardinalityconstraint to the objective as a regularization term as explained in §4.5;

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334 CHAPTER 4. SEMIDEFINITE PROGRAMMING

0 100 200 300 400 500−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 100 200 300 400 500−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

(a)

(b)

flatline and g

f and g

f

s + η

ηs + η

dropout (s = 0)

Figure 99: (a) Signal dropout in signal s corrupted by noise η (SNR =10dB,g = s + η). Flatline indicates duration of signal dropout. (b) Reconstructedsignal f (red) overlaid with corrupted signal g .

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4.5. CONSTRAINING CARDINALITY 335

0 100 200 300 400 500−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 100 200 300 400 500−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

(a)

(b)

f − s

f and s

Figure 100: (a) Error signal power (reconstruction f less original noiselesssignal s) is 36dB below s . (b) Original signal s overlaid withreconstruction f (red) from signal g having dropout plus noise.

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336 CHAPTER 4. SEMIDEFINITE PROGRAMMING

id est, by iteration of two convex problems until convergence:

minimizex∈R

n〈x , y〉 +

[

f1:ℓ − g1:ℓ

fn−ℓ+1:n − gn−ℓ+1:n

]∥

subject to f = Ψ xx º 0

(750)

and

minimizey∈R

n〈x⋆, y〉

subject to 0 ¹ y ¹ 1

yT1 = n − k

(503)

Signal cardinality 2ℓ is implicit to the problem statement. When numberof samples in the dropout region exceeds half the window size, then thatdeficient cardinality of signal remaining becomes a source of degradationto reconstruction in presence of noise. Thus, by observation, we divinea reconstruction rule for this signal dropout problem to attain goodnoise suppression: ℓ must exceed a maximum of cardinality bounds;2ℓ ≥ max2k , n/2.

Figure 99 and Figure 100 show one realization of this dropout problem.Original signal s is created by adding four (k = 4) randomly selected DCTbasis vectors, from Ψ (n = 500 in this example), whose amplitudes arerandomly selected from a uniform distribution above the noise floor; in theinterval [10−10/20, 1]. Then a 240-sample dropout is realized (ℓ = 130) andGaussian noise η added to make corrupted signal g (from which a bestapproximation f will be made) having 10dB signal to noise ratio (748).The time gap contains much noise, as apparent from Figure 99a. But inonly a few iterations (750) (503), original signal s is recovered with relativeerror power 36dB down; illustrated in Figure 100. Correct cardinality isalso recovered (card x = card z) along with the basis vector indices used tomake original signal s . Approximation error is due to DCT basis coefficientestimate error. When this experiment is repeated 1000 times on noisy signalsaveraging 10dB SNR, the correct cardinality and indices are recovered 99%of the time with average relative error power 30dB down. Without noise, weget perfect reconstruction in one iteration. [Matlab code: Wıκımization]

2

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4.5. CONSTRAINING CARDINALITY 337

R3

A

F

cS = s | s º 0 , 1Ts ≤ c

Figure 101: Line A intersecting two-dimensional (cardinality-2) face F ofnonnegative simplex S from Figure 52 emerges at a (cardinality-1) vertex.S is first quadrant of 1-norm ball B1 from Figure 69. Kissing point (• onedge) is achieved as ball contracts or expands under optimization.

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338 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.5.1.5 Compressed sensing geometry with a nonnegative variable

It is well known that cardinality problem (508), on page 225, is easier to solveby linear programming when variable x is nonnegatively constrained thanwhen not; e.g., Figure 70, Figure 98. We postulate a simple geometricalexplanation: Figure 69 illustrates 1-norm ball B1 in R3 and affine subsetA= x |Ax=b. Prototypical compressed sensing problem, for A∈Rm×n

minimizex

‖x‖1

subject to Ax = b(496)

is solved when the 1-norm ball B1 kisses affine subset A . If variable x is constrained to the nonnegative orthant

minimizex

‖x‖1

subject to Ax = bx º 0

(501)

then 1-norm ball B1 (Figure 69) becomes simplex S in Figure 101.

Whereas 1-norm ball B1 has only six vertices in R3 corresponding tocardinality-1 solutions, simplex S has three edges (along the Cartesian axes)containing an infinity of cardinality-1 solutions. And whereas B1 has twelveedges containing cardinality-2 solutions, S has three (out of total four)facets constituting cardinality-2 solutions. In other words, likelihood of alow-cardinality solution is higher by kissing nonnegative simplex S (501) thanby kissing 1-norm ball B1 (496) because facial dimension (corresponding togiven cardinality) is higher in S .

Empirically, this conclusion also holds in other Euclidean dimensions(Figure 70, Figure 98).

4.5.1.6 cardinality-1 compressed sensing problem always solvable

In the special case of cardinality-1 solution to prototypical compressedsensing problem (501), there is a geometrical interpretation that leadsto an algorithm more efficient than convex iteration. Figure 101 showsa vertex-solution to problem (501) when desired cardinality is 1. Butfirst-quadrant S of 1-norm ball B1 does not kiss line A ; which requiresexplanation: Under the assumption that nonnegative cardinality-1 solutionsexist in feasible set A , it so happens,

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4.5. CONSTRAINING CARDINALITY 339 columns of measurement matrix A , corresponding to any particularsolution (to (501)) of cardinality greater than 1, may be deprecated andthe problem solved again with those columns missing. Such columnsare recursively removed from A until a cardinality-1 solution is found.

Either a solution to problem (501) is cardinality-1 or column indices of A ,corresponding to a higher cardinality solution, do not intersect that indexcorresponding to the cardinality-1 solution. When problem (501) is firstsolved, in the example of Figure 101, solution is cardinality-2 at the kissingpoint • on the indicated edge of simplex S . Imagining that the correspondingcardinality-2 face F has been removed, then the simplex collapses to a linesegment along one of the Cartesian axes. When that line segment kissesline A , then the cardinality-1 vertex-solution illustrated has been found. Asimilar argument holds for any orientation of line A and point of entry tosimplex S .

Because signed compressed sensing problem (496) can be equivalentlyexpressed in a nonnegative variable, as we learned in Example 3.3.0.0.1(p.222), and because a cardinality-1 constraint in (496) transforms toa cardinality-1 constraint in its nonnegative equivalent (500), then thiscardinality-1 recursive reconstruction algorithm continues to hold for a signedvariable as in (496). This cardinality-1 reconstruction algorithm also holdsmore generally when affine subset A has any higher dimension n−m .

Although it is more efficient to search over the columns of matrix A fora cardinality-1 solution known a priori to exist, a combinatorial approach,tables are turned when cardinality exceeds 1 :

4.5.1.7 cardinality-k geometric presolver

This idea of deprecating columns has foundation in convex cone theory.(§2.13.4.3) Removing columns (and rows)4.34 from A∈Rm×n in a linearprogram like (501) (§3.3) is known in the industry as presolving ;4.35 theelimination of redundant constraints and identically zero variables prior tonumerical solution. We offer a different and geometric presolver:

Two interpretations of the constraints from problem (501) are realizedin Figure 102. Assuming that a cardinality-k solution exists and matrix A

4.34Rows of matrix A are removed based upon linear dependence. Assuming b∈R(A) ,corresponding entries of vector b may also be removed without loss of generality.4.35The conic technique proposed here can exceed performance of the best industrialpresolvers in terms of number of columns removed, but not in execution time.

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340 CHAPTER 4. SEMIDEFINITE PROGRAMMING

AP

Rn+

0

Rn (a)

K

b

0

Rm

(b)

Figure 102: Constraint interpretations: (a) Halfspace-description of feasibleset in problem (501) is a polyhedron P formed by intersection of nonnegativeorthant Rn

+ with hyperplanes A prescribed by equality constraint. (Drawingby Pedro Sanchez) (b) Vertex-description of constraints in problem (501):point b belongs to polyhedral cone K= Ax | xº 0. Number of extremedirections in K may exceed dimensionality of ambient space.

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4.5. CONSTRAINING CARDINALITY 341

describes a pointed polyhedral cone K= Ax | xº 0 , columns are removedfrom A if they do not belong to the smallest face of K containing vector b .Columns so removed correspond to 0-entries in variable vector x .

Benefit accrues when vector b does not belong to the relative interiorof K ; there would be no columns to remove were b∈rel intK since thesmallest face becomes cone K itself (Example 4.5.1.7.1). Were b an extremedirection, at the other end of our spectrum, then the smallest face is an edgethat is a ray containing b ; this is a geometrical description of a cardinality-1case where all columns, save one, would be removed from A .

When vector b resides in a face of K that is not K itself, benefit isrealized as a reduction in computational intensity because the consequentequivalent problem has smaller dimension. Number of columns removeddepends completely on particular geometry of a given problem; particularly,location of b within K .

There are always exactly n linear feasibility problems to solve in orderto discern generators of the smallest face of K containing b ; the topic of§2.13.4.3. So comparison of computational intensity for this conic approach

pits combinatorial complexity ∝(

nk

)

a binomial coefficient versus n linear

feasibility problems plus numerical solution of the presolved problem.

4.5.1.7.1 Example. Presolving for cardinality-2 solution to Ax = b .(confer Example 4.5.1.4.1) Again taking data from Example 4.2.3.1.1, form=3, n=6, k=2 (A∈ Rm×n, desired cardinality of x is k)

A =

−1 1 8 1 1 0

−3 2 8 12

13

12− 1

3

−9 4 8 14

19

14− 1

9

, b =

11214

(658)

proper cone K= Ax | xº 0 is pointed as proven by method of §2.12.2.2.A cardinality-2 solution is known to exist; sum of the last two columns ofmatrix A . Generators of the smallest face containing vector b , found by themethod in Example 2.13.4.3.1, comprise the entire A matrix because b∈ intK(§2.13.4.2.4). So geometry of this particular problem does not permit numberof generators to be reduced below n by discerning the smallest face.4.36

2

4.36But a canonical set of conically independent generators of K comprise only the firsttwo and last two columns of A .

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342 CHAPTER 4. SEMIDEFINITE PROGRAMMING

There is wondrous bonus to presolving when a constraint matrix is sparse.After columns are removed by theory of convex cones (finding the smallestface), some remaining rows may become 0T, identical to other rows, ornonnegative. When nonnegative rows appear in an equality constraint to 0,all nonnegative variables corresponding to nonnegative entries in those rowsmust vanish (§A.7.1); meaning, more columns may be removed. Once rowsand columns have been removed from a constraint matrix, even more rowsand columns may be removed by repeating the presolver procedure.

4.5.2 constraining cardinality of signed variable

Now consider a feasibility problem equivalent to the classical problem fromlinear algebra Ax = b , but with an upper bound k on cardinality ‖x‖0 : forvector b∈R(A)

find x ∈ Rn

subject to Ax = b

‖x‖0 ≤ k

(751)

where ‖x‖0 ≤ k means vector x has at most k nonzero entries; such a vectoris presumed existent in the feasible set. Convex iteration (§4.5.1) works witha nonnegative variable; so absolute value |x| is needed here. We propose thatnonconvex problem (751) can be equivalently written as a sequence of convexproblems that move the cardinality constraint to the objective:

minimizex∈R

n〈|x| , y〉

subject to Ax = b≡

minimizex∈R

n , t∈Rn

〈t , y + ε1〉subject to Ax = b

−t ¹ x ¹ t

(752)

minimizey∈R

n〈t⋆, y + ε1〉

subject to 0 ¹ y ¹ 1

yT1 = n − k

(503)

where ε is a relatively small positive constant. This sequence is iterated untila direction vector y is found that makes |x⋆|Ty⋆ vanish. The term 〈t , ε1〉in (752) is necessary to determine absolute value |x⋆|= t⋆ (§3.3) becausevector y can have zero-valued entries. By initializing y to (1−ε)1, the

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4.5. CONSTRAINING CARDINALITY 343

first iteration of problem (752) is a 1-norm problem (492); id est,

minimizex∈R

n, t∈R

n〈t , 1〉

subject to Ax = b

−t ¹ x ¹ t

≡minimize

x∈Rn

‖x‖1

subject to Ax = b(496)

Subsequent iterations of problem (752) engaging cardinality term 〈t , y〉 canbe interpreted as corrections to this 1-norm problem leading to a 0-normsolution; vector y can be interpreted as a direction of search.

4.5.2.1 local convergence

As before (§4.5.1.2), convex iteration (752) (503) always converges to a locallyoptimal solution; a fixed point of possibly infeasible cardinality.

4.5.2.2 simple variations on a signed variable

Several useful equivalents to linear programs (752) (503) are easily devised,but their geometrical interpretation is not as apparent: e.g., equivalent inthe limit ε→0+

minimizex∈R

n , t∈Rn

〈t , y〉subject to Ax = b

−t ¹ x ¹ t

(753)

minimizey∈R

n〈|x⋆| , y〉

subject to 0 ¹ y ¹ 1

yT1 = n − k

(503)

We get another equivalent to linear programs (752) (503), in the limit, byinterpreting problem (496) as infimum to a vertex-description of the 1-normball (Figure 69, Example 3.3.0.0.1, confer (495)):

minimizea∈R2n

〈a , y〉subject to [ A −A ]a = b

a º 0

(754)

minimizey∈R2n

〈a⋆, y〉subject to 0 ¹ y ¹ 1

yT1 = 2n − k

(503)

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344 CHAPTER 4. SEMIDEFINITE PROGRAMMING

where x⋆ = [ I −I ]a⋆ ; from which it may be construed that any vector1-norm minimization problem has equivalent expression in a nonnegativevariable.

4.6 Cardinality and rank constraint examples

4.6.0.0.1 Example. Projection on ellipsoid boundary. [50] [144, §5.1][243, §2] Consider classical linear equation Ax = b but with constraint onnorm of solution x , given matrices C , fat A , and vector b∈R(A)

find x ∈ RN

subject to Ax = b

‖Cx‖ = 1

(755)

The set x | ‖Cx‖=1 (2) describes an ellipsoid boundary (Figure 13). Thisis a nonconvex problem because solution is constrained to that boundary.Assign

G =

[

Cx1

]

[xTCT 1 ]=

[

X CxxTCT 1

]

,

[

CxxTCT CxxTCT 1

]

∈ SN+1

(756)

Any rank-1 solution must have this form. (§B.1.0.2) Ellipsoidally constrainedfeasibility problem (755) is equivalent to:

findX∈SN

x ∈ RN

subject to Ax = b

G =

[

X CxxTCT 1

]

(º 0)

rank G = 1

tr X = 1

(757)

This is transformed to an equivalent convex problem by moving the rankconstraint to the objective: We iterate solution of

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 345

minimizeX∈SN , x∈RN

〈G , Y 〉subject to Ax = b

G =

[

X CxxTCT 1

]

º 0

tr X = 1

(758)

withminimize

Y ∈ SN+1〈G⋆, Y 〉

subject to 0 ¹ Y ¹ I

tr Y = N

(759)

until convergence. Direction matrix Y ∈ SN+1, initially 0, regulates rank.(1663a) Singular value decomposition G⋆ = UΣQT∈ SN+1

+ (§A.6) provides anew direction matrix Y = U(: , 2:N+1)U(: , 2:N+1)T that optimally solves(759) at each iteration. An optimal solution to (755) is thereby found in afew iterations, making convex problem (758) its equivalent.

It remains possible for the iteration to stall; were a rank-1 G matrix notfound. In that case, the current search direction is momentarily reversedwith an added randomized element:

Y = −U(: , 2 :N+1) ∗ (U(: , 2 :N+1)′ + randn(N , 1) ∗ U(: , 1)′) (760)

in Matlab notation. This heuristic is quite effective for problem (755) whichis exceptionally easy to solve by convex iteration.

When b /∈R(A) then problem (755) must be restated as a projection:

minimizex∈RN

‖Ax − b‖

subject to ‖Cx‖ = 1(761)

This is a projection of point b on an ellipsoid boundary because any affinetransformation of an ellipsoid remains an ellipsoid. Problem (758) in turnbecomes

minimizeX∈SN , x∈RN

〈G , Y 〉 + ‖Ax − b‖

subject to G =

[

X CxxTCT 1

]

º 0

tr X = 1

(762)

We iterate this with calculation (759) of direction matrix Y as before untila rank-1 G matrix is found. 2

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346 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.6.0.0.2 Example. Procrustes problem. [50]Example 4.6.0.0.1 is extensible. An orthonormal matrix Q∈ Rn×p ischaracterized QTQ = I . Consider the particular case Q = [ x y ]∈ Rn×2 asvariable to a Procrustes problem (§C.3): given A∈ Rm×n and B∈ Rm×2

minimizeQ∈R

n×2

‖AQ − B‖F

subject to QTQ = I(763)

which is nonconvex. By vectorizing matrix Q we can make the assignment:

G =

xy

1

[xT yT 1 ]=

X Z xZT Y yxT yT 1

,

xxT xyT xyxT yyT yxT yT 1

∈ S2n+1 (764)

Now orthonormal Procrustes problem (763) can be equivalently restated:

minimizeX , Y ∈ S , Z , x , y

‖A[ x y ] − B‖F

subject to G =

X Z xZT Y yxT yT 1

(º 0)

rank G = 1

tr X = 1

tr Y = 1

tr Z = 0

(765)

To solve this, we form the convex problem sequence:

minimizeX , Y , Z , x , y

‖A[ x y ]−B‖F + 〈G , W 〉

subject to G =

X Z xZT Y yxT yT 1

º 0

tr X = 1

tr Y = 1

tr Z = 0

(766)

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 347

andminimizeW∈ S2n+1

〈G⋆, W 〉subject to 0 ¹ W ¹ I

tr W = 2n

(767)

which has an optimal solution W that is known in closed form (p.646).These two problems are iterated until convergence and a rank-1 G matrix isfound. A good initial value for direction matrix W is 0. Optimal Q⋆ equals[ x⋆ y⋆ ].

Numerically, this Procrustes problem is easy to solve; a solution seemsalways to be found in one or few iterations. This problem formulation isextensible, of course, to orthogonal (square) matrices Q . 2

4.6.0.0.3 Example. Combinatorial Procrustes problem.In case A ,B∈ Rn, when vector A = ΞB is known to be a permutation ofvector B , solution to orthogonal Procrustes problem

minimizeX∈R

n×n‖A − XB‖F

subject to XT = X−1(1675)

is not necessarily a permutation matrix Ξ even though an optimal objectivevalue of 0 is found by the known analytical solution (§C.3). The simplestmethod of solution finds permutation matrix X⋆ = Ξ simply by sortingvector B with respect to A .

Instead of sorting, we design two different convex problems each of whoseoptimal solution is a permutation matrix: one design is based on rankconstraint, the other on cardinality. Because permutation matrices are sparseby definition, we depart from a traditional Procrustes problem by insteaddemanding a vector 1-norm which is known to produce solutions more sparsethan Frobenius’ norm.

There are two principal facts exploited by the first convex iteration design(§4.4.1) we propose. Permutation matrices Ξ constitute:

1) the set of all nonnegative orthogonal matrices,

2) all points extreme to the polyhedron (99) of doubly stochastic matrices.

That means:

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348 CHAPTER 4. SEMIDEFINITE PROGRAMMING

X(: , i) | 1TX(: , i) = 1

X(: , i) | X(: , i)TX(: , i) = 1

Figure 103: Permutation matrix ith column-sum and column-norm constraintin two dimensions when rank constraint is satisfied. Optimal solutions resideat intersection of hyperplane with unit circle.

1) norm of each row and column is 1 ,4.37

‖Ξ(: , i)‖ = 1 , ‖Ξ(i , :)‖ = 1 , i=1 . . . n (768)

2) sum of each nonnegative row and column is 1, (§2.3.2.0.4)

ΞT1=1 , Ξ1=1 , Ξ≥ 0 (769)

solution via rank constraintThe idea is to individually constrain each column of variable matrix X tohave unity norm. Matrix X must also belong to that polyhedron, (99) inthe nonnegative orthant, implied by constraints (769); so each row-sum and

4.37This fact would be superfluous were the objective of minimization linear, because thepermutation matrices reside at the extreme points of a polyhedron (99) implied by (769).But as posed, only either rows or columns need be constrained to unit norm becausematrix orthogonality implies transpose orthogonality. (§B.5.1) Absence of vanishing innerproduct constraints that help define orthogonality, like trZ = 0 from Example 4.6.0.0.2,is a consequence of nonnegativity; id est, the only orthogonal matrices having exclusivelynonnegative entries are permutations of the identity.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 349

column-sum of X must also be unity. It is this combination of nonnegativity,sum, and sum square constraints that extracts the permutation matrices:(Figure 103) given nonzero vectors A , B

minimizeX∈R

n×n, Gi∈ S

n+1‖A − XB‖1 + w

n∑

i=1

〈Gi , Wi〉

subject toGi =

[

Gi(1 :n , 1:n) X(: , i)X(: , i)T 1

]

º 0

tr Gi = 2

, i=1 . . . n

XT1 = 1X1 = 1X ≥ 0

(770)

where w≈ 10 positively weights the rank regularization term. Optimalsolutions G⋆

i are key to finding direction matrices Wi for the next iterationof semidefinite programs (770) (771):

minimizeWi∈ S

n+1〈G⋆

i , Wi 〉subject to 0 ¹ Wi ¹ I

tr Wi = n

, i=1 . . . n (771)

Direction matrices thus found lead toward rank-1 matrices G⋆i on subsequent

iterations. Constraint on trace of G⋆i normalizes the ith column of X⋆ to unity

because (confer p.409)

G⋆i =

[

X⋆(: , i)1

]

[X⋆(: , i)T 1 ](772)

at convergence. Binary-valued X⋆ column entries result from the furthersum constraint X1=1. Columnar orthogonality is a consequence of thefurther transpose-sum constraint XT1=1 in conjunction with nonnegativityconstraint X≥ 0 ; but we leave proof of orthogonality an exercise. Theoptimal objective value is 0 for both semidefinite programs when vectors Aand B are related by permutation. In any case, optimal solution X⋆ becomesa permutation matrix Ξ .

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350 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Because there are n direction matrices Wi to find, it can be advantageousto invoke a known closed-form solution for each from page 646. What makesthis combinatorial problem more tractable are relatively small semidefiniteconstraints in (770). (confer (766)) When a permutation A of vector B exists,number of iterations can be as small as 1. But this combinatorial Procrustesproblem can be made even more challenging when vector A has repeatedentries.

solution via cardinality constraintNow the idea is to force solution at a vertex of permutation polyhedron (99)by finding a solution of desired sparsity. Because permutation matrix X isn-sparse by assumption, this combinatorial Procrustes problem may insteadbe formulated as a compressed sensing problem with convex iteration oncardinality of vectorized X (§4.5.1): given nonzero vectors A , B

minimizeX∈R

n×n‖A − XB‖1 + w〈X , Y 〉

subject to XT1 = 1X1 = 1X ≥ 0

(773)

where direction vector Y is an optimal solution to

minimizeY ∈R

n×n〈X⋆, Y 〉

subject to 0 ≤ Y ≤ 1

1TY 1 = n2− n

(503)

each a linear program. In this circumstance, use of closed-form solution fordirection vector Y is discouraged. When vector A is a permutation of B ,both linear programs have objectives that converge to 0. When vectors A andB are permutations and no entries of A are repeated, optimal solution X⋆

can be found as soon as the first iteration.In any case, X⋆ = Ξ is a permutation matrix. 2

4.6.0.0.4 Exercise. Combinatorial Procrustes constraints.Assume that the objective of semidefinite program (770) is 0 at optimality.Prove that the constraints in program (770) are necessary and sufficientto produce a permutation matrix as optimal solution. Alternatively andequivalently, prove those constraints necessary and sufficient to optimallyproduce a nonnegative orthogonal matrix. H

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4.6.0.0.5 Example. Tractable polynomial constraint.The ability to handle rank constraints makes polynomial constraints(generally nonconvex) transformable to convex constraints. All optimizationproblems having polynomial objective and polynomial constraints can bereformulated as a semidefinite program with a rank-1 constraint. [282]Suppose we require

3 + 2x − xy ≤ 0 (774)

Identify

G =

xy

1

[x y 1 ]=

x2 xy xxy y2 y

x y 1

∈ S3 (775)

Then nonconvex polynomial constraint (774) is equivalent to constraint set

tr(GA) ≤ 0G33 = 1(G º 0)rank G = 1

(776)

with direct correspondence to sense of trace inequality where G is assumedsymmetric (§B.1.0.2) and

A =

0 −12

1−1

20 0

1 0 3

∈ S3 (777)

Then the method of convex iteration from §4.4.1 is applied to implement therank constraint. 2

4.6.0.0.6 Exercise. Binary Pythagorean theorem.Technique of Example 4.6.0.0.5 is extensible to any quadratic constraint; e.g.,xTAx + 2bTx + c ≤ 0 , xTAx + 2bTx + c ≥ 0 , and xTAx + 2bTx + c = 0.Write a rank-constrained semidefinite program to solve (Figure 103)

x + y = 1x2 + y2 = 1

(778)

whose feasible set is not connected. Implement this system in cvx [163] byconvex iteration. H

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352 CHAPTER 4. SEMIDEFINITE PROGRAMMING

4.6.0.0.7 Example. High-order polynomials.Consider nonconvex problem from Canadian Mathematical Olympiad 1999:

findx , y , z∈R

x , y , z

subject to x2y + y2z + z2x = 22

33

x + y + z = 1x , y , z ≥ 0

(779)

We wish to solve for what is known to be a tight upper bound 22

33

on the constrained polynomial x2y + y2z + z2x by transformation to arank-constrained semidefinite program; identify

G =

xyz1

[x y z 1 ]

=

x2 xy zx xxy y2 yz yzx yz z2 zx y z 1

∈ S4 (780)

X =

x2

y2

z2

xyz1

[x2 y2 z2 x y z 1 ]

=

x4 x2y2 z2x2 x3 x2y zx2 x2

x2y2 y4 y2z2 xy2 y3 y2z y2

z2x2 y2z2 z4 z2x yz2 z3 z2

x3 xy2 z2x x2 xy zx xx2y y3 yz2 xy y2 yz yzx2 y2z z3 zx yz z2 zx2 y2 z2 x y z 1

∈ S7

(781)then apply convex iteration (§4.4.1) to implement rank constraints:

findA , C∈S , b

b

subject to tr(XE) = 22

33

G =

[

A bbT 1

]

(º 0)

X =

C

[

δ(A)b

]

[

δ(A)T bT]

1

(º 0)

1Tb = 1b º 0rank G = 1rank X = 1

(782)

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 353

where

E =

0 0 0 0 1 0 00 0 0 0 0 1 00 0 0 1 0 0 00 0 1 0 0 0 01 0 0 0 0 0 00 1 0 0 0 0 00 0 0 0 0 0 0

1

2∈ S7 (783)

(Matlab code on Wıκımization) Positive semidefiniteness is optional onlywhen rank-1 constraints are explicit by Theorem A.3.1.0.7. Optimal solutionto problem (779) is not unique: (x , y , z) = (0 , 2

3, 1

3). 2

4.6.0.0.8 Example. Boolean vector satisfying Ax ¹ b . (confer §4.2.3.1.1)Now we consider solution to a discrete problem whose only known analyticalmethod of solution is combinatorial in complexity: given A∈ RM×N andb∈RM

find x ∈ RN

subject to Ax¹ b

δ(xxT) = 1

(784)

This nonconvex problem demands a Boolean solution [xi =±1, i=1 . . . N ].

Assign a rank-1 matrix of variables; symmetric variable matrix X andsolution vector x :

G =

[

x1

]

[xT 1 ]=

[

X xxT 1

]

,

[

xxT xxT 1

]

∈ SN+1 (785)

Then design an equivalent semidefinite feasibility problem to find a Booleansolution to Ax¹ b :

findX∈SN

x ∈ RN

subject to Ax¹ b

G =

[

X xxT 1

]

(º 0)

rank G = 1

δ(X) = 1

(786)

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354 CHAPTER 4. SEMIDEFINITE PROGRAMMING

where x⋆i ∈ −1, 1 , i=1 . . . N . The two variables X and x are made

dependent via their assignment to rank-1 matrix G . By (1585), an optimalrank-1 matrix G⋆ must take the form (785).

As before, we regularize the rank constraint by introducing a directionmatrix Y into the objective:

minimizeX∈SN , x∈RN

〈G , Y 〉subject to Ax¹ b

G =

[

X xxT 1

]

º 0

δ(X) = 1

(787)

Solution of this semidefinite program is iterated with calculation of thedirection matrix Y from semidefinite program (759). At convergence, in thesense (713), convex problem (787) becomes equivalent to nonconvex Booleanproblem (784).

Direction matrix Y can be an orthogonal projector having closed-formexpression, by (1663a), although convex iteration is not a projection method.(§4.4.1.1) Given randomized data A and b for a large problem, we find thatstalling becomes likely (convergence of the iteration to a positive objective〈G⋆, Y 〉). To overcome this behavior, we introduce a heuristic into theimplementation on Wıκımization that momentarily reverses direction ofsearch (like (760)) upon stall detection. We find that rate of convergencecan be sped significantly by detecting stalls early. 2

4.6.0.0.9 Example. Variable-vector normalization.Suppose, within some convex optimization problem, we want vector variablesx , y∈RN constrained by a nonconvex equality:

x‖y‖ = y (788)

id est, ‖x‖= 1 and x points in the same direction as y 6=0 ; e.g.,

minimizex , y

f(x , y)

subject to (x , y)∈ Cx‖y‖ = y

(789)

where f is some convex function and C is some convex set. We can realizethe nonconvex equality by constraining rank and adding a regularization

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 355

term to the objective. Make the assignment:

G =

xy

1

[xT yT 1 ]=

X Z xZ Y yxT yT 1

,

xxT xyT xyxT yyT yxT yT 1

∈ S2N+1 (790)

where X , Y ∈ SN , also Z∈ SN [sic] . Any rank-1 solution must take theform of (790). (§B.1) The problem statement equivalent to (789) is thenwritten

minimizeX , Y ∈ S , Z , x , y

f(x , y) + ‖X − Y ‖F

subject to (x , y)∈ C

G =

X Z xZ Y yxT yT 1

(º 0)

rank G = 1

tr(X) = 1

δ(Z) º 0

(791)

The trace constraint on X normalizes vector x while the diagonal constrainton Z maintains sign between respective entries of x and y . Regularizationterm ‖X−Y ‖F then makes x equal to y to within a real scalar; (§C.2.0.0.2)in this case, a positive scalar. To make this program solvable by convexiteration, as explained in Example 4.4.1.2.4 and other previous examples, wemove the rank constraint to the objective

minimizeX , Y , Z , x , y

f(x , y) + ‖X − Y ‖F + 〈G , W 〉subject to (x , y)∈ C

G =

X Z xZ Y yxT yT 1

º 0

tr(X) = 1

δ(Z) º 0

(792)

by introducing a direction matrix W found from (1663a):

minimizeW∈ S2N+1

〈G⋆, W 〉subject to 0 ¹ W ¹ I

tr W = 2N

(793)

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356 CHAPTER 4. SEMIDEFINITE PROGRAMMING

This semidefinite program has an optimal solution that is known inclosed form. Iteration (792) (793) terminates when rank G = 1 and linearregularization 〈G , W 〉 vanishes to within some numerical tolerance in (792);typically, in two iterations. If function f competes too much with theregularization, positively weighting each regularization term will becomerequired. At convergence, problem (792) becomes a convex equivalent tothe original nonconvex problem (789). 2

4.6.0.0.10 Example. fast max cut. [110]

Let Γ be an n-node graph, and let the arcs (i , j) of the graph beassociated with [ ] weights aij . The problem is to find a cut of thelargest possible weight, i.e., to partition the set of nodes into twoparts S , S ′ in such a way that the total weight of all arcs linkingS and S ′ (i.e., with one incident node in S and the other onein S ′ [Figure 104]) is as large as possible. [33, §4.3.3]

Literature on the max cut problem is vast because this problem has elegantprimal and dual formulation, its solution is very difficult, and there existmany commercial applications; e.g., semiconductor design [122], quantumcomputing [384].

Our purpose here is to demonstrate how iteration of two simple convexproblems can quickly converge to an optimal solution of the max cut

problem with a 98% success rate, on average.4.38 max cut is stated:

maximizex

1≤i<j≤n

aij(1 − xi xj)12

subject to δ(xxT) = 1(794)

where [aij] are real arc weights, and binary vector x = [xi]∈ Rn correspondsto the n nodes; specifically,

node i ∈ S ⇔ xi = 1node i ∈ S ′ ⇔ xi = −1

(795)

If nodes i and j have the same binary value xi and xj , then they belongto the same partition and contribute nothing to the cut. Arc (i , j) traversesthe cut, otherwise, adding its weight aij to the cut.

4.38We term our solution to max cut fast because we sacrifice a little accuracy to achievespeed; id est, only about two or three convex iterations, achieved by heavily weighting arank regularization term.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 357

-1

-35

1

-3

2

1

4

CUT

S

S ′

1

2 3 4

5

6

7

8

9

10

1112

13

14

15

16

Figure 104: A cut partitions nodes i=1 . . . 16 of this graph into S and S ′.Linear arcs have circled weights. The problem is to find a cut maximizingtotal weight of all arcs linking partitions made by the cut.

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358 CHAPTER 4. SEMIDEFINITE PROGRAMMING

max cut statement (794) is the same as, for A = [aij]∈ Sn

maximizex

14〈11T− xxT, A〉

subject to δ(xxT) = 1(796)

Because of Boolean assumption δ(xxT) = 1

〈11T− xxT, A〉 = 〈xxT, δ(A1) − A〉 (797)

so problem (796) is the same as

maximizex

14〈xxT, δ(A1) − A〉

subject to δ(xxT) = 1(798)

This max cut problem is combinatorial (nonconvex).

Because an estimate of upper bound to max cut is needed to ascertainconvergence when vector x has large dimension, we digress to derive thedual problem: Directly from (798), its Lagrangian is [59, §5.1.5] (1394)

L(x , ν) = 14〈xxT, δ(A1) − A〉 + 〈ν , δ(xxT) − 1〉

= 14〈xxT, δ(A1) − A〉 + 〈δ(ν) , xxT〉 − 〈ν , 1〉

= 14〈xxT, δ(A1 + 4ν) − A〉 − 〈ν , 1〉

(799)

where quadratic xT(δ(A1+ 4ν)−A)x has supremum 0 if δ(A1+ 4ν)−A isnegative semidefinite, and has supremum ∞ otherwise. The finite supremumof dual function

g(ν) = supx

L(x , ν) =

−〈ν , 1〉 , A − δ(A1 + 4ν) º 0∞ otherwise

(800)

is chosen to be the objective of minimization to dual (convex) problem

minimizeν

−νT1

subject to A − δ(A1 + 4ν) º 0(801)

whose optimal value provides a least upper bound to max cut, but is nottight (1

4〈xxT, δ(A1)−A〉< g(ν) , duality gap is nonzero). [153] In fact, we

find that the bound’s variance with problem instance is too large to beuseful for this problem; thus ending our digression.4.39

4.39Taking the dual of dual problem (801) would provide (802) but without the rankconstraint. [146] Dual of a dual of even a convex primal problem is not necessarily thesame primal problem; although, optimal solution of one can be obtained from the other.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 359

To transform max cut to its convex equivalent, first define

X = xxT ∈ Sn (806)

then max cut (798) becomes

maximizeX∈ S

n

14〈X , δ(A1) − A〉

subject to δ(X) = 1(X º 0)rank X = 1

(802)

whose rank constraint can be regularized as in

maximizeX∈ S

n

14〈X , δ(A1) − A〉 − w〈X , W 〉

subject to δ(X) = 1X º 0

(803)

where w≈1000 is a nonnegative fixed weight, and W is a direction matrixdetermined from

n∑

i=2

λ(X⋆)i = minimizeW∈ S

n〈X⋆, W 〉

subject to 0 ¹ W ¹ I

tr W = n − 1

(1663a)

which has an optimal solution that is known in closed form. These twoproblems (803) and (1663a) are iterated until convergence as defined onpage 301.

Because convex problem statement (803) is so elegant, it is numericallysolvable for large binary vectors within reasonable time.4.40 To test ourconvex iterative method, we compare an optimal convex result to anactual solution of the max cut problem found by performing a brute forcecombinatorial search of (798)4.41 for a tight upper bound. Search-time limitsbinary vector lengths to 24 bits (about five days CPU time). Accuracy

4.40We solved for a length-250 binary vector in only a few minutes and convex iterationson a 2006 vintage laptop Core 2 CPU (Intel [email protected], 666MHz FSB).4.41 more computationally intensive than the proposed convex iteration by many ordersof magnitude. Solving max cut by searching over all binary vectors of length 100, forexample, would occupy a contemporary supercomputer for a million years.

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360 CHAPTER 4. SEMIDEFINITE PROGRAMMING

obtained, 98%, is independent of binary vector length (12, 13, 20, 24)when averaged over more than 231 problem instances including planar,randomized, and toroidal graphs.4.42 When failure occurred, large andsmall errors were manifest. That same 98% average accuracy is presumedmaintained when binary vector length is further increased. A Matlab

program is provided on Wıκımization. 2

4.6.0.0.11 Example. Cardinality/rank problem.d’Aspremont, El Ghaoui, Jordan, & Lanckriet [93] propose approximating apositive semidefinite matrix A∈ SN

+ by a rank-one matrix having constrainton cardinality c : for 0 < c < N

minimizez

‖A − zzT‖F

subject to card z ≤ c(804)

which, they explain, is a hard problem equivalent to

maximizex

xTAx

subject to ‖x‖ = 1

card x ≤ c

(805)

where z ,√

λx and where optimal solution x⋆ is a principal eigenvector(1657) (§A.5) of A and λ = x⋆TAx⋆ is the principal eigenvalue [155, p.331]when c is true cardinality of that eigenvector. This is principal componentanalysis with a cardinality constraint which controls solution sparsity. Definethe matrix variable

X , xxT ∈ SN (806)

whose desired rank is 1, and whose desired diagonal cardinality

card δ(X) ≡ card x (807)

4.42Existence of a polynomial-time approximation to max cut with accuracy better than94.11% would refute proof of NP-hardness, which Hastad believes to be highly unlikely.[177, thm.8.2] [178]

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 361

is equivalent to cardinality c of vector x . Then we can transform cardinalityproblem (805) to an equivalent problem in new variable X :4.43

maximizeX∈SN

〈X , A〉subject to 〈X , I 〉 = 1

(X º 0)

rank X = 1

card δ(X) ≤ c

(808)

We transform problem (808) to an equivalent convex problem byintroducing two direction matrices into regularization terms: W to achievedesired cardinality card δ(X) , and Y to find an approximating rank-onematrix X :

maximizeX∈SN

〈X , A − w1Y 〉 − w2〈δ(X) , δ(W )〉subject to 〈X , I 〉 = 1

X º 0

(809)

where w1 and w2 are positive scalars respectively weighting tr(XY ) andδ(X)Tδ(W ) just enough to insure that they vanish to within some numericalprecision, where direction matrix Y is an optimal solution to semidefiniteprogram

minimizeY ∈ SN

〈X⋆, Y 〉subject to 0 ¹ Y ¹ I

tr Y = N − 1

(810)

and where diagonal direction matrix W ∈ SN optimally solves linear program

minimizeW=δ2(W )

〈δ(X⋆) , δ(W )〉subject to 0 ¹ δ(W ) ¹ 1

tr W = N − c

(811)

Both direction matrix programs are derived from (1663a) whose analyticalsolution is known but is not necessarily unique. We emphasize (confer p.301):because this iteration (809) (810) (811) (initial Y,W = 0) is not a projection

4.43A semidefiniteness constraint Xº 0 is not required, theoretically, because positivesemidefiniteness of a rank-1 matrix is enforced by symmetry. (Theorem A.3.1.0.7)

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362 CHAPTER 4. SEMIDEFINITE PROGRAMMING

phantom(256)

Figure 105: Shepp-Logan phantom from Matlab image processing toolbox.

method (§4.4.1.1), success relies on existence of matrices in the feasible setof (809) having desired rank and diagonal cardinality. In particular, thefeasible set of convex problem (809) is a Fantope (91) whose extreme pointsconstitute the set of all normalized rank-one matrices; among those are foundrank-one matrices of any desired diagonal cardinality.

Convex problem (809) is neither a relaxation of cardinality problem (805);instead, problem (809) is a convex equivalent to (805) at convergence ofiteration (809) (810) (811). Because the feasible set of convex problem (809)contains all normalized (§B.1) symmetric rank-one matrices of every nonzerodiagonal cardinality, a constraint too low or high in cardinality c willnot prevent solution. An optimal rank-one solution X⋆, whose diagonalcardinality is equal to cardinality of a principal eigenvector of matrix A ,will produce the lowest residual Frobenius norm (to within machine noiseprocesses) in the original problem statement (804). 2

4.6.0.0.12 Example. Compressive sampling of a phantom.In summer of 2004, Candes, Romberg, & Tao [68] and Donoho [115]released papers on perfect signal reconstruction from samples that stand inviolation of Shannon’s classical sampling theorem. These defiant signals areassumed sparse inherently or under some sparsifying affine transformation.Essentially, they proposed sparse sampling theorems asserting averagesample rate independent of signal bandwidth and less than Shannon’s rate.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 363

Minimum sampling rate: of Ω-bandlimited signal: 2Ω (Shannon) of k-sparse length-n signal: k log2(1+n/k) (Candes/Donoho)

(Figure 98). Certainly, much was already known about nonuniformor random sampling [35] [204] and about subsampling or multiratesystems [89] [354]. Vetterli, Marziliano, & Blu [363] had congealed atheory of noiseless signal reconstruction, in May 2001, from samples thatviolate the Shannon rate. They anticipated the sparsifying transformby recognizing: it is the innovation (onset) of functions constituting a(not necessarily bandlimited) signal that determines minimum samplingrate for perfect reconstruction. Average onset (sparsity) Vetterli et aliicall rate of innovation. The vector inner-products that Candes/Donohocall samples or measurements, Vetterli calls projections. From thoseprojections Vetterli demonstrates reconstruction (by digital signal processingand “root finding”) of a Dirac comb, the very same prototypicalsignal from which Candes probabilistically derives minimum samplingrate (Compressive Sampling and Frontiers in Signal Processing , Universityof Minnesota, June 6, 2007). Combining their terminology, we paraphrase asparse sampling theorem: Minimum sampling rate, asserted by Candes/Donoho, is proportional

to Vetterli’s rate of innovation (a.k.a, information rate, degrees offreedom, ibidem June 5, 2007).

What distinguishes these researchers are their methods of reconstruction.Properties of the 1-norm were also well understood by June 2004

finding applications in deconvolution of linear systems [81], penalized linearregression (Lasso) [314], and basis pursuit [209]. But never before hadthere been a formalized and rigorous sense that perfect reconstruction werepossible by convex optimization of 1-norm when information lost in asubsampling process became nonrecoverable by classical methods. Donohonamed this discovery compressed sensing to describe a nonadaptive perfectreconstruction method by means of linear programming. By the time Candes’and Donoho’s landmark papers were finally published by IEEE in 2006,compressed sensing was old news that had spawned intense research whichstill persists; notably, from prominent members of the wavelet community.

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364 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Reconstruction of the Shepp-Logan phantom (Figure 105), from aseverely aliased image (Figure 107) obtained by Magnetic ResonanceImaging (MRI), was the impetus driving Candes’ quest for a sparse samplingtheorem. He realized that line segments appearing in the aliased imagewere regions of high total variation. There is great motivation, in themedical community, to apply compressed sensing to MRI because it translatesto reduced scan-time which brings great technological and physiologicalbenefits. MRI is now about 35 years old, beginning in 1973 with Nobellaureate Paul Lauterbur from Stony Brook USA. There has been muchprogress in MRI and compressed sensing since 2004, but there have alsobeen indications of 1-norm abandonment (indigenous to reconstruction bycompressed sensing) in favor of criteria closer to 0-norm because of acorrespondingly smaller number of measurements required to accuratelyreconstruct a sparse signal:4.44

5481 complex samples (22 radial lines, ≈256 complex samples per) wererequired in June 2004 to reconstruct a noiseless 256×256-pixel Shepp-Loganphantom by 1-norm minimization of an image-gradient integral estimatecalled total variation; id est, 8.4% subsampling of 65536 data. [68, §1.1][67, §3.2] It was soon discovered that reconstruction of the Shepp-Loganphantom were possible with only 2521 complex samples (10 radial lines,Figure 106); 3.8% subsampled data input to a (nonconvex) 1

2-norm

total-variation minimization. [74, §IIIA] The closer to 0-norm, the fewer thesamples required for perfect reconstruction.

Passage of a few years witnessed an algorithmic speedup and dramaticreduction in minimum number of samples required for perfect reconstructionof the noiseless Shepp-Logan phantom. But minimization of total variationis ideally suited to recovery of any piecewise-constant image, like a phantom,because the gradient of such images is highly sparse by design.

There is no inherent characteristic of real-life MRI images that wouldmake reasonable an expectation of sparse gradient. Sparsification of adiscrete image-gradient tends to preserve edges. Then minimization of totalvariation seeks an image having fewest edges. There is no deeper theoreticalfoundation than that. When applied to human brain scan or angiogram,

4.44Efficient techniques continually emerge urging 1-norm criteria abandonment; [78] [353][352, §IID] e.g., five techniques for compressed sensing are compared in [36] demonstratingthat 1-norm performance limits for cardinality minimization can be reliably exceeded.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 365

with as much as 20% of 256×256 Fourier samples, we have observed4.45 a30dB image/reconstruction-error ratio4.46 barrier that seems impenetrableby the total-variation objective. Total-variation minimization has met withmoderate success, in retrospect, only because some medical images aremoderately piecewise-constant signals. One simply hopes a reconstruction,that is in some sense equal to a known subset of samples and whose gradientis most sparse, is that unique image we seek.4.47

The total-variation objective, operating on an image, is expressible asnorm of a linear transformation (830). It is natural to ask whether thereexist other sparsifying transforms that might break the real-life 30dB barrier(any sampling pattern @20% 256×256 data) in MRI. There has beenmuch research into application of wavelets, discrete cosine transform (DCT),randomized orthogonal bases, splines, etcetera, but with suspiciously littlefocus on objective measures like image/error or illustration of differenceimages; the predominant basis of comparison instead being subjectivelyvisual (Duensing & Huang, ISMRM Toronto 2008).4.48 Despite choice oftransform, there seems yet to have been a breakthrough of the 30dB barrier.Application of compressed sensing to MRI, therefore, remains fertile in 2008for continued research.

4.45Experiments with real-life images were performed by Christine Law at Lucas Centerfor Imaging, Stanford University.4.46Noise considered here is due only to the reconstruction process itself; id est, noisein excess of that produced by the best reconstruction of an image from a complete setof samples in the sense of Shannon. At less than 30dB image/error, artifacts generallyremain visible to the naked eye. We estimate about 50dB is required to eliminate noticeabledistortion in a visual A/B comparison.4.47In vascular radiology, diagnoses are almost exclusively based on morphology of vesselsand, in particular, presence of stenoses. There is a compelling argument for total-variationreconstruction of magnetic resonance angiogram because it helps isolate structures ofparticular interest.4.48I have never calculated the PSNR of these reconstructed images [of Barbara].

−Jean-Luc StarckThe sparsity of the image is the percentage of transform coefficients sufficient for

diagnostic-quality reconstruction. Of course the term “diagnostic quality” is subjective.. . . I have yet to see an “objective” measure of image quality. Difference images, in myexperience, definitely do not tell the whole story. Often I would show people some of myresults and get mixed responses, but when I add artificial Gaussian noise to an image,often people say that it looks better. −Michael Lustig

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366 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Lagrangian form of compressed sensing in imagingWe now repeat Candes’ image reconstruction experiment from 2004 whichled to discovery of sparse sampling theorems. [68, §1.2] But we achieveperfect reconstruction with an algorithm based on vanishing gradient of acompressed sensing problem’s Lagrangian, which is computationally efficient.Our contraction method (p.372) is fast also because matrix multiplications arereplaced by fast Fourier transforms and number of constraints is cut in halfby sampling symmetrically. Convex iteration for cardinality minimization(§4.5) is incorporated which allows perfect reconstruction of a phantom at4.1% subsampling rate; 50% Candes’ rate. By making neighboring-pixelselection adaptive, convex iteration reduces discrete image-gradient sparsityof the Shepp-Logan phantom to 1.9% ; 33% lower than previously reported.

We demonstrate application of discrete image-gradient sparsification tothe n×n=256×256 Shepp-Logan phantom, simulating idealized acquisitionof MRI data by radial sampling in the Fourier domain (Figure 106).4.49

Define a Nyquist-centric discrete Fourier transform (DFT) matrix

F ,

1 1 1 1 · · · 11 e−2π/n e−4π/n e−6π/n · · · e−(n−1)2π/n

1 e−4π/n e−8π/n e−12π/n · · · e−(n−1)4π/n

1 e−6π/n e−12π/n e−18π/n · · · e−(n−1)6π/n

......

......

. . ....

1 e−(n−1)2π/n e−(n−1)4π/n e−(n−1)6π/n · · · e−(n−1)22π/n

1√n∈ Cn×n

(812)

a symmetric (nonHermitian) unitary matrix characterized

F = FT

F−1 = FH (813)

Denoting an unknown image U ∈ Rn×n, its two-dimensional discrete Fouriertransform F is

F(U) , F UF (814)

hence the inverse discrete transform

U = FHF(U)FH (815)

4.49 k-space is conventional acquisition terminology indicating domain of the continuousraw data provided by an MRI machine. An image is reconstructed by inverse discreteFourier transform of that data interpolated on a Cartesian grid in two dimensions.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 367

Φ

Figure 106: MRI radial sampling pattern, in DC-centric Fourier domain,representing 4.1% (10 lines) subsampled data. Only half of these complexsamples, in any halfspace about the origin in theory, need be acquiredfor a real image because of conjugate symmetry. Due to MRI machineimperfections, samples are generally taken over full extent of each radialline segment. MRI acquisition time is proportional to number of lines.

From §A.1.1 no.31 we have a vectorized two-dimensional DFT via Kroneckerproduct ⊗

vec F(U) , (F⊗F ) vecU (816)

and from (815) its inverse [162, p.24]

vecU = (FH⊗FH)(F⊗F ) vecU = (FHF ⊗ FHF ) vecU (817)

Idealized radial sampling in the Fourier domain can be simulated byHadamard product with a binary mask Φ∈Rn×n whose nonzero entriescould, for example, correspond with the radial line segments in Figure 106.To make the mask Nyquist-centric, like DFT matrix F , define a circulant[164] symmetric permutation matrix4.50

Θ ,

[

0 II 0

]

∈ Sn (818)

Then given subsampled Fourier domain (MRI k-space) measurements in

4.50Matlab fftshift()

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368 CHAPTER 4. SEMIDEFINITE PROGRAMMING

incomplete K∈ Cn×n, we might constrain F(U) thus:

ΘΦΘ F UF = K (819)

and in vector form, (42) (1750)

δ(vec ΘΦΘ)(F⊗F ) vecU = vec K (820)

Because measurements K are complex, there are actually twice the numberof equality constraints as there are measurements.

We can cut that number of constraints in half via vertical and horizontalmask Φ symmetry which forces the imaginary inverse transform to 0 : Theinverse subsampled transform in matrix form is

FH(ΘΦΘ F UF )FH = FHKFH (821)

and in vector form

(FH⊗FH)δ(vec ΘΦΘ)(F⊗F ) vecU = (FH⊗FH) vec K (822)

later abbreviated

P vecU = f (823)

where

P , (FH⊗FH)δ(vec ΘΦΘ)(F⊗F ) ∈ Cn2×n2

(824)

Because of idempotence P = P 2, P is a projection matrix. Because of itsHermitian symmetry [162, p.24]

P = (FH⊗FH)δ(vec ΘΦΘ)(F⊗F ) = (F⊗F )Hδ(vec ΘΦΘ)(FH⊗FH)H = PH

(825)

P is an orthogonal projector.4.51 P vecU is real when P is real; id est, whenfor positive even integer n

Φ =

[

Φ11 Φ(1 , 2:n)ΞΞΦ(2:n , 1) ΞΦ(2:n , 2:n)Ξ

]

∈ Rn×n (826)

where Ξ∈Sn−1 is the order-reversing permutation matrix (1691). In words,this necessary and sufficient condition on Φ (for a real inverse subsampled

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 369

vec−1 f

Figure 107: Aliasing of Shepp-Logan phantom in Figure 105 resulting fromk-space subsampling pattern in Figure 106. This image is real becausebinary mask Φ is vertically and horizontally symmetric. It is remarkablethat the phantom can be reconstructed, by convex iteration, given onlyU0 = vec−1f .

transform [281, p.53]) demands vertical symmetry about row n2+1 and

horizontal symmetry4.52 about column n2+1.

Define

∆ ,

1 0 0

−1 1 0

−1 1. . .

. . . . . . . . .. . . 1 0

0T −1 1

∈ Rn×n (827)

Express an image-gradient estimate

∇U ,

U ∆U ∆T

∆ U∆T U

∈ R4n×n (828)

4.51 (824) is a diagonalization of matrix P whose binary eigenvalues are δ(vec ΘΦΘ) whilethe corresponding eigenvectors constitute the columns of unitary matrix FH⊗FH.4.52This condition on Φ applies to both DC- and Nyquist-centric DFT matrices.

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that is a simple first-order difference of neighboring pixels (Figure 108) tothe right, left, above, and below.4.53 By §A.1.1 no.31, its vectorization:for Ψi∈ Rn2×n2

vec∇U =

∆T⊗ I∆ ⊗ II ⊗ ∆I ⊗ ∆T

vecU ,

Ψ1

ΨT1

Ψ2

ΨT2

vecU , Ψ vecU ∈ R4n2

(829)

where Ψ∈ R4n2×n2

. A total-variation minimization for reconstructing MRIimage U , that is known suboptimal [203] [69], may be concisely posed

minimizeU

‖Ψ vecU‖1

subject to P vecU = f(830)

wheref = (FH⊗FH) vec K ∈ Cn2

(831)

is the known inverse subsampled Fourier data (a vectorized aliased image,Figure 107), and where a norm of discrete image-gradient ∇U is equivalentlyexpressed as norm of a linear transformation Ψ vecU .

Although this simple problem statement (830) is equivalent to a linearprogram (§3.3), its numerical solution is beyond the capability of eventhe most highly regarded of contemporary commercial solvers.4.54 Ouronly recourse is to recast the problem in Lagrangian form (§3.2) and writecustomized code to solve it:

minimizeU

〈|Ψ vecU| , y〉subject to P vecU = f

(a)

≡minimize

U〈|Ψ vecU| , y〉 + 1

2λ‖P vecU − f‖2

2 (b)

(832)

4.53There is significant improvement in reconstruction quality by augmentation of anominally two-point discrete image-gradient estimate to four points per pixel by inclusionof two polar directions. Improvement is due to centering; symmetry of discrete differencesabout a central pixel. We find small improvement on real-life images, ≈1dB empirically,by further augmentation with diagonally adjacent pixel differences.4.54 for images as small as 128×128 pixels. Obstacle to numerical solution is not acomputer resource: e.g., execution time, memory. The obstacle is, in fact, inadequatenumerical precision. Even when all dependent equality constraints are manually removed,the best commercial solvers fail simply because computer numerics become nonsense;id est, numerical errors enter significant digits and the algorithm exits prematurely, loopsindefinitely, or produces an infeasible solution.

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Figure 108: Neighboring-pixel stencil [353] for image-gradient estimation onCartesian grid. Implementation selects adaptively from darkest four • aboutcentral. Continuous image-gradient from two pixels holds only in a limit. Fordiscrete differences, better practical estimates are obtained when centered.

where multiobjective parameter λ∈R+ is quite large (λ≈1E8) so as to enforcethe equality constraint: P vecU−f = 0 ⇔ ‖P vecU−f‖2

2 =0 (§A.7.1). We

introduce a direction vector y∈ R4n2

+ as part of a convex iteration (§4.5.2)to overcome that known suboptimal minimization of discrete image-gradientcardinality: id est, there exists a vector y⋆ with entries y⋆

i ∈ 0 , 1 such that

minimizeU

‖Ψ vecU‖0

subject to P vecU = f≡ minimize

U〈|Ψ vecU| , y⋆〉 + 1

2λ‖P vecU − f‖2

2

(833)

Existence of such a y⋆, complementary to an optimal vector Ψ vecU⋆, isobvious by definition of global optimality 〈|Ψ vecU⋆| , y⋆〉= 0 (734) underwhich a cardinality-c optimal objective ‖Ψ vecU⋆‖0 is assumed to exist.

Because (832b) is an unconstrained convex problem, a zero in theobjective function’s gradient is necessary and sufficient for optimality(§2.13.3); id est, (§D.2.1)

ΨTδ(y) sgn(Ψ vecU ) + λPH(P vecU − f ) = 0 (834)

Because of P idempotence and Hermitian symmetry and sgn(x)= x/|x| ,

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372 CHAPTER 4. SEMIDEFINITE PROGRAMMING

this is equivalent to

limǫ→0

(

ΨTδ(y)δ(|Ψ vecU| + ǫ1)−1 Ψ + λP)

vecU = λPf (835)

where small positive constant ǫ∈ R+ has been introduced for invertibility.The analytical effect of introducing ǫ is to make the objective’s gradient valideverywhere in the function domain; id est, it transforms absolute value in(832b) from a function differentiable almost everywhere into a differentiablefunction. An example of such a transformation in one dimension is illustratedin Figure 109. When small enough for practical purposes4.55 (ǫ≈1E-3), wemay ignore the limiting operation. Then the mapping, for 0¹ y¹ 1

vecU t+1 =(

ΨTδ(y)δ(|Ψ vecU t| + ǫ1)−1 Ψ + λP)−1

λPf (836)

is a contraction in U t that can be solved recursively in t for itsunique fixed point ; id est, until U t+1→ U t . [223, p.300] [200, p.155]Calculating this inversion directly is not possible for large matrices oncontemporary computers because of numerical precision, so instead we applythe conjugate gradient method of solution to

(

ΨTδ(y)δ(|Ψ vecU t| + ǫ1)−1 Ψ + λP)

vecU t+1 = λPf (837)

which is linear in U t+1 at each recursion in the Matlab program.4.56

Observe that P (824), in the equality constraint from problem (832a), isnot a fat matrix.4.57 Although number of Fourier samples taken is equal tothe number of nonzero entries in binary mask Φ , matrix P is square butnever actually formed during computation. Rather, a two-dimensional fastFourier transform of U is computed followed by masking with ΘΦΘ andthen an inverse fast Fourier transform. This technique significantly reducesmemory requirements and, together with contraction method of solution, isthe principal reason for relatively fast computation.

4.55We are looking for at least 50dB image/error ratio from only 4.1% subsampled data(10 radial lines in k-space). With this setting of ǫ , we actually attain in excess of 100dBfrom a simple Matlab program in about a minute on a 2006 vintage laptop Core 2 CPU(Intel [email protected], 666MHz FSB). By trading execution time and treating discreteimage-gradient cardinality as a known quantity for this phantom, over 160dB is achievable.4.56Conjugate gradient method requires positive definiteness. [149, §4.8.3.2]4.57Fat is typical of compressed sensing problems; e.g., [67] [74].

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 373

∫ x

−1y

|y|+ǫdy

|x|

Figure 109: Real absolute value function f2(x)= |x| on x∈ [−1, 1] fromFigure 67b superimposed upon integral of its derivative at ǫ=0.05 whichsmooths objective function.

convex iterationBy convex iteration we mean alternation of solution to (832a) and (838)until convergence. Direction vector y is initialized to 1 until the first fixedpoint is found; which means, the contraction recursion begins calculatinga (1-norm) solution U⋆ to (830) via problem (832b). Once U⋆ is found,vector y is updated according to an estimate of discrete image-gradientcardinality c : Sum of the 4n2− c smallest entries of |Ψ vecU⋆|∈ R4n2

isthe optimal objective value from a linear program, for 0≤c≤ 4n2− 1 (503)

4n2∑

i=c+1

π(|Ψ vecU⋆|)i = minimizey∈R4n2

〈|Ψ vecU⋆| , y〉subject to 0 ¹ y ¹ 1

yT1 = 4n2− c

(838)

where π is the nonlinear permutation-operator sorting its vector argumentinto nonincreasing order. An optimal solution y to (838), that is anextreme point of its feasible set, is known in closed form: it has 1 in eachentry corresponding to the 4n2− c smallest entries of |Ψ vecU⋆| and has 0elsewhere (page 327). Updated image U⋆ is assigned to U t , the contractionis recomputed solving (832b), direction vector y is updated again, andso on until convergence which is guaranteed by virtue of a monotonicallynonincreasing real sequence of objective values in (832a) and (838).

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374 CHAPTER 4. SEMIDEFINITE PROGRAMMING

There are two features that distinguish problem formulation (832b) andour particular implementation of it [Matlab on Wıκımization]:

1) An image-gradient estimate may engage any combination of fouradjacent pixels. In other words, the algorithm is not lockedinto a four-point gradient estimate (Figure 108); number of pointsconstituting an estimate is directly determined by direction vectory . 4.58 Indeed, we find only c = 5092 zero entries in y⋆ for theShepp-Logan phantom; meaning, discrete image-gradient sparsity isactually closer to 1.9% than the 3% reported elsewhere; e.g., [352, §IIB].

2) Numerical precision of the fixed point of contraction (836) (≈1E-2

for almost perfect reconstruction @−103dB error) is a parameter tothe implementation; meaning, direction vector y is updated aftercontraction begins but prior to its culmination. Impact of thisidiosyncrasy tends toward simultaneous optimization in variables Uand y while insuring y settles on a boundary point of the feasible set(nonnegative hypercube slice) in (838) at every iteration; for only aboundary point4.59 can yield the sum of smallest entries in |Ψ vecU⋆|.

Almost perfect reconstruction of the Shepp-Logan phantom at 103dBimage/error is achieved in a Matlab minute with 4.1% subsampled data(2671 complex samples); well below an 11% least lower bound predicted bythe sparse sampling theorem. Because reconstruction approaches optimalsolution to a 0-norm problem, minimum number of Fourier-domain samplesis bounded below by cardinality of discrete image-gradient at 1.9%. 2

4.6.0.0.13 Exercise. Contraction operator.Determine conditions on λ and ǫ under which (836) is a contraction andΨTδ(y)δ(|Ψ vecU t| + ǫ1)−1 Ψ + λP from (837) is positive definite. H

4.58This adaptive gradient was not contrived. It is an artifact of the convex iterationmethod for minimum cardinality solution; in this case, cardinality minimization of adiscrete image-gradient.4.59Simultaneous optimization of these two variables U and y should never be a pinnacleof aspiration; for then, optimal y might not attain a boundary point.

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 375

4.6.1 Constraining rank of indefinite matrices

Example 4.6.1.0.1, which follows, demonstrates that convex iteration is moregenerally applicable: to indefinite or nonsquare matrices X∈ Rm×n ; not onlyto positive semidefinite matrices. Indeed,

findX∈R

m×nX

subject to X ∈ Crank X ≤ k

findX, Y, Z

X

subject to X ∈ C

G =

[

Y XXT Z

]

rank G ≤ k

(839)

Proof. rank G ≤ k ⇒ rank X ≤ k because X is the projection of compositematrix G on subspace Rm×n. For symmetric Y and Z , any rank-k positivesemidefinite composite matrix G can be factored into rank-k terms R ;G = RTR where R , [ B C ] and rank B, rank C ≤ rank R and B∈ Rk×m

and C∈ Rk×n. Because Y and Z and X = BTC are variable, (1441)rank X ≤ rank B, rank C ≤ rank R = rank G is tight. ¨

So there must exist an optimal direction vector W ⋆ such that

findX, Y, Z

X

subject to X ∈ C

G =

[

Y XXT Z

]

rank G ≤ k

minimizeX, Y, Z

〈G , W ⋆〉

subject to X ∈ C

G =

[

Y XXT Z

]

º 0

(840)

Were W ⋆ = I , by (1651) the optimal resulting trace objective would beequivalent to the minimization of nuclear norm of X over C . This means: (confer p.222) Any nuclear norm minimization problem may have its

variable replaced with a composite semidefinite variable.

Then Figure 83 becomes an accurate geometrical description of a consequentcomposite semidefinite problem objective. But there are better directionvectors than identity I which occurs only under special conditions:

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376 CHAPTER 4. SEMIDEFINITE PROGRAMMING

Figure 110: Massachusetts Institute of Technology (MIT) logo, including itswhite boundary, may be interpreted as a rank-5 matrix. (Stanford Universitylogo rank is much higher;) This constitutes Scene Y observed by theone-pixel camera in Figure 111 for Example 4.6.1.0.1.

4.6.1.0.1 Example. Compressed sensing, compressive sampling. [296]As our modern technology-driven civilization acquires and exploitsever-increasing amounts of data, everyone now knows that most of thedata we acquire can be thrown away with almost no perceptual loss − witnessthe broad success of lossy compression formats for sounds, images, andspecialized technical data. The phenomenon of ubiquitous compressibilityraises very natural questions: Why go to so much effort to acquire all thedata when most of what we get will be thrown away? Can’t we just directlymeasure the part that won’t end up being thrown away? −David Donoho [115]

Lossy data compression techniques like JPEG are popular, but it is alsowell known that compression artifacts become quite perceptible with signalpostprocessing that goes beyond mere playback of a compressed signal. [216][241] Spatial or audio frequencies presumed masked by a simultaneity, sonot encoded for example, can be rendered imperceptible with significantpostfiltering (of the compressed signal) that is meant to reveal them; id est,desirable artifacts are forever lost, so highly compressed data is not amenableto analysis and postprocessing: e.g., sound effects [95] [96] [97] or imageenhancement (Photoshop). Further, there can be no universally acceptableunique metric of perception for gauging exactly how much data can be tossed.For these reasons, there will always be need for raw (uncompressed) data.

In this example we throw out only so much information as to leave perfectreconstruction within reach. Specifically, the MIT logo in Figure 110 is

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 377

Y

yi

Figure 111: One-pixel camera. [341] [367] Compressive imaging camerablock diagram. Incident lightfield (corresponding to the desired image Y )is reflected off a digital micromirror device (DMD) array whose mirrororientations are modulated in the pseudorandom pattern supplied by therandom number generators (RNG). Each different mirror pattern producesa voltage at the single photodiode that corresponds to one measurement yi .

perfectly reconstructed from 700 time-sequential samples yi acquired bythe one-pixel camera illustrated in Figure 111. The MIT-logo image in thisexample impinges a 46×81 array micromirror device. This mirror arrayis modulated by a pseudonoise source that independently positions all theindividual mirrors. A single photodiode (one pixel) integrates incidentlight from all mirrors. After stabilizing the mirrors to a fixed butpseudorandom pattern, light so collected is then digitized into one sampleyi by analog-to-digital (A/D) conversion. This sampling process is repeatedwith the micromirror array modulated to a new pseudorandom pattern.

The most important questions are: How many samples do we need forperfect reconstruction? Does that number of samples represent compressionof the original data?

We claim that perfect reconstruction of the MIT logo can be reliablyachieved with as few as 700 samples y=[yi]∈ R700 from this one-pixelcamera. That number represents only 19% of information obtainable from3726 micromirrors.4.60 (Figure 112)

4.60That number (700 samples) is difficult to achieve, as reported in [296, §6]. If a minimalbasis for the MIT logo were instead constructed, only five rows or columns worth of

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378 CHAPTER 4. SEMIDEFINITE PROGRAMMING

0 1000 2000 3000 3726

1 2 3 4 5

samples

Figure 112: Estimates of compression for various encoding methods:1) linear interpolation (140 samples),2) minimal columnar basis (311 samples),3) convex iteration (700 samples) can achieve lower bound predicted bycompressed sensing (670 samples, n=46×81, k =140, Figure 98) whereasnuclear norm minimization alone does not [296, §6],4) JPEG @100% quality (2588 samples),5) no compression (3726 samples).

Our approach to reconstruction is to look for low-rank solution to anunderdetermined system:

findX∈R46×81

X

subject to A vec X = yrank X ≤ 5

(841)

where vec X is the vectorized (37) unknown image matrix. Each row offat matrix A is one realization of a pseudorandom pattern applied to themicromirrors. Since these patterns are deterministic (known), then the ith

sample yi equals A(i , :) vec Y ; id est, y = A vec Y . Perfect reconstructionhere means optimal solution X⋆ equals scene Y ∈ R46×81 to within machineprecision.

Because variable matrix X is generally not square or positive semidefinite,we constrain its rank by rewriting the problem equivalently

findW1∈R46×46, W2∈R81×81, X∈R46×81

X

subject to A vec X = y

rank

[

W1 XXT W2

]

≤ 5

(842)

data (from a 46×81 matrix) are linearly independent. This means a lower bound onachievable compression is about 5×46 = 230 samples plus 81 samples column encoding;which corresponds to 8% of the original information. (Figure 112)

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4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 379

This rank constraint on the composite (block) matrix insures rank X≤ 5for any choice of dimensionally compatible matrices W1 and W2 . But tosolve this problem by convex iteration, we alternate solution of semidefiniteprogram

minimizeW1∈ S46, W2∈ S81, X∈R46×81

tr

([

W1 XXT W2

]

Z

)

subject to A vec X = y[

W1 XXT W2

]

º 0

(843)

with semidefinite program

minimizeZ∈ S46+81

tr

([

W1 XXT W2

]⋆

Z

)

subject to 0 ¹ Z ¹ I

tr Z = 46 + 81 − 5

(844)

(which has an optimal solution that is known in closed form, p.646) until arank-5 composite matrix is found.

With 1000 samples yi , convergence occurs in two iterations; 700samples require more than ten iterations but reconstruction remainsperfect. Iterating more admits taking of fewer samples. Reconstruction isindependent of pseudorandom sequence parameters; e.g., binary sequencessucceed with the same efficiency as Gaussian or uniformly distributedsequences. 2

4.6.2 rank-constraint midsummary

We find that this direction matrix idea works well and quite independentlyof desired rank or affine dimension. This idea of direction matrix is goodprincipally because of its simplicity: When confronted with a problemotherwise convex if not for a rank or cardinality constraint, then thatconstraint becomes a linear regularization term in the objective.

There exists a common thread through all these Examples; that being,convex iteration with a direction matrix as normal to a linear regularization(a generalization of the well-known trace heuristic). But each problem type(per Example) possesses its own idiosyncrasies that slightly modify how arank-constrained optimal solution is actually obtained: The ball packing

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380 CHAPTER 4. SEMIDEFINITE PROGRAMMING

problem in Chapter 5.4.2.3.4, for example, requires a problem sequence ina progressively larger number of balls to find a good initial value for thedirection matrix, whereas many of the examples in the present chapterrequire an initial value of 0. Finding a Boolean solution in Example 4.6.0.0.8requires a procedure to detect stalls, while other problems have no suchrequirement. The combinatorial Procrustes problem in Example 4.6.0.0.3allows use of a known closed-form solution for direction vector when solvedvia rank constraint, but not when solved via cardinality constraint. Someproblems require a careful weighting of the regularization term, whereas otherproblems do not, and so on. It would be nice if there were a universallyapplicable method for constraining rank; one that is less susceptible to quirksof a particular problem type.

Poor initialization of the direction matrix from the regularization canlead to an erroneous result. We speculate one reason to be a simple dearthof optimal solutions of desired rank or cardinality;4.61 an unfortunate choiceof initial search direction leading astray. Ease of solution by convex iterationoccurs when optimal solutions abound. With this speculation in mind, wenow propose a further generalization of convex iteration for constraining rankthat attempts to ameliorate quirks and unify problem types:

4.7 Convex Iteration rank-1

We now develop a general method for constraining rank that first decomposesa given problem via standard diagonalization of matrices (§A.5). Thismethod is motivated by observation (§4.4.1.1) that an optimal directionmatrix can be simultaneously diagonalizable with an optimal variablematrix. This suggests minimization of an objective function directly interms of eigenvalues. A second motivating observation is that variableorthogonal matrices seem easily found by convex iteration; e.g., ProcrustesExample 4.6.0.0.2.

It turns out that this general method always requires solution to a rank-1constrained problem regardless of desired rank from the original problem. To

4.61Recall that, in Convex Optimization, an optimal solution generally comes from aconvex set of optimal solutions that can be large.

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4.7. CONVEX ITERATION RANK-1 381

demonstrate, we pose a semidefinite feasibility problem

find X ∈ Sn

subject to A svec X = b

X º 0

rank X ≤ ρ

(845)

given an upper bound 0 < ρ < n on rank, vector b∈ Rm, and typically fatfull-rank

A =

svec(A1)T

...svec(Am)T

∈ Rm×n(n+1)/2 (626)

where Ai∈ Sn, i=1 . . . m are also given. Symmetric matrix vectorizationsvec is defined in (56). Thus

A svec X =

tr(A1 X )...

tr(Am X )

(627)

This program (845) states the classical problem of finding a matrix ofspecified rank in the intersection of the positive semidefinite cone witha number m of hyperplanes in the subspace of symmetric matrices Sn.[26, §II.13] [24, §2.2]

Express the nonincreasingly ordered diagonalization of variable matrix

X , QΛQT =n

i=1

λi Qii ∈ Sn (846)

which is a sum of rank-1 orthogonal projection matrices Qii weighted byeigenvalues λi where Qij , qiq

Tj ∈ Rn×n, Q = [ q1 · · · qn ]∈ Rn×n, QT = Q−1,

Λii = λi ∈ R , and

Λ =

λ1 0λ2

. . .

0T λn

∈ Sn (847)

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382 CHAPTER 4. SEMIDEFINITE PROGRAMMING

The fact

Λ º 0 ⇔ X º 0 (1426)

allows splitting semidefinite feasibility problem (845) into two parts:

minimizeΛ

A svec

(

ρ∑

i=1

λi Qii

)

− b

subject to δ(Λ) ∈[

Rρ+

0

] (848)

minimizeQ

A svec

(

ρ∑

i=1

λ⋆i Qii

)

− b

subject to QT = Q−1

(849)

The linear equality constraint A svec X = b has been moved to the objectivewithin a norm because these two problems (848) (849) are iterated; equalitymight only become attainable near convergence. This iteration alwaysconverges to a local minimum because the sequence of objective values ismonotonic and nonincreasing; any monotonically nonincreasing real sequenceconverges. [254, §1.2] [41, §1.1] A rank-ρ matrix X solving the originalproblem (845) is found when the objective converges to 0 ; a certificate ofglobal optimality for the iteration. Positive semidefiniteness of matrix Xwith an upper bound ρ on rank is assured by the constraint on eigenvaluematrix Λ in convex problem (848); it means, the main diagonal of Λ mustbelong to the nonnegative orthant in a ρ-dimensional subspace of Rn.

The second problem (849) in the iteration is not convex. We proposesolving it by convex iteration: Make the assignment

G =

q1...qρ

[ qT1 · · · qT

ρ ]∈ Snρ

=

Q11 · · · Q1ρ...

. . ....

QT1ρ · · · Qρρ

,

q1qT1 · · · q1q

.... . .

...qρq

T1 · · · qρq

(850)

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4.7. CONVEX ITERATION RANK-1 383

Given ρ eigenvalues λ⋆i , then a problem equivalent to (849) is

minimizeQii∈S

n, Qij∈R

n×n

A svec

(

ρ∑

i=1

λ⋆i Qii

)

− b

subject to tr Qii = 1 , i=1 . . . ρtr Qij = 0 , i<j = 2 . . . ρ

G =

Q11 · · · Q1ρ...

. . ....

QT1ρ · · · Qρρ

(º 0)

rank G = 1

(851)

This new problem always enforces a rank-1 constraint on matrix G ; id est,regardless of upper bound on rank ρ of variable matrix X , this equivalentproblem (851) always poses a rank-1 constraint. Given some fixed weightingfactor w , we propose solving (851) by iteration of the convex sequence

minimizeQij∈R

n×n

A svec

(

ρ∑

i=1

λ⋆i Qii

)

− b

+ w tr(GW )

subject to tr Qii = 1 , i=1 . . . ρtr Qij = 0 , i<j = 2 . . . ρ

G =

Q11 · · · Q1ρ...

. . ....

QT1ρ · · · Qρρ

º 0

(852)

withminimize

W∈ Snρ

tr(G⋆ W )

subject to 0 ¹ W ¹ I

tr W = nρ − 1

(853)

the latter providing direction of search W for a rank-1 matrix G in (852).An optimal solution to (853) has closed form (p.646). These convex problems(852) (853) are iterated with (848) until a rank-1 G matrix is found and theobjective of (852) vanishes.

For a fixed number of equality constraints m and upper bound ρ , thefeasible set in rank-constrained semidefinite feasibility problem (845) growswith matrix dimension n . Empirically, this semidefinite feasibility problembecomes easier to solve by method of convex iteration as n increases. We

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384 CHAPTER 4. SEMIDEFINITE PROGRAMMING

find a good value of weight w≈ 5 for randomized problems. Initial value ofdirection matrix W is not critical. With dimension n = 26 and numberof equality constraints m = 10, convergence to a rank-ρ = 2 solution tosemidefinite feasibility problem (845) is achieved for nearly all realizationsin less than 10 iterations.4.62 For small n , stall detection is required; onenumerical implementation is disclosed on Wıκımization.

This diagonalization decomposition technique is extensible to otherproblem types, of course, including optimization problems having nontrivialobjectives. Because of a greatly expanded feasible set, for example,

find X ∈ Sn

subject to A svec X º b

X º 0

rank X ≤ ρ

(854)

this relaxation of (845) [221, §III] is more easily solved by convex iterationrank-1 .4.63 Because of the nonconvex nature of a rank-constrained problem,more generally, there can be no proof of global convergence of convex iterationfrom an arbitrary initialization; although, local convergence is guaranteed byvirtue of monotonically nonincreasing real objective sequences. [254, §1.2][41, §1.1]

4.62This finding is significant in so far as Barvinok’s Proposition 2.9.3.0.1 predictsexistence of matrices having rank 4 or less in this intersection, but his upper bound canbe no tighter than 3.4.63The inequality in A remains in the constraints after diagonalization.

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Chapter 5

Euclidean Distance Matrix

These results [(886)] were obtained by Schoenberg (1935), asurprisingly late date for such a fundamental property ofEuclidean geometry.

−John Clifford Gower [158, §3]

By itself, distance information between many points in Euclidean space islacking. We might want to know more; such as, relative or absolute positionor dimension of some hull. A question naturally arising in some fields(e.g., geodesy, economics, genetics, psychology, biochemistry, engineering)[101] asks what facts can be deduced given only distance information. Whatcan we know about the underlying points that the distance informationpurports to describe? We also ask what it means when given distanceinformation is incomplete; or suppose the distance information is not reliable,available, or specified only by certain tolerances (affine inequalities). Thesequestions motivate a study of interpoint distance, well represented in anyspatial dimension by a simple matrix from linear algebra.5.1 In what follows,we will answer some of these questions via Euclidean distance matrices.

5.1 e.g.,

√D∈RN×N , a classical two-dimensional matrix representation of absolute

interpoint distance because its entries (in ordered rows and columns) can be written neatlyon a piece of paper. Matrix D will be reserved throughout to hold distance-square.© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

385

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386 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

√5

2

1

D =

1 2 3

0 1 51 0 45 4 0

1

2

3

Figure 113: Convex hull of three points (N = 3) is shaded in R3 (n=3).Dotted lines are imagined vectors to points whose affine dimension is 2.

5.1 EDM

Euclidean space Rn is a finite-dimensional real vector space having an innerproduct defined on it, inducing a metric. [223, §3.1] A Euclidean distancematrix, an EDM in RN×N

+ , is an exhaustive table of distance-square dij

between points taken by pair from a list of N points xℓ , ℓ=1 . . . N in Rn ;the squared metric, the measure of distance-square:

dij = ‖xi − xj‖22 , 〈xi − xj , xi − xj〉 (855)

Each point is labelled ordinally, hence the row or column index of an EDM,i or j =1 . . . N , individually addresses all the points in the list.

Consider the following example of an EDM for the case N = 3 :

D = [dij] =

d11 d12 d13

d21 d22 d23

d31 d32 d33

=

0 d12 d13

d12 0 d23

d13 d23 0

=

0 1 51 0 45 4 0

(856)

Matrix D has N 2 entries but only N(N−1)/2 pieces of information. InFigure 113 are shown three points in R3 that can be arranged in a list

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5.2. FIRST METRIC PROPERTIES 387

to correspond to D in (856). But such a list is not unique because anyrotation, reflection, or translation (§5.5) of those points would produce thesame EDM D .

5.2 First metric properties

For i,j =1 . . . N , distance between points xi and xj must satisfy thedefining requirements imposed upon any metric space: [223, §1.1] [254, §1.7]namely, for Euclidean metric

dij (§5.4) in Rn

1.√

dij ≥ 0 , i 6= j nonnegativity

2.√

dij = 0 ⇔ xi = xj self-distance

3.√

dij =√

dji symmetry

4.√

dij ≤√

dik +√

dkj , i 6=j 6=k triangle inequality

Then all entries of an EDM must be in concord with these Euclidean metricproperties: specifically, each entry must be nonnegative,5.2 the main diagonalmust be 0 ,5.3 and an EDM must be symmetric. The fourth propertyprovides upper and lower bounds for each entry. Property 4 is true moregenerally when there are no restrictions on indices i,j,k , but furnishes nonew information.

5.3 ∃ fifth Euclidean metric property

The four properties of the Euclidean metric provide information insufficientto certify that a bounded convex polyhedron more complicated than atriangle has a Euclidean realization. [158, §2] Yet any list of points or thevertices of any bounded convex polyhedron must conform to the properties.

5.2Implicit from the terminology,√

dij ≥ 0 ⇔ dij ≥ 0 is always assumed.5.3What we call self-distance, Marsden calls nondegeneracy. [254, §1.6] Kreyszig calls

these first metric properties axioms of the metric; [223, p.4] Blumenthal refers to them aspostulates. [48, p.15]

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388 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.3.0.0.1 Example. Triangle.Consider the EDM in (856), but missing one of its entries:

D =

0 1 d13

1 0 4d31 4 0

(857)

Can we determine unknown entries of D by applying the metric properties?Property 1 demands

√d13 ,

√d31 ≥ 0, property 2 requires the main diagonal

be 0, while property 3 makes√

d31 =√

d13 . The fourth property tells us

1 ≤√

d13 ≤ 3 (858)

Indeed, described over that closed interval [1, 3] is a family of triangularpolyhedra whose angle at vertex x2 varies from 0 to π radians. So, yes wecan determine the unknown entries of D , but they are not unique; nor shouldthey be from the information given for this example. 2

5.3.0.0.2 Example. Small completion problem, I.Now consider the polyhedron in Figure 114b formed from an unknownlist x1 , x2 , x3 , x4. The corresponding EDM less one critical piece ofinformation, d14 , is given by

D =

0 1 5 d14

1 0 4 15 4 0 1

d14 1 1 0

(859)

From metric property 4 we may write a few inequalities for the two trianglescommon to d14 ; we find

√5−1 ≤

d14 ≤ 2 (860)

We cannot further narrow those bounds on√

d14 using only the four metricproperties (§5.8.3.1.1). Yet there is only one possible choice for

√d14 because

points x2 , x3 , x4 must be collinear. All other values of√

d14 in the interval[√

5−1, 2] specify impossible distances in any dimension; id est, in thisparticular example the triangle inequality does not yield an interval for

√d14

over which a family of convex polyhedra can be reconstructed. 2

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5.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY 389

x1 x2

x3

x4

√5

1

1

1

2

x1 x2

x3

x4 (b)(a)

Figure 114: (a) Complete dimensionless EDM graph. (b) Emphasizingobscured segments x2x4 , x4x3 , and x2x3 , now only five (2N−3) absolutedistances are specified. EDM so represented is incomplete, missing d14 asin (859), yet the isometric reconstruction (§5.4.2.3.7) is unique as proved in§5.9.3.0.1 and §5.14.4.1.1. First four properties of Euclidean metric are nota recipe for reconstruction of this polyhedron.

We will return to this simple Example 5.3.0.0.2 to illustrate more elegantmethods of solution in §5.8.3.1.1, §5.9.3.0.1, and §5.14.4.1.1. Until then, wecan deduce some general principles from the foregoing examples: Unknown dij of an EDM are not necessarily uniquely determinable. The triangle inequality does not produce necessarily tight bounds.5.4 Four Euclidean metric properties are insufficient for reconstruction.

5.4The term tight with reference to an inequality means equality is achievable.

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390 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.3.1 Lookahead

There must exist at least one requirement more than the four properties ofthe Euclidean metric that makes them altogether necessary and sufficient tocertify realizability of bounded convex polyhedra. Indeed, there are infinitelymany more; there are precisely N +1 necessary and sufficient Euclideanmetric requirements for N points constituting a generating list (§2.3.2). Hereis the fifth requirement:

5.3.1.0.1 Fifth Euclidean metric property. Relative-angle inequality.(confer §5.14.2.1.1) Augmenting the four fundamental properties of theEuclidean metric in Rn, for all i, j, ℓ 6= k∈1 . . . N , i<j<ℓ , and forN ≥ 4 distinct points xk , the inequalities

cos(θikℓ + θℓkj) ≤ cos θikj ≤ cos(θikℓ − θℓkj)

0 ≤ θikℓ , θℓkj , θikj ≤ π(861)

where θikj = θjki represents angle between vectors at vertex xk (930)(Figure 115), must be satisfied at each point xk regardless of affinedimension. ⋄

We will explore this in §5.14. One of our early goals is to determinematrix criteria that subsume all the Euclidean metric properties and anyfurther requirements. Looking ahead, we will find (1211) (886) (890)

−zTDz ≥ 01Tz = 0

(∀ ‖z‖ = 1)

D ∈ SNh

⇔ D ∈ EDMN (862)

where the convex cone of Euclidean distance matrices EDMN ⊆ SNh belongs

to the subspace of symmetric hollow5.5 matrices (§2.2.3.0.1). [Numerical testisedm(D) is provided on Wıκımization.] Having found equivalent matrixcriteria, we will see there is a bridge from bounded convex polyhedra toEDMs in §5.9 .5.6

5.5 0 main diagonal.5.6From an EDM, a generating list (§2.3.2, §2.12.2) for a polyhedron can be found (§5.12)

correct to within a rotation, reflection, and translation (§5.5).

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5.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY 391

θikj θjkℓ

θikℓ

k

i

j

Figure 115: Fifth Euclidean metric property nomenclature. Each angle θ ismade by a vector pair at vertex k while i, j, k, ℓ index four points at thevertices of a generally irregular tetrahedron. The fifth property is necessaryfor realization of four or more points; a reckoning by three angles in anydimension. Together with the first four Euclidean metric properties, thisfifth property is necessary and sufficient for realization of four points.

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392 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Now we develop some invaluable concepts, moving toward a link of theEuclidean metric properties to matrix criteria.

5.4 EDM definition

Ascribe points in a list xℓ ∈ Rn, ℓ=1 . . . N to the columns of a matrix

X = [ x1 · · · xN ] ∈ Rn×N (76)

where N is regarded as cardinality of list X . When matrix D = [dij] is anEDM, its entries must be related to those points constituting the list by theEuclidean distance-square: for i, j =1 . . . N (§A.1.1 no.33)

dij = ‖xi − xj‖2 = (xi − xj)T(xi − xj) = ‖xi‖2 + ‖xj‖2 − 2xT

i xj

=[

xTi xT

j

]

[

I −I−I I

] [

xi

xj

]

= vec(X)T(Φij ⊗ I ) vec X = 〈Φij , XTX 〉

(863)

where

vec X =

x1

x2...

xN

∈ RnN (864)

and where ⊗ signifies Kronecker product (§D.1.2.1). Φij ⊗ I is positivesemidefinite (1456) having I∈ Sn in its iith and jj th block of entries while−I∈ Sn fills its ij th and jith block; id est,

Φij , δ((eieTj + eje

Ti )1) − (eie

Tj + eje

Ti ) ∈ SN

+

= eieTi + eje

Tj − eie

Tj − eje

Ti

= (ei − ej)(ei − ej)T

(865)

where ei∈ RN , i=1 . . . N is the set of standard basis vectors. Thus eachentry dij is a convex quadratic function (§A.4.0.0.1) of vec X (37). [303, §6]

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5.4. EDM DEFINITION 393

The collection of all Euclidean distance matrices EDMN is a convex subsetof RN×N

+ called the EDM cone (§6, Figure 150 p.540);

0 ∈ EDMN ⊆ SNh ∩ RN×N

+ ⊂ SN (866)

An EDM D must be expressible as a function of some list X ; id est, it musthave the form

D(X) , δ(XTX)1T + 1δ(XTX)T− 2XTX ∈ EDMN (867)

= [vec(X)T(Φij ⊗ I ) vec X , i, j =1 . . . N ] (868)

Function D(X) will make an EDM given any X∈ Rn×N , conversely, butD(X) is not a convex function of X (§5.4.1). Now the EDM cone may bedescribed:

EDMN =

D(X) | X ∈ RN−1×N

(869)

Expression D(X) is a matrix definition of EDM and so conforms to theEuclidean metric properties:

Nonnegativity of EDM entries (property 1, §5.2) is obvious from thedistance-square definition (863), so holds for any D expressible in the formD(X) in (867).

When we say D is an EDM, reading from (867), it implicitly meansthe main diagonal must be 0 (property 2, self-distance) and D must besymmetric (property 3); δ(D) = 0 and DT = D or, equivalently, D∈ SN

h

are necessary matrix criteria.

5.4.0.1 homogeneity

Function D(X) is homogeneous in the sense, for ζ ∈ R

D(ζX) = |ζ| √

D(X) (870)

where the positive square root is entrywise ().

Any nonnegatively scaled EDM remains an EDM; id est, the matrix classEDM is invariant to nonnegative scaling (αD(X) for α≥0) because allEDMs of dimension N constitute a convex cone EDMN (§6, Figure 137).

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394 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.1 −V TN D(X)VN convexity

We saw that EDM entries dij

([

xi

xj

])

are convex quadratic functions. Yet

−D(X) (867) is not a quasiconvex function of matrix X ∈ Rn×N because thesecond directional derivative (§3.9)

− d2

dt2

t=0

D(X+ t Y ) = 2(

−δ(Y TY )1T − 1δ(Y TY )T + 2Y TY)

(871)

is indefinite for any Y ∈ Rn×N since its main diagonal is 0. [155, §4.2.8][199, §7.1, prob.2] Hence −D(X) can neither be convex in X .

The outcome is different when instead we consider

−V TN D(X)VN = 2V T

NXTXVN (872)

where we introduce the full-rank skinny Schoenberg auxiliary matrix (§B.4.2)

VN ,1√2

−1 −1 · · · −11 0

1. . .

0 1

=1√2

[

−1T

I

]

∈ RN×N−1 (873)

(N (VN )=0) having range

R(VN ) = N (1T) , V TN 1 = 0 (874)

Matrix-valued function (872) meets the criterion for convexity in §3.8.3.0.2over its domain that is all of Rn×N ; videlicet, for any Y ∈ Rn×N

− d2

dt2V TN D(X + t Y )VN = 4V T

N Y TY VN º 0 (875)

Quadratic matrix-valued function −V TN D(X)VN is therefore convex in X

achieving its minimum, with respect to a positive semidefinite cone (§2.7.2.2),at X = 0. When the penultimate number of points exceeds the dimensionof the space n < N−1, strict convexity of the quadratic (872) becomesimpossible because (875) could not then be positive definite.

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5.4. EDM DEFINITION 395

5.4.2 Gram-form EDM definition

Positive semidefinite matrix XTX in (867), formed from inner product oflist X , is known as a Gram matrix ; [246, §3.6]

G , XTX =

xT1...

xTN

[x1 · · · xN ]

=

‖x1‖2 xT1x2 xT

1x3 · · · xT1xN

xT2x1 ‖x2‖2 xT

2x3 · · · xT2xN

xT3x1 xT

3x2 ‖x3‖2 . . . xT3xN

......

. . . . . ....

xTNx1 xT

Nx2 xTNx3 · · · ‖xN‖2

∈ SN+

= δ

‖x1‖‖x2‖

...‖xN‖

1 cos ψ12 cos ψ13 · · · cos ψ1N

cos ψ12 1 cos ψ23 · · · cos ψ2N

cos ψ13 cos ψ23 1. . . cos ψ3N

......

. . . . . ....

cos ψ1N cos ψ2N cos ψ3N · · · 1

δ

‖x1‖‖x2‖

...‖xN‖

,√

δ2(G) Ψ√

δ2(G) (876)

where ψij (895) is angle between vectors xi and xj , and where δ2 denotesa diagonal matrix in this case. Positive semidefiniteness of interpoint anglematrix Ψ implies positive semidefiniteness of Gram matrix G ;

G º 0 ⇐ Ψ º 0 (877)

When δ2(G) is nonsingular, then Gº 0 ⇔ Ψº 0. (§A.3.1.0.5)Distance-square dij (863) is related to Gram matrix entries GT = G , [gij]

dij = gii + gjj − 2gij

= 〈Φij , G〉(878)

where Φij is defined in (865). Hence the linear EDM definition

D(G) , δ(G)1T + 1δ(G)T− 2G ∈ EDMN

= [〈Φij , G〉 , i, j =1 . . . N ]

⇐ G º 0 (879)

The EDM cone may be described, (confer (965)(971))

EDMN =

D(G) | G ∈ SN+

(880)

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396 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.2.1 First point at origin

Assume the first point x1 in an unknown list X resides at the origin;

Xe1 = 0 ⇔ Ge1 = 0 (881)

Consider the symmetric translation (I − 1eT1 )D(G)(I − e11

T) that shiftsthe first row and column of D(G) to the origin; setting Gram-form EDMoperator D(G) = D for convenience,

−(

D − (De11T + 1eT

1D) + 1eT1De11

T)

12

= G− (Ge11T+ 1eT

1G) + 1eT1Ge11

T

(882)where

e1 ,

10...0

(883)

is the first vector from the standard basis. Then it follows for D∈ SNh

G = −(

D − (De11T + 1eT

1D))

12

, x1 = 0

= −[

0√

2VN]T

D[

0√

2VN]

12

=

[

0 0T

0 −V TNDVN

]

V TN GVN = −V T

NDVN12

∀X

(884)

whereI − e11

T =[

0√

2VN]

(885)

is a projector nonorthogonally projecting (§E.1) on subspace

SN1 = G∈ SN | Ge1 = 0

=

[

0√

2VN]T

Y[

0√

2VN]

| Y ∈ SN (1964)

in the Euclidean sense. From (884) we get sufficiency of the first matrixcriterion for an EDM proved by Schoenberg in 1935; [308]5.7

D ∈ EDMN ⇔

−V TNDVN ∈ SN−1

+

D ∈ SNh

(886)

5.7From (874) we know R(VN )=N (1T) , so (886) is the same as (862). In fact, anymatrix V in place of VN will satisfy (886) whenever R(V )=R(VN )=N (1T). But VN isthe matrix implicit in Schoenberg’s seminal exposition.

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5.4. EDM DEFINITION 397

We provide a rigorous complete more geometric proof of this fundamentalSchoenberg criterion in §5.9.1.0.4. [isedm(D) on Wıκımization]

By substituting G =

[

0 0T

0 −V TNDVN

]

(884) into D(G) (879),

D =

[

0δ(

−V TNDVN

)

]

1T+ 1[

0 δ(

−V TNDVN

)T]

− 2

[

0 0T

0 −V TNDVN

]

(985)

assuming x1 = 0. Details of this bijection are provided in §5.6.2.

5.4.2.2 0 geometric center

Assume the geometric center (§5.5.1.0.1) of an unknown list X is the origin;

X1 = 0 ⇔ G1 = 0 (887)

Now consider the calculation (I − 1N11T)D(G)(I − 1

N11T) , a geometric

centering or projection operation. (§E.7.2.0.2) Setting D(G) = D forconvenience as in §5.4.2.1,

G = −(

D − 1N

(D11T + 11TD) + 1N211TD11T

)

12

, X1 = 0

= −V DV 12

V GV = −V DV 12

∀X

(888)

where more properties of the auxiliary (geometric centering, projection)matrix

V , I − 1

N11T ∈ SN (889)

are found in §B.4. V GV may be regarded as a covariance matrix of means 0.From (888) and the assumption D∈ SN

h we get sufficiency of the more popularform of Schoenberg’s criterion:

D ∈ EDMN ⇔

−V DV ∈ SN+

D ∈ SNh

(890)

Of particular utility when D∈ EDMN is the fact, (§B.4.2 no.20) (863)

tr(

−V DV 12

)

= 12N

i,j

dij = 12N

vec(X)T

(

i,j

Φij ⊗ I

)

vec X

= tr(V GV ) , G º 0

= tr G =N∑

ℓ=1

‖xℓ‖2 = ‖X‖2F , X1 = 0

(891)

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398 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

where∑

Φij ∈ SN+ (865), therefore convex in vec X . We will find this trace

useful as a heuristic to minimize affine dimension of an unknown list arrangedcolumnar in X (§7.2.2), but it tends to facilitate reconstruction of a listconfiguration having least energy; id est, it compacts a reconstructed list byminimizing total norm-square of the vertices.

By substituting G=−V DV 12

(888) into D(G) (879), assuming X1=0

D = δ(

−V DV 12

)

1T + 1δ(

−V DV 12

)T − 2(

−V DV 12

)

(975)

Details of this bijection can be found in §5.6.1.1.

5.4.2.3 hypersphere

These foregoing relationships allow combination of distance and Gramconstraints in any optimization problem we might pose: Interpoint angle Ψ can be constrained by fixing all the individual point

lengths δ(G)1/2 ; then

Ψ = −1

2δ2(G)−1/2V DV δ2(G)−1/2 (892) (confer §5.9.1.0.3, (1074) (1220)) Constraining all main diagonal entries

gii of a Gram matrix to 1, for example, is equivalent to the constraintthat all points lie on a hypersphere of radius 1 centered at the origin.

D = 2(g1111T− G) ∈ EDMN (893)

Requiring 0 geometric center then becomes equivalent to the constraint:D1 = 2N1. [87, p.116] Any further constraint on that Gram matrixapplies only to interpoint angle matrix Ψ = G .

Because any point list may be constrained to lie on a hypersphere boundarywhose affine dimension exceeds that of the list, a Gram matrix may alwaysbe constrained to have equal positive values along its main diagonal.(Laura Klanfer 1933 [308, §3]) This observation renewed interest in theelliptope (§5.9.1.0.1).

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5.4. EDM DEFINITION 399

5.4.2.3.1 Example. List-member constraints via Gram matrix.Capitalizing on identity (888) relating Gram and EDM D matrices, aconstraint set such as

tr(

−12V DV eie

Ti

)

= ‖xi‖2

tr(

−12V DV (eie

Tj + eje

Ti )1

2

)

= xTi xj

tr(

−12V DV eje

Tj

)

= ‖xj‖2

(894)

relates list member xi to xj to within an isometry through inner-productidentity [377, §1-7]

cos ψij =xT

i xj

‖xi‖ ‖xj‖(895)

where ψij is angle between the two vectors as in (876). For M list members,there total M(M+1)/2 such constraints. Angle constraints are incorporatedin Example 5.4.2.3.3 and Example 5.4.2.3.10. 2

Consider the academic problem of finding a Gram matrix subject toconstraints on each and every entry of the corresponding EDM:

findD∈SN

h

−V DV 12∈ SN

subject to⟨

D , (eieTj + eje

Ti )1

2

= dij , i, j =1 . . . N , i < j

−V DV º 0

(896)

where the dij are given nonnegative constants. EDM D can, of course,be replaced with the equivalent Gram-form (879). Requiring only theself-adjointness property (1394) of the main-diagonal linear operator δ weget, for A∈ SN

〈D , A〉 =⟨

δ(G)1T + 1δ(G)T− 2G , A⟩

= 〈G , δ(A1) − A〉 2 (897)

Then the problem equivalent to (896) becomes, for G∈ SNc ⇔ G1=0

findG∈SN

c

G ∈ SN

subject to⟨

G , δ(

(eieTj + eje

Ti )1

)

− (eieTj + eje

Ti )

= dij , i, j =1 . . . N , i < j

G º 0 (898)

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400 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Figure 116: Rendering of Fermat point in acrylic on canvas by Suman Vaze.Three circles intersect at Fermat point of minimum total distance from threevertices of (and interior to) red/black/white triangle.

Barvinok’s Proposition 2.9.3.0.1 predicts existence for either formulation(896) or (898) such that implicit equality constraints induced by subspacemembership are ignored

rank G , rank V DV ≤⌊

8(N(N−1)/2) + 1 − 1

2

= N − 1 (899)

because, in each case, the Gram matrix is confined to a face of positivesemidefinite cone SN

+ isomorphic with SN−1+ (§6.7.1). (§E.7.2.0.2) This bound

is tight (§5.7.1.1) and is the greatest upper bound.5.8

5.4.2.3.2 Example. First duality.Kuhn reports that the first dual optimization problem5.9 to be recorded inthe literature dates back to 1755. [Wıκımization] Perhaps more intriguing

5.8 −V DV |N←1 = 0 (§B.4.1)5.9By dual problem is meant: in the strongest sense, the optimal objective achieved by

a maximization problem dual to a given minimization problem (related to each otherby a Lagrangian function) is always equal to the optimal objective achieved by theminimization. (Figure 57 Example 2.13.1.0.3) A dual problem is always convex.

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5.4. EDM DEFINITION 401

is the fact: this earliest instance of duality is a two-dimensional Euclideandistance geometry problem known as a Fermat point (Figure 116) namedafter the French mathematician. Given N distinct points in the planexi∈ R2, i=1 . . . N , the Fermat point y is an optimal solution to

minimizey

N∑

i=1

‖y − xi‖ (900)

a convex minimization of total distance. The historically first dual problemformulation asks for the smallest equilateral triangle encompassing (N = 3)three points xi . Another problem dual to (900) (Kuhn 1967)

maximizezi

N∑

i=1

〈zi , xi〉

subject toN∑

i=1

zi = 0

‖zi‖ ≤ 1 ∀ i

(901)

has interpretation as minimization of work required to balance potentialenergy in an N -way tug-of-war between equally matched opponents situatedat xi. [371]

It is not so straightforward to write the Fermat point problem (900)equivalently in terms of a Gram matrix from this section. Squaring instead

minimizeα

N∑

i=1

‖α−xi‖2 ≡minimize

D∈SN+1〈−V , D〉

subject to 〈D , eieTj + eje

Ti 〉1

2= dij ∀(i, j)∈ I

−V DV º 0 (902)

yields an inequivalent convex geometric centering problem whoseequality constraints comprise EDM D main-diagonal zeros and knowndistances-square.5.10 Going the other way, a problem dual to totaldistance-square maximization (Example 6.4.0.0.1) is a penultimate minimumeigenvalue problem having application to PageRank calculation by searchengines [227, §4]. [337]

Fermat function (900) is empirically compared with (902) in [59, §8.7.3],but for multiple unknowns in R2, where propensity of (900) for producing

5.10 α⋆ is geometric center of points xi (949). For three points, I = 1, 2, 3 ; optimalaffine dimension (§5.7) must be 2 because a third dimension can only increase totaldistance. Minimization of 〈−V,D〉 is a heuristic for rank minimization. (§7.2.2)

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402 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

x1

x2

x3

x4

x5

x6

Figure 117: Arbitrary hexagon in R3 whose vertices are labelled clockwise.

zero distance between unknowns is revealed. An optimal solution to (900)gravitates toward gradient discontinuities (§D.2.1), as in Figure 71, whereasoptimal solution to (902) is less compact in the unknowns.5.11

2

5.4.2.3.3 Example. Hexagon.Barvinok [25, §2.6] poses a problem in geometric realizability of an arbitraryhexagon (Figure 117) having:

1. prescribed (one-dimensional) face-lengths l

2. prescribed angles ϕ between the three pairs of opposing faces

3. a constraint on the sum of norm-square of each and every vertex x ;

ten affine equality constraints in all on a Gram matrix G∈ S6 (888). Let’srealize this as a convex feasibility problem (with constraints written in thesame order) also assuming 0 geometric center (887):

findD∈S6

h

−V DV 12∈ S6

subject to tr(

D(eieTj + eje

Ti )1

2

)

= l2ij , j−1 = (i = 1 . . . 6) mod 6

tr(

−12V DV (Ai + AT

i )12

)

= cos ϕi , i = 1, 2, 3

tr(−12V DV ) = 1

−V DV º 0 (903)

5.11Optimal solution to (900) has mechanical interpretation in terms of interconnectingsprings with constant force when distance is nonzero; otherwise, 0 force. Problem (902) isinterpreted instead using linear springs.

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5.4. EDM DEFINITION 403

Figure 118: Sphere-packing illustration from [370, kissing number ].Translucent balls illustrated all have the same diameter.

where, for Ai∈ R6×6 (895)

A1 = (e1 − e6)(e3 − e4)T/(l61l34)

A2 = (e2 − e1)(e4 − e5)T/(l12l45)

A3 = (e3 − e2)(e5 − e6)T/(l23l56)

(904)

and where the first constraint on length-square l2ij can be equivalently written

as a constraint on the Gram matrix −V DV 12

via (897). We show how tonumerically solve such a problem by alternating projection in §E.10.2.1.1.Barvinok’s Proposition 2.9.3.0.1 asserts existence of a list, correspondingto Gram matrix G solving this feasibility problem, whose affine dimension(§5.7.1.1) does not exceed 3 because the convex feasible set is bounded bythe third constraint tr(−1

2V DV ) = 1 (891). 2

5.4.2.3.4 Example. Kissing number of sphere packing.Two nonoverlapping Euclidean balls are said to kiss if they touch. Anelementary geometrical problem can be posed: Given hyperspheres, eachhaving the same diameter 1, how many hyperspheres can simultaneouslykiss one central hypersphere? [391] The noncentral hyperspheres are allowed,but not required, to kiss.

As posed, the problem seeks the maximal number of spheres K kissinga central sphere in a particular dimension. The total number of spheres isN = K + 1. In one dimension the answer to this kissing problem is 2. In twodimensions, 6. (Figure 7)

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404 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

The question was presented, in three dimensions, to Isaac Newton byDavid Gregory in the context of celestial mechanics. And so was borna controversy between the two scholars on the campus of Trinity CollegeCambridge in 1694. Newton correctly identified the kissing number as 12(Figure 118) while Gregory argued for 13. Their dispute was finally resolvedin 1953 by Schutte & van der Waerden. [294] In 2003, Oleg Musin tightenedthe upper bound on kissing number K in four dimensions from 25 to K = 24by refining a method of Philippe Delsarte from 1973. Delsarte’s methodprovides an infinite number [16] of linear inequalities necessary for convertinga rank-constrained semidefinite program5.12 to a linear program.5.13 [269]

There are no proofs known for kissing number in higher dimensionexcepting dimensions eight and twenty four. Interest persists [82] becausesphere packing has found application to error correcting codes from thefields of communications and information theory; specifically to quantumcomputing. [90]

Translating this problem to an EDM graph realization (Figure 114,Figure 119) is suggested by Pfender & Ziegler. Imagine the centers of eachsphere are connected by line segments. Then the distance between centersmust obey simple criteria: Each sphere touching the central sphere has a linesegment of length exactly 1 joining its center to the central sphere’s center.All spheres, excepting the central sphere, must have centers separated by adistance of at least 1.

From this perspective, the kissing problem can be posed as a semidefiniteprogram. Assign index 1 to the central sphere, and assume a total of Nspheres:

minimizeD∈SN

− tr W V TNDVN

subject to D1j = 1 , j = 2 . . . N

Dij ≥ 1 , 2 ≤ i < j = 3 . . . N

D ∈ EDMN

(905)

Then the kissing number

K = Nmax − 1 (906)

5.12 whose feasible set belongs to that subset of an elliptope (§5.9.1.0.1) bounded aboveby some desired rank.5.13Simplex-method solvers for linear programs produce numerically better results thancontemporary log-barrier (interior-point method) solvers for semidefinite programs byabout 7 orders of magnitude; and they are far more predisposed to vertex solutions.

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5.4. EDM DEFINITION 405

is found from the maximal number of spheres N that solve this semidefiniteprogram in a given affine dimension r . Matrix W can be interpreted as thedirection of search through the positive semidefinite cone SN−1

+ for a rank-roptimal solution −V T

ND⋆VN ; it is constant, in this program, determined bya method disclosed in §4.4.1. In two dimensions,

W =

4 1 2 −1 −1 11 4 −1 −1 2 12 −1 4 1 1 −1

−1 −1 1 4 1 2−1 2 1 1 4 −1

1 1 −1 2 −1 4

1

6(907)

In three dimensions,

W =

9 1 −2 −1 3 −1 −1 1 2 1 −2 11 9 3 −1 −1 1 1 −2 1 2 −1 −1

−2 3 9 1 2 −1 −1 2 −1 −1 1 2−1 −1 1 9 1 −1 1 −1 3 2 −1 1

3 −1 2 1 9 1 1 −1 −1 −1 1 −1−1 1 −1 −1 1 9 2 −1 2 −1 2 3−1 1 −1 1 1 2 9 3 −1 1 −2 −1

1 −2 2 −1 −1 −1 3 9 2 −1 1 12 1 −1 3 −1 2 −1 2 9 −1 1 −11 2 −1 2 −1 −1 1 −1 −1 9 3 1

−2 −1 1 −1 1 2 −2 1 1 3 9 −11 −1 2 1 −1 3 −1 1 −1 1 −1 9

1

12(908)

A four-dimensional solution has rational direction matrix as well, but thesedirection matrices are not unique and their precision not critical. Here is anoptimal point list5.14 in Matlab output format:

5.14An optimal five-dimensional point list is known: The answer was known at least 175years ago. I believe Gauss knew it. Moreover, Korkine & Zolotarev proved in 1882 that D5

is the densest lattice in five dimensions. So they proved that if a kissing arrangement infive dimensions can be extended to some lattice, then k(5)= 40. Of course, the conjecturein the general case also is: k(5)= 40. You would like to see coordinates? Easily.Let A=

√2 . Then p(1)=(A,A, 0, 0, 0), p(2)=(−A,A, 0, 0, 0), p(3)=(A,−A, 0, 0, 0), . . .

p(40)=(0, 0, 0,−A,−A) ; i.e., we are considering points with coordinates that have twoA and three 0 with any choice of signs and any ordering of the coordinates; the same

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406 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Columns 1 through 6

X = 0 -0.1983 -0.4584 0.1657 0.9399 0.7416

0 0.6863 0.2936 0.6239 -0.2936 0.3927

0 -0.4835 0.8146 -0.6448 0.0611 -0.4224

0 0.5059 0.2004 -0.4093 -0.1632 0.3427

Columns 7 through 12

-0.4815 -0.9399 -0.7416 0.1983 0.4584 -0.2832

0 0.2936 -0.3927 -0.6863 -0.2936 -0.6863

-0.8756 -0.0611 0.4224 0.4835 -0.8146 -0.3922

-0.0372 0.1632 -0.3427 -0.5059 -0.2004 -0.5431

Columns 13 through 18

0.2832 -0.2926 -0.6473 0.0943 0.3640 -0.3640

0.6863 0.9176 -0.6239 -0.2313 -0.0624 0.0624

0.3922 0.1698 -0.2309 -0.6533 -0.1613 0.1613

0.5431 -0.2088 0.3721 0.7147 -0.9152 0.9152

Columns 19 through 25

-0.0943 0.6473 -0.1657 0.2926 -0.5759 0.5759 0.4815

0.2313 0.6239 -0.6239 -0.9176 0.2313 -0.2313 0

0.6533 0.2309 0.6448 -0.1698 -0.2224 0.2224 0.8756

-0.7147 -0.3721 0.4093 0.2088 -0.7520 0.7520 0.0372

This particular optimal solution was found by solving a problem sequence inincreasing number of spheres.

Numerical problems begin to arise with matrices of this size due tointerior-point methods of solution. By eliminating some equality constraintsfor this particular problem, matrix size can be reduced: From §5.8.3 we have

−V TNDVN = 11T− [0 I ] D

[

0T

I

]

1

2(909)

(which does not hold more generally) where identity matrix I∈ SN−1 has

coordinates-expression in dimensions 3 and 4.The first miracle happens in dimension 6. There are better packings than D6

(Conjecture: k(6)=72). It’s a real miracle how dense the packing is in eight dimensions(E8=Korkine & Zolotarev packing that was discovered in 1880s) and especially indimension 24, that is the so-called Leech lattice.

Actually, people in coding theory have conjectures on the kissing numbers for dimensionsup to 32 (or even greater? ). However, sometimes they found better lower bounds. I knowthat Ericson & Zinoviev a few years ago discovered (by hand, no computer) in dimensions13 and 14 better kissing arrangements than were known before. −Oleg Musin

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5.4. EDM DEFINITION 407

one less dimension than EDM D . By defining an EDM principal submatrix

D , [0 I ] D

[

0T

I

]

∈ SN−1h (910)

we get a convex problem equivalent to (905)

minimizeD∈SN−1

− tr(WD)

subject to Dij ≥ 1 , 1 ≤ i < j = 2 . . . N−1

11T− D 12º 0

δ(D) = 0

(911)

Any feasible solution 11T− D 12

belongs to an elliptope (§5.9.1.0.1). 2

This next example shows how finding the common point of intersectionfor three circles in a plane, a nonlinear problem, has convex expression.

5.4.2.3.5 Example. Trilateration in wireless sensor network.Given three known absolute point positions in R2 (three anchors x2 , x3 , x4)and only one unknown point (one sensor x1), the sensor’s absolute position isdetermined from its noiseless measured distance-square di1 to each of threeanchors (Figure 2, Figure 119a). This trilateration can be expressed as aconvex optimization problem in terms of list X , [ x1 x2 x3 x4 ]∈R2×4 andGram matrix G∈ S4 (876):

minimizeG∈S4, X∈R2×4

tr G

subject to tr(GΦi1) = di1 , i = 2, 3, 4

tr(

GeieTi

)

= ‖xi‖2 , i = 2, 3, 4

tr(G(eieTj + eje

Ti )/2) = xT

i xj , 2≤ i < j = 3, 4

X(: , 2:4) = [ x2 x3 x4 ][

I XXT G

]

º 0

(912)

whereΦij = (ei − ej)(ei − ej)

T ∈ SN+ (865)

and where the constraint on distance-square di1 is equivalently written asa constraint on the Gram matrix via (878). There are 9 independent affine

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408 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

(a) (b)

(c) (d)

x1x1

x1x1

x2

x3 x4

x2x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x6

√d12

√d13

√d14

Figure 119: (a) Given three distances indicated with absolute pointpositions x2 , x3 , x4 known and noncollinear, absolute position of x1 in R2

can be precisely and uniquely determined by trilateration; solution to asystem of nonlinear equations. Dimensionless EDM graphs (b) (c) (d)represent EDMs in various states of completion. Line segments representknown absolute distances and may cross without vertex at intersection.(b) Four-point list must always be embeddable in affine subset havingdimension rank V T

NDVN not exceeding 3. To determine relative position ofx2 , x3 , x4 , triangle inequality is necessary and sufficient (§5.14.1). Knowingall distance information, then (by injectivity of D (§5.6)) point position x1

is uniquely determined to within an isometry in any dimension. (c) Whenfifth point is introduced, only distances to x3 , x4 , x5 are required todetermine relative position of x2 in R2. Graph represents first instanceof missing distance information;

√d12 . (d) Three distances are absent

(√

d12 ,√

d13 ,√

d23 ) from complete set of interpoint distances, yet uniqueisometric reconstruction (§5.4.2.3.7) of six points in R2 is certain.

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5.4. EDM DEFINITION 409

equality constraints on that Gram matrix while the sensor is constrainedonly by dimensioning to lie in R2. Although the objective trG ofminimization5.15 insures a solution on the boundary of positive semidefinitecone S4

+ , for this problem, we claim that the set of feasible Gram matricesforms a line (§2.5.1.1) in isomorphic R10 tangent (§2.1.7.1.2) to the positivesemidefinite cone boundary. (§5.4.2.3.6, confer §4.2.1.3)

By Schur complement (§A.4, §2.9.1.0.1), any feasible G and X provide

G º XTX (913)

which is a convex relaxation of the desired (nonconvex) equality constraint

[

I XXT G

]

=

[

IXT

]

[ I X ](914)

expected positive semidefinite rank-2 under noiseless conditions. But,by (1464), the relaxation admits

(3 ≥) rank G ≥ rank X (915)

(a third dimension corresponding to an intersection of three spheres,not circles, were there noise). If rank of an optimal solution equals 2,

rank

[

I X⋆

X⋆T G⋆

]

= 2 (916)

then G⋆ = X⋆TX⋆ by Theorem A.4.0.0.5.As posed, this localization problem does not require affinely independent

(Figure 27, three noncollinear) anchors. Assuming the anchors exhibitno rotational or reflective symmetry in their affine hull (§5.5.2) andassuming the sensor x1 lies in that affine hull, then sensor position solutionx⋆

1 = X⋆(: , 1) is unique under noiseless measurement. [316] 2

This preceding transformation of trilateration to a semidefinite programworks all the time ((916) holds) despite relaxation (913) because the optimalsolution set is a unique point.

5.15Trace (tr G = 〈I , G〉) minimization is a heuristic for rank minimization. (§7.2.2.1)It may be interpreted as squashing G which is bounded below by XTX as in (913);id est, G−XTXº 0 ⇒ trG ≥ tr XTX (1462). δ(G−XTX)= 0 ⇔ G=XTX (§A.7.2)⇒ tr G = tr XTX which is a condition necessary for equality.

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410 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.2.3.6 Proof (sketch). Only the sensor location x1 is unknown.The objective function together with the equality constraints make a linearsystem of equations in Gram matrix variable G

tr G = ‖x1‖2 + ‖x2‖2 + ‖x3‖2 + ‖x4‖2

tr(GΦi1) = di1 , i = 2, 3, 4

tr(

GeieTi

)

= ‖xi‖2 , i = 2, 3, 4

tr(G(eieTj + eje

Ti )/2) = xT

i xj , 2≤ i < j = 3, 4

(917)

which is invertible:

svec G =

svec(I )T

svec(Φ21)T

svec(Φ31)T

svec(Φ41)T

svec(e2eT2 )T

svec(e3eT3 )T

svec(e4eT4 )T

svec(

(e2eT3 + e3e

T2 )/2

)T

svec(

(e2eT4 + e4e

T2 )/2

)T

svec(

(e3eT4 + e4e

T3 )/2

)T

−1

‖x1‖2 + ‖x2‖2 + ‖x3‖2 + ‖x4‖2

d21

d31

d41

‖x2‖2

‖x3‖2

‖x4‖2

xT2x3

xT2x4

xT3x4

(918)

That line in the ambient space S4 of G , claimed on page 409, is traced by‖x1‖2∈ R on the right-hand side, as it turns out. One must show this line tobe tangential (§2.1.7.1.2) to S4

+ in order to prove uniqueness. Tangency ispossible for affine dimension 1 or 2 while its occurrence depends completelyon the known measurement data. ¥

But as soon as significant noise is introduced or whenever distance data isincomplete, such problems can remain convex although the set of all optimalsolutions generally becomes a convex set bigger than a single point (and stillcontaining the noiseless solution).

5.4.2.3.7 Definition. Isometric reconstruction. (confer §5.5.3)Isometric reconstruction from an EDM means building a list X correct towithin a rotation, reflection, and translation; in other terms, reconstructionof relative position, correct to within an isometry, or to within a rigidtransformation.

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5.4. EDM DEFINITION 411

How much distance information is needed to uniquely localize a sensor(to recover actual relative position)? The narrative in Figure 119 helpsdispel any notion of distance data proliferation in low affine dimension(r<N−2) .5.16 Huang, Liang, and Pardalos [202, §4.2] claim O(2N )distances is a least lower bound (independent of affine dimension r) forunique isometric reconstruction; achievable under certain noiseless conditionson graph connectivity and point position. Alfakih shows how to ascertainuniqueness over all affine dimensions via Gale matrix. [9] [4] [5] Figure 114b(p.389, from small completion problem Example 5.3.0.0.2) is an example inR2 requiring only 2N− 3 = 5 known symmetric entries for unique isometricreconstruction, although the four-point example in Figure 119b will not yielda unique reconstruction when any one of the distances is left unspecified.

The list represented by the particular dimensionless EDM graph inFigure 120, having only 2N− 3 = 9 absolute distances specified, has onlyone realization in R2 but has more realizations in higher dimensions. Forsake of reference, we provide the complete corresponding EDM:

D =

0 50641 56129 8245 18457 2664550641 0 49300 25994 8810 2061256129 49300 0 24202 31330 91608245 25994 24202 0 4680 5290

18457 8810 31330 4680 0 665826645 20612 9160 5290 6658 0

(919)

We consider paucity of distance information in this next example whichshows it is possible to recover exact relative position given incompletenoiseless distance information. An ad hoc method for recovery of thelowest-rank optimal solution under noiseless conditions is introduced:

5.4.2.3.8 Example. Tandem trilateration in wireless sensor network.Given three known absolute point-positions in R2 (three anchors x3 , x4 , x5),two unknown sensors x1 , x2∈ R2 have absolute position determinable fromtheir noiseless distances-square (as indicated in Figure 121) assuming theanchors exhibit no rotational or reflective symmetry in their affine hull

5.16When affine dimension r reaches N− 2, then all distances-square in the EDM mustbe known for unique isometric reconstruction in Rr ; going the other way, when r = 1 thenthe condition that the dimensionless EDM graph be connected is necessary and sufficient.[184, §2.2]

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412 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

x5

x2x3

x1

x4

x6

Figure 120: Incomplete EDM corresponding to this dimensionless EDMgraph provides unique isometric reconstruction in R2. (drawn freehand, nosymmetry intended)

x4

x5x3

x1

x2

Figure 121: (Ye) Two sensors • and three anchors in R2. Connectingline-segments denote known absolute distances. Incomplete EDMcorresponding to this dimensionless EDM graph provides unique isometricreconstruction in R2.

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5.4. EDM DEFINITION 413

−6 −4 −2 0 2 4 6 8−15

−10

−5

0

5

10

x4

x5x3

x2

x1

Figure 122: Given in red # are two discrete linear trajectories of sensors x1

and x2 in R2 localized by algorithm (920) as indicated by blue bullets • .Anchors x3 , x4 , x5 corresponding to Figure 121 are indicated by ⊗ . Whentargets # and bullets • coincide under these noiseless conditions, localizationis successful. On this run, two visible localization errors are due to rank-3Gram optimal solutions. These errors can be corrected by choosing a differentnormal in objective of minimization.

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414 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

(§5.5.2). This example differs from Example 5.4.2.3.5 in so far as trilaterationof each sensor is now in terms of one unknown position: the other sensor. Weexpress this localization as a convex optimization problem (a semidefiniteprogram, §4.1) in terms of list X , [ x1 x2 x3 x4 x5 ]∈ R2×5 and Grammatrix G∈ S5 (876) via relaxation (913):

minimizeG∈S5, X∈R2×5

tr G

subject to tr(GΦi1) = di1 , i = 2, 4, 5

tr(GΦi2) = di2 , i = 3, 5

tr(

GeieTi

)

= ‖xi‖2 , i = 3, 4, 5

tr(G(eieTj + eje

Ti )/2) = xT

i xj , 3≤ i < j = 4, 5

X(: , 3:5) = [ x3 x4 x5 ][

I XXT G

]

º 0

(920)

whereΦij = (ei − ej)(ei − ej)

T ∈ SN+ (865)

This problem realization is fragile because of the unknown distances betweensensors and anchors. Yet there is no more information we may include beyondthe 11 independent equality constraints on the Gram matrix (nonredundantconstraints not antithetical) to reduce the feasible set.5.17

Exhibited in Figure 122 are two mistakes in solution X⋆(: , 1:2) dueto a rank-3 optimal Gram matrix G⋆. The trace objective is a heuristicminimizing convex envelope of quasiconcave function5.18 rank G . (§2.9.2.6.2,§7.2.2.1) A rank-2 optimal Gram matrix can be found and the errorscorrected by choosing a different normal for the linear objective function,now implicitly the identity matrix I ; id est,

tr G = 〈G , I 〉 ← 〈G , δ(u)〉 (921)

where vector u ∈ R5 is randomly selected. A random search for a goodnormal δ(u) in only a few iterations is quite easy and effective because

5.17By virtue of their dimensioning, the sensors are already constrained to R2 the affinehull of the anchors.5.18Projection on that nonconvex subset of all N×N -dimensional positive semidefinitematrices, in an affine subset, whose rank does not exceed 2 is a problem considered difficultto solve. [350, §4]

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5.4. EDM DEFINITION 415

the problem is small, an optimal solution is known a priori to exist in twodimensions, a good normal direction is not necessarily unique, and (wespeculate) because the feasible affine-subset slices the positive semidefinitecone thinly in the Euclidean sense.5.19

2

We explore ramifications of noise and incomplete data throughout; theirindividual effect being to expand the optimal solution set, introducing moresolutions and higher-rank solutions. Hence our focus shifts in §4.4 todiscovery of a reliable means for diminishing the optimal solution set byintroduction of a rank constraint.

Now we illustrate how a problem in distance geometry can be solvedwithout equality constraints representing measured distance; instead, wehave only upper and lower bounds on distances measured:

5.4.2.3.9 Example. Wireless location in a cellular telephone network.Utilizing measurements of distance, time of flight, angle of arrival, or signalpower in the context of wireless telephony, multilateration is the processof localizing (determining absolute position of) a radio signal source • byinferring geometry relative to multiple fixed base stations whose locationsare known.

We consider localization of a cellular telephone by distance geometry,so we assume distance to any particular base station can be inferred fromreceived signal power. On a large open flat expanse of terrain, signal-powermeasurement corresponds well with inverse distance. But it is not uncommonfor measurement of signal power to suffer 20 decibels in loss caused by factorssuch as multipath interference (signal reflections), mountainous terrain,man-made structures, turning one’s head, or rolling the windows up in anautomobile. Consequently, contours of equal signal power are no longercircular; their geometry is irregular and would more aptly be approximatedby translated ellipsoids of graduated orientation and eccentricity as inFigure 124.

Depicted in Figure 123 is one cell phone x1 whose signal power isautomatically and repeatedly measured by 6 base stations nearby.5.20

5.19The log det rank-heuristic from §7.2.2.4 does not work here because it chooses thewrong normal. Rank reduction (§4.1.1.2) is unsuccessful here because Barvinok’s upperbound (§2.9.3.0.1) on rank of G⋆ is 4.5.20Cell phone signal power is typically encoded logarithmically with 1-decibel incrementand 64-decibel dynamic range.

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416 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

x1

x2

x3

x4

x5x6

x7

Figure 123: Regions of coverage by base stations in a cellular telephonenetwork. The term cellular arises from packing of regions best coveredby neighboring base stations. Illustrated is a pentagonal cell best coveredby base station x2 . Like a Voronoi diagram, cell geometry depends onbase-station arrangement. In some US urban environments, it is not unusualto find base stations spaced approximately 1 mile apart. There can be asmany as 20 base-station antennae capable of receiving signal from any givencell phone • ; practically, that number is closer to 6.

Figure 124: Some fitted contours of equal signal power in R2 transmittedfrom a commercial cellular telephone • over about 1 mile suburban terrainoutside San Francisco in 2005. Data courtesy Polaris Wireless. [319]

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5.4. EDM DEFINITION 417

Those signal power measurements are transmitted from that cell phone tobase station x2 who decides whether to transfer (hand-off or hand-over)responsibility for that call should the user roam outside its cell.5.21

Due to noise, at least one distance measurement more than the minimumnumber of measurements is required for reliable localization in practice;3 measurements are minimum in two dimensions, 4 in three.5.22 Existenceof noise precludes measured distance from the input data. We instead assignmeasured distance to a range estimate specified by individual upper andlower bounds: di1 is the upper bound on distance-square from the cell phoneto ith base station, while di1 is the lower bound. These bounds become theinput data. Each measurement range is presumed different from the others.

Then convex problem (912) takes the form:

minimizeG∈S7, X∈R2×7

tr G

subject to di1 ≤ tr(GΦi1) ≤ di1 , i = 2 . . . 7

tr(

GeieTi

)

= ‖xi‖2 , i = 2 . . . 7

tr(G(eieTj + eje

Ti )/2) = xT

i xj , 2≤ i < j = 3 . . . 7

X(: , 2:7) = [ x2 x3 x4 x5 x6 x7 ][

I XXT G

]

º 0 (922)

whereΦij = (ei − ej)(ei − ej)

T ∈ SN+ (865)

This semidefinite program realizes the wireless location problem illustrated inFigure 123. Location X⋆(: , 1) is taken as solution, although measurementnoise will often cause rank G⋆ to exceed 2. Randomized search for a rank-2optimal solution is not so easy here as in Example 5.4.2.3.8. We introduce amethod in §4.4 for enforcing the stronger rank-constraint (916). To formulatethis same problem in three dimensions, point list X is simply redimensionedin the semidefinite program. 2

5.21Because distance to base station is quite difficult to infer from signal powermeasurements in an urban environment, localization of a particular cell phone • bydistance geometry would be far easier were the whole cellular system instead conceived socell phone x1 also transmits (to base station x2) its signal power as received by all othercell phones within range.5.22In Example 4.4.1.2.4, we explore how this convex optimization algorithm fares in theface of measurement noise.

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418 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Figure 125: A depiction of molecular conformation. [113]

5.4.2.3.10 Example. (Biswas, Nigam, Ye) Molecular Conformation.The subatomic measurement technique called nuclear magnetic resonancespectroscopy (NMR) is employed to ascertain physical conformation ofmolecules; e.g., Figure 3, Figure 125. From this technique, distance, angle,and dihedral angle measurements can be obtained. Dihedral angles ariseconsequent to a phenomenon where atom subsets are physically constrainedto Euclidean planes.

In the rigid covalent geometry approximation, the bond lengthsand angles are treated as completely fixed, so that a given spatialstructure can be described very compactly indeed by a list oftorsion angles alone. . . These are the dihedral angles betweenthe planes spanned by the two consecutive triples in a chain offour covalently bonded atoms.

−G. M. Crippen & T. F. Havel (1988) [86, §1.1]

Crippen & Havel recommend working exclusively with distance data becausethey consider angle data to be mathematically cumbersome. The presentexample shows instead how inclusion of dihedral angle data into a problemstatement can be made elegant and convex.

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5.4. EDM DEFINITION 419

As before, ascribe position information to the matrix

X = [ x1 · · · xN ] ∈ R3×N (76)

and introduce a matrix ℵ holding normals η to planes respecting dihedralangles ϕ :

ℵ , [ η1 · · · ηM ] ∈ R3×M (923)

As in the other examples, we preferentially work with Gram matrices Gbecause of the bridge they provide between other variables; we define

[

Gℵ ZZT GX

]

,

[

ℵTℵ ℵTXXTℵ XTX

]

=

[

ℵT

XT

]

[ℵ X ] ∈ RN+M×N+M (924)

whose rank is 3 by assumption. So our problem’s variables are the twoGram matrices GX and Gℵ and matrix Z = ℵTX of cross products. Thenmeasurements of distance-square d can be expressed as linear constraints onGX as in (922), dihedral angle ϕ measurements can be expressed as linearconstraints on Gℵ by (895), and normal-vector η conditions can be expressedby vanishing linear constraints on cross-product matrix Z : Consider threepoints x labelled 1 , 2 , 3 assumed to lie in the ℓth plane whose normal is ηℓ .There might occur, for example, the independent constraints

ηTℓ (x1 − x2) = 0

ηTℓ (x2 − x3) = 0

(925)

which are expressible in terms of constant matrices A∈ RM×N ;

〈Z , Aℓ12〉 = 0〈Z , Aℓ23〉 = 0

(926)

Although normals η can be constrained exactly to unit length,

δ(Gℵ) = 1 (927)

NMR data is noisy; so measurements are given as upper and lower bounds.Given bounds on dihedral angles respecting 0≤ ϕj ≤ π and bounds ondistances di and given constant matrices Ak (926) and symmetric matricesΦi (865) and Bj per (895), then a molecular conformation problem can beexpressed:

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420 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

findGℵ∈SM , GX∈SN , Z∈RM×N

GX

subject to di ≤ tr(GX Φi) ≤ di , ∀ i ∈ I1

cos ϕj ≤ tr(GℵBj) ≤ cos ϕj , ∀ j ∈ I2

〈Z , Ak〉 = 0 , ∀ k ∈ I3

GX1 = 0

δ(Gℵ) = 1[

Gℵ ZZT GX

]

º 0

rank

[

Gℵ ZZT GX

]

= 3

(928)

where GX1=0 provides a geometrically centered list X (§5.4.2.2). Ignoringthe rank constraint would tend to force cross-product matrix Z to zero.What binds these three variables is the rank constraint; we show how tosatisfy it in §4.4. 2

5.4.3 Inner-product form EDM definition

We might, for example, want to realize a constellation given onlyinterstellar distance (or, equivalently, parsecs from our Sun andrelative angular measurement; the Sun as vertex to two distantstars); called stellar cartography. (p.22)

Equivalent to (863) is [377, §1-7] [328, §3.2]

dij = dik + dkj − 2√

dikdkj cos θikj

=[√

dik

dkj

]

[

1 −eıθikj

−e−ıθikj 1

]

[√

dik√

dkj

]

(929)

called law of cosines where ı ,√−1 , i , j , k are positive integers, and θikj

is the angle at vertex xk formed by vectors xi − xk and xj − xk ;

cos θikj =12(dik + dkj − dij)

dikdkj

=(xi − xk)

T(xj − xk)

‖xi − xk‖ ‖xj − xk‖(930)

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5.4. EDM DEFINITION 421

where the numerator forms an inner product of vectors. Distance-square

dij

([√

dik√

dkj

])

is a convex quadratic function5.23 on R2

+ whereas dij(θikj) is

quasiconvex (§3.9) minimized over domain −π≤ θikj ≤π by θ⋆

ikj =0, weget the Pythagorean theorem when θikj =±π/2, and dij(θikj) is maximizedwhen θ

ikj =±π ;

dij =(√

dik +√

dkj

)2, θikj = ±π

dij = dik + dkj , θikj = ±π2

dij =(√

dik −√

dkj

)2, θikj = 0

(931)

so|√

dik −√

dkj | ≤√

dij ≤√

dik +√

dkj (932)

Hence the triangle inequality, Euclidean metric property 4, holds for anyEDM D .

We may construct an inner-product form of the EDM definition formatrices by evaluating (929) for k=1 : By defining

ΘTΘ ,

d12

d12d13 cos θ213

d12d14 cos θ214 · · ·√

d12d1N cos θ21N√

d12d13 cos θ213 d13

d13d14 cos θ314 · · ·√

d13d1N cos θ31N√

d12d14 cos θ214

d13d14 cos θ314 d14. . .

d14d1N cos θ41N...

.... . . . . .

...√

d12d1N cos θ21N

d13d1N cos θ31N

d14d1N cos θ41N · · · d1N

∈ SN−1

(933)then any EDM may be expressed

D(Θ) ,

[

0δ(ΘTΘ)

]

1T + 1[

0 δ(ΘTΘ)T]

− 2

[

0 0T

0 ΘTΘ

]

∈ EDMN

=

[

0 δ(ΘTΘ)T

δ(ΘTΘ) δ(ΘTΘ)1T+ 1δ(ΘTΘ)T− 2ΘTΘ

]

(934)

EDMN =

D(Θ) | Θ ∈ RN−1×N−1

(935)

for which all Euclidean metric properties hold. Entries of ΘTΘ result fromvector inner-products as in (930); id est,

5.23

[

1 −eıθikj

−e−ıθikj 1

]

º 0, having eigenvalues 0, 2. Minimum is attained for[ √

dik√

dkj

]

=

µ1 , µ ≥ 0 , θikj = 00 , −π ≤ θikj ≤ π , θikj 6= 0

. (§D.2.1, [59, exmp.4.5])

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422 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Θ = [ x2 − x1 x3 − x1 · · · xN − x1 ] = X√

2VN ∈ Rn×N−1 (936)

Inner product ΘTΘ is obviously related to a Gram matrix (876),

G =

[

0 0T

0 ΘTΘ

]

, x1 = 0 (937)

For D = D(Θ) and no condition on the list X (confer (884) (888))

ΘTΘ = −V TNDVN ∈ RN−1×N−1 (938)

5.4.3.1 Relative-angle form

The inner-product form EDM definition is not a unique definition ofEuclidean distance matrix; there are approximately five flavors distinguishedby their argument to operator D . Here is another one:

Like D(X) (867), D(Θ) will make an EDM given any Θ∈Rn×N−1, it isneither a convex function of Θ (§5.4.3.2), and it is homogeneous in the sense(870). Scrutinizing ΘTΘ (933) we find that because of the arbitrary choicek = 1, distances therein are all with respect to point x1 . Similarly, relativeangles in ΘTΘ are between all vector pairs having vertex x1 . Yet pickingarbitrary θi1j to fill ΘTΘ will not necessarily make an EDM; inner product(933) must be positive semidefinite.

ΘTΘ =√

δ(d) Ω√

δ(d) ,

√d12 0√

d13

. . .

0√

d1N

1 cos θ213 · · · cos θ21N

cos θ213 1. . . cos θ31N

.... . . . . .

...cos θ21N cos θ31N · · · 1

√d12 0√

d13

. . .

0√

d1N

(939)

Expression D(Θ) defines an EDM for any positive semidefinite relative-anglematrix

Ω = [cos θi1j , i, j = 2 . . . N ] ∈ SN−1 (940)

and any nonnegative distance vector

d = [d1j , j = 2 . . . N ] = δ(ΘTΘ) ∈ RN−1 (941)

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5.4. EDM DEFINITION 423

because (§A.3.1.0.5)

Ω º 0 ⇒ ΘTΘ º 0 (942)

Decomposition (939) and the relative-angle matrix inequality Ωº 0 lead toa different expression of an inner-product form EDM definition (934)

D(Ω , d) ,

[

0d

]

1T + 1[

0 dT]

− 2

δ

([

0d

])[

0 0T

0 Ω

]

δ

([

0d

])

=

[

0 dT

d d1T + 1dT− 2√

δ(d) Ω√

δ(d)

]

∈ EDMN

(943)

and another expression of the EDM cone:

EDMN =

D(Ω , d) | Ω º 0 ,√

δ(d) º 0

(944)

In the particular circumstance x1 = 0, we can relate interpoint anglematrix Ψ from the Gram decomposition in (876) to relative-angle matrixΩ in (939). Thus,

Ψ ≡[

1 0T

0 Ω

]

, x1 = 0 (945)

5.4.3.2 Inner-product form −V TN D(Θ)VN convexity

On page 421 we saw that each EDM entry dij is a convex quadratic function

of

[√

dik√

dkj

]

and a quasiconvex function of θikj . Here the situation for

inner-product form EDM operator D(Θ) (934) is identical to that in §5.4.1for list-form D(X) ; −D(Θ) is not a quasiconvex function of Θ by the samereasoning, and from (938)

−V TN D(Θ)VN = ΘTΘ (946)

is a convex quadratic function of Θ on domain Rn×N−1 achieving its minimumat Θ = 0.

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424 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.4.3.3 Inner-product form, discussion

We deduce that knowledge of interpoint distance is equivalent to knowledgeof distance and angle from the perspective of one point, x1 in our chosencase. The total amount of information N(N−1)/2 in ΘTΘ is unchanged5.24

with respect to EDM D .

5.5 Invariance

When D is an EDM, there exist an infinite number of corresponding N -pointlists X (76) in Euclidean space. All those lists are related by isometrictransformation: rotation, reflection, and translation (offset or shift).

5.5.1 Translation

Any translation common among all the points xℓ in a list will be cancelled inthe formation of each dij . Proof follows directly from (863). Knowing thattranslation α in advance, we may remove it from the list constituting thecolumns of X by subtracting α1T. Then it stands to reason by list-formdefinition (867) of an EDM, for any translation α∈Rn

D(X − α1T) = D(X) (947)

In words, interpoint distances are unaffected by offset; EDM D is translationinvariant. When α = x1 in particular,

[ x2−x1 x3−x1 · · · xN −x1 ] = X√

2VN ∈ Rn×N−1 (936)

and so

D(X−x11T) = D(X−Xe11

T) = D(

X[

0√

2VN])

= D(X) (948)

5.24The reason for amount O(N 2) information is because of the relative measurements.Use of a fixed reference in measurement of angles and distances would reduce requiredinformation but is antithetical. In the particular case n = 2, for example, ordering allpoints xℓ (in a length-N list) by increasing angle of vector xℓ − x1 with respect to

x2 − x1 , θi1j becomes equivalent toj−1∑

k=i

θk,1,k+1 ≤ 2π and the amount of information is

reduced to 2N−3 ; rather, O(N ).

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5.5. INVARIANCE 425

5.5.1.0.1 Example. Translating geometric center to origin.We might choose to shift the geometric center αc of an N -point list xℓ(arranged columnar in X) to the origin; [349] [159]

α = αc , Xbc , X1 1N

∈ P ⊆ A (949)

where A represents the list’s affine hull. If we were to associate a point-massmℓ with each of the points xℓ in the list, then their center of mass(or gravity) would be (

xℓ mℓ) /∑

mℓ . The geometric center is the sameas the center of mass under the assumption of uniform mass density acrosspoints. [215] The geometric center always lies in the convex hull P of the list;id est, αc∈ P because bT

c 1=1 and bcº 0 .5.25 Subtracting the geometriccenter from every list member,

X − αc1T = X − 1

NX11T = X(I − 1

N11T) = XV ∈ Rn×N (950)

where V is the geometric centering matrix (889). So we have (confer (867))

D(X) = D(XV ) = δ(V TXTXV )1T+1δ(V TXTXV )T−2V TXTXV ∈ EDMN

(951)2

5.5.1.1 Gram-form invariance

Following from (951) and the linear Gram-form EDM operator (879):

D(G) = D(V GV ) = δ(V GV )1T + 1δ(V GV )T− 2V GV ∈ EDMN (952)

The Gram-form consequently exhibits invariance to translation by a doublet(§B.2) u1T + 1uT

D(G) = D(G − (u1T + 1uT)) (953)

because, for any u∈RN , D(u1T + 1uT)=0. The collection of all suchdoublets forms the nullspace (969) to the operator; the translation-invariantsubspace SN⊥

c (1962) of the symmetric matrices SN . This means matrix Gis not unique and can belong to an expanse more broad than a positivesemidefinite cone; id est, G∈ SN

+− SN⊥c . So explains Gram matrix sufficiency

in EDM definition (879).5.26

5.25Any b from α = Xb chosen such that bT1 = 1, more generally, makes an auxiliaryV -matrix. (§B.4.5)5.26A constraint G1=0 would prevent excursion into the translation-invariant subspace(numerical unboundedness).

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426 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.5.2 Rotation/Reflection

Rotation of the list X∈ Rn×N about some arbitrary point α∈Rn, orreflection through some affine subset containing α , can be accomplishedvia Q(X−α1T) where Q is an orthogonal matrix (§B.5).

We rightfully expect

D(

Q(X − α1T))

= D(QX − β1T) = D(QX) = D(X) (954)

Because list-form D(X) is translation invariant, we may safely ignoreoffset and consider only the impact of matrices that premultiply X .Interpoint distances are unaffected by rotation or reflection; we say,EDM D is rotation/reflection invariant. Proof follows from the fact,QT=Q−1 ⇒ XTQTQX =XTX . So (954) follows directly from (867).

The class of premultiplying matrices for which interpoint distances areunaffected is a little more broad than orthogonal matrices. Looking at EDMdefinition (867), it appears that any matrix Qp such that

XTQTpQp X = XTX (955)

will have the property

D(QpX) = D(X) (956)

An example is skinny Qp∈ Rm×n (m>n) having orthonormal columns. Wecall such a matrix orthonormal.

5.5.2.0.1 Example. Reflection prevention and quadrature rotation.Consider the EDM graph in Figure 126b representing known distancebetween vertices (Figure 126a) of a tilted-square diamond in R2. Supposesome geometrical optimization problem were posed where isometrictransformation is allowed excepting reflection, and where rotation mustbe quantized so that only quadrature rotations are allowed; only multiplesof π/2.

In two dimensions, a counterclockwise rotation of any vector about theorigin by angle θ is prescribed by the orthogonal matrix

Q =

[

cos θ − sin θsin θ cos θ

]

(957)

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5.5. INVARIANCE 427

x1 x1

x2 x2

x3 x3

x4 x4

(a) (b) (c)

Figure 126: (a) Four points in quadrature in two dimensions about theirgeometric center. (b) Complete EDM graph of diamond-shaped vertices.(c) Quadrature rotation of a Euclidean body in R2 first requires a shroudthat is the smallest Cartesian square containing it.

whereas reflection of any point through a hyperplane containing the origin

∂H =

x∈R2

[

cos θsin θ

]T

x = 0

(958)

is accomplished via multiplication with symmetric orthogonal matrix (§B.5.2)

R =

[

sin(θ)2− cos(θ)2 −2 sin(θ) cos(θ)−2 sin(θ) cos(θ) cos(θ)2− sin(θ)2

]

(959)

Rotation matrix Q is characterized by identical diagonal entries and byantidiagonal entries equal but opposite in sign, whereas reflection matrix Ris characterized in the reverse sense.

Assign the diamond vertices

xℓ ∈ R2, ℓ=1 . . . 4

to columns of a matrix

X = [ x1 x2 x3 x4 ] ∈ R2×4 (76)

Our scheme to prevent reflection enforces a rotation matrix characteristicupon the coordinates of adjacent points themselves: First shift the geometriccenter of X to the origin; for geometric centering matrix V ∈ S4 (§5.5.1.0.1),define

Y , XV ∈ R2×4 (960)

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428 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

To maintain relative quadrature between points (Figure 126a) and to preventreflection, it is sufficient that all interpoint distances be specified and thatadjacencies Y (: , 1: 2) , Y (: , 2:3) , and Y (: , 3:4) be proportional to 2×2rotation matrices; any clockwise rotation would ascribe a reflection matrixcharacteristic. Counterclockwise rotation is thereby enforced by constrainingequality among diagonal and antidiagonal entries as prescribed by (957);

Y (: , 1: 3) =

[

0 1−1 0

]

Y (: , 2:4) (961)

Quadrature quantization of rotation can be regarded as a constrainton tilt of the smallest Cartesian square containing the diamond as inFigure 126c. Our scheme to quantize rotation requires that all squarevertices be described by vectors whose entries are nonnegative when thesquare is translated anywhere interior to the nonnegative orthant. Wecapture the four square vertices as columns of a product Y C where

C =

1 0 0 11 1 0 00 1 1 00 0 1 1

(962)

Then, assuming a unit-square shroud, the affine constraint

Y C +

[

1/21/2

]

1T ≥ 0 (963)

quantizes rotation, as desired. 2

5.5.2.1 Inner-product form invariance

Likewise, D(Θ) (934) is rotation/reflection invariant;

D(QpΘ) = D(QΘ) = D(Θ) (964)

so (955) and (956) similarly apply.

5.5.3 Invariance conclusion

In the making of an EDM, absolute rotation, reflection, and translationinformation is lost. Given an EDM, reconstruction of point position (§5.12,the list X) can be guaranteed correct only in affine dimension r and relativeposition. Given a noiseless complete EDM, this isometric reconstruction isunique in so far as every realization of a corresponding list X is congruent :

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5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 429

5.6 Injectivity of D & unique reconstruction

Injectivity implies uniqueness of isometric reconstruction (§5.4.2.3.7); hence,we endeavor to demonstrate it.

EDM operators list-form D(X) (867), Gram-form D(G) (879), andinner-product form D(Θ) (934) are many-to-one surjections (§5.5) onto thesame range; the EDM cone (§6): (confer (880) (971))

EDMN =

D(X) : RN−1×N → SNh | X∈ RN−1×N

=

D(G) : SN → SNh | G ∈ SN

+− SN⊥c

=

D(Θ) : RN−1×N−1 → SNh | Θ ∈ RN−1×N−1

(965)

where (§5.5.1.1)

SN⊥c = u1T + 1uT | u∈RN ⊆ SN (1962)

5.6.1 Gram-form bijectivity

Because linear Gram-form EDM operator

D(G) = δ(G)1T + 1δ(G)T− 2G (879)

has no nullspace [83, §A.1] on the geometric center subspace5.27 (§E.7.2.0.2)

SNc , G∈ SN | G1 = 0 (1960)

= G∈ SN | N (G) ⊇ 1 = G∈ SN | R(G) ⊆ N (1T)= V Y V | Y ∈ SN ⊂ SN (1961)

≡ VNAV TN | A ∈ SN−1

(966)

then D(G) on that subspace is injective.

5.27The equivalence ≡ in (966) follows from the fact: Given B = V Y V = VNAV TN ∈ SN

c

with only matrix A∈ SN−1 unknown, then V †NBV †T

N = A or V †NY V †T

N = A .

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430 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

basis SN⊥c

∈ basis SN⊥h

∈ basis SN⊥h

SNc

SNh

dim SNc = dim SN

h = N(N−1)2

in RN(N+1)/2

dim SN⊥c = dim SN⊥

h = N in RN(N+1)/2

basis SNc = V EijV (confer (59))

Figure 127: Orthogonal complements in SN abstractly oriented inisometrically isomorphic RN(N+1)/2. Case N = 2 accurately illustrated in R3.Orthogonal projection of basis for SN⊥

h on SN⊥c yields another basis for SN⊥

c .(Basis vectors for SN⊥

c are illustrated lying in a plane orthogonal to SNc in this

dimension. Basis vectors for each ⊥ space outnumber those for its respectiveorthogonal complement; such is not the case in higher dimension.)

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5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 431

To prove injectivity of D(G) on SNc : Any matrix Y ∈ SN can be

decomposed into orthogonal components in SN ;

Y = V Y V + (Y − V Y V ) (967)

where V Y V ∈ SNc and Y −V Y V ∈ SN⊥

c (1962). Because of translationinvariance (§5.5.1.1) and linearity, D(Y−V Y V )=0 hence N (D)⊇ SN⊥

c . Itremains only to show

D(V Y V ) = 0 ⇔ V Y V = 0 (968)(

⇔ Y = u1T + 1uT for some u∈RN)

. D(V Y V ) will vanish whenever

2V Y V = δ(V Y V )1T + 1δ(V Y V )T. But this implies R(1) (§B.2) were asubset of R(V Y V ) , which is contradictory. Thus we have

N (D) = Y | D(Y )=0 = Y | V Y V = 0 = SN⊥c (969)

¨

Since G1=0 ⇔ X1=0 (887) simply means list X is geometricallycentered at the origin, and because the Gram-form EDM operator D istranslation invariant and N (D) is the translation-invariant subspace SN⊥

c ,then EDM definition D(G) (965) on5.28 (confer §6.6.1, §6.7.1, §A.7.4.0.1)

SNc ∩ SN

+ = V Y V º 0 | Y ∈ SN ≡ VNAV TN | A∈ SN−1

+ ⊂ SN (970)

must be surjective onto EDMN ; (confer (880))

EDMN =

D(G) | G ∈ SNc ∩ SN

+

(971)

5.6.1.1 Gram-form operator D inversion

Define the linear geometric centering operator V ; (confer (888))

V(D) : SN → SN , −V DV 12

(972)

[87, §4.3]5.29 This orthogonal projector V has no nullspace on

SNh = aff EDMN (1225)

5.28Equivalence ≡ in (970) follows from the fact: Given B = V Y V = VNAV TN ∈ SN

+ with

only matrix A unknown, then V †NBV †T

N = A and A∈ SN−1+ must be positive semidefinite

by positive semidefiniteness of B and Corollary A.3.1.0.5.5.29Critchley cites Torgerson (1958) [346, ch.11, §2] for a history and derivation of (972).

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432 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

because the projection of −D/2 on SNc (1960) can be 0 if and only if

D∈ SN⊥c ; but SN⊥

c ∩ SNh = 0 (Figure 127). Projector V on SN

h is thereforeinjective hence uniquely invertible. Further, −V SN

h V/2 is equivalent to thegeometric center subspace SN

c in the ambient space of symmetric matrices; asurjection,

SNc = V(SN) = V

(

SNh ⊕ SN⊥

h

)

= V(

SNh

)

(973)

because (72)V

(

SNh

)

⊇ V(

SN⊥h

)

= V(

δ2(SN))

(974)

Because D(G) on SNc is injective, and aff D

(

V(EDMN))

= D(

V(aff EDMN))

by property (126) of the affine hull, we find for D∈ SNh

D(−V DV 12) = δ(−V DV 1

2)1T + 1δ(−V DV 1

2)T − 2(−V DV 1

2) (975)

id est,D = D

(

V(D))

(976)

−V DV = V(

D(−V DV ))

(977)

orSN

h = D(

V(SNh )

)

(978)

−V SNh V = V

(

D(−V SNh V )

)

(979)

These operators V and D are mutual inverses.The Gram-form D

(

SNc

)

(879) is equivalent to SNh ;

D(

SNc

)

= D(

V(SNh ⊕ SN⊥

h ))

= SNh + D

(

V(SN⊥h )

)

= SNh (980)

because SNh ⊇ D

(

V(SN⊥h )

)

. In summary, for the Gram-form we have theisomorphisms [88, §2] [87, p.76, p.107] [7, §2.1]5.30 [6, §2] [8, §18.2.1] [2, §2.1]

SNh = D(SN

c ) (981)

SNc = V(SN

h ) (982)

and from the bijectivity results in §5.6.1,

EDMN = D(SNc ∩ SN

+ ) (983)

SNc ∩ SN

+ = V(EDMN) (984)

5.30In [7, p.6, line 20], delete sentence: Since G is also . . . not a singleton set.[7, p.10, line 11] x3 =2 (not 1).

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5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 433

5.6.2 Inner-product form bijectivity

The Gram-form EDM operator D(G)= δ(G)1T + 1δ(G)T− 2G (879) is aninjective map, for example, on the domain that is the subspace of symmetricmatrices having all zeros in the first row and column

SN1 = G∈ SN | Ge1 = 0

=

[

0 0T

0 I

]

Y

[

0 0T

0 I

]

| Y ∈ SN

(1964)

because it obviously has no nullspace there. Since Ge1 = 0 ⇔ Xe1 = 0 (881)means the first point in the list X resides at the origin, then D(G) on SN

1 ∩ SN+

must be surjective onto EDMN .Substituting ΘTΘ ← −V T

NDVN (946) into inner-product form EDMdefinition D(Θ) (934), it may be further decomposed:

D(D) =

[

0δ(

−V TNDVN

)

]

1T + 1[

0 δ(

−V TNDVN

)T]

− 2

[

0 0T

0 −V TNDVN

]

(985)

This linear operator D is another flavor of inner-product form and an injectivemap of the EDM cone onto itself. Yet when its domain is instead the entiresymmetric hollow subspace SN

h = aff EDMN , D(D) becomes an injectivemap onto that same subspace. Proof follows directly from the fact: linear Dhas no nullspace [83, §A.1] on SN

h = aff D(EDMN)= D(aff EDMN) (126).

5.6.2.1 Inversion of D(

−V TNDVN

)

Injectivity of D(D) suggests inversion of (confer (884))

VN (D) : SN → SN−1 , −V TNDVN (986)

a linear surjective5.31 mapping onto SN−1 having nullspace5.32 SN⊥c ;

VN (SNh ) = SN−1 (987)

5.31Surjectivity of VN (D) is demonstrated via the Gram-form EDM operator D(G) :

Since SNh = D(SN

c ) (980), then for any Y ∈ SN−1, −V TN D(V †T

N Y V †N /2)VN = Y .

5.32 N (VN ) ⊇ SN⊥c is apparent. There exists a linear mapping

T (VN (D)) , V †TN VN (D)V †

N = −V DV 12 = V(D)

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434 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

injective on domain SNh because SN⊥

c ∩ SNh = 0. Revising the argument of

this inner-product form (985), we get another flavor

D(

−V TNDVN

)

=

[

0δ(

−V TNDVN

)

]

1T + 1[

0 δ(

−V TNDVN

)T]

− 2

[

0 0T

0 −V TNDVN

]

(988)and we obtain mutual inversion of operators VN and D , for D∈ SN

h

D = D(

VN (D))

(989)

−V TNDVN = VN

(

D(−V TNDVN )

)

(990)

orSN

h = D(

VN (SNh )

)

(991)

−V TN SN

h VN = VN(

D(−V TN SN

h VN ))

(992)

Substituting ΘTΘ ← Φ into inner-product form EDM definition (934),any EDM may be expressed by the new flavor

D(Φ) ,

[

0δ(Φ)

]

1T + 1[

0 δ(Φ)T]

− 2

[

0 0T

0 Φ

]

∈ EDMN

⇔Φ º 0

(993)

where this D is a linear surjective operator onto EDMN by definition,injective because it has no nullspace on domain SN−1

+ . More broadly,aff D(SN−1

+ )= D(aff SN−1+ ) (126),

SNh = D(SN−1)

SN−1 = VN (SNh )

(994)

demonstrably isomorphisms, and by bijectivity of this inner-product form:

EDMN = D(SN−1+ ) (995)

SN−1+ = VN (EDMN) (996)

such that

N (T (VN )) = N (V) ⊇ N (VN ) ⊇ SN⊥c = N (V)

where the equality SN⊥c =N (V) is known (§E.7.2.0.2). ¨

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5.7. EMBEDDING IN AFFINE HULL 435

5.7 Embedding in affine hull

The affine hull A (78) of a point list xℓ (arranged columnar in X∈ Rn×N

(76)) is identical to the affine hull of that polyhedron P (86) formed from allconvex combinations of the xℓ ; [59, §2] [303, §17]

A = aff X = aff P (997)

Comparing hull definitions (78) and (86), it becomes obvious that the xℓ

and their convex hull P are embedded in their unique affine hull A ;

A ⊇ P ⊇ xℓ (998)

Recall: affine dimension r is a lower bound on embedding, equal todimension of the subspace parallel to that nonempty affine set A in whichthe points are embedded. (§2.3.1) We define dimension of the convex hull Pto be the same as dimension r of the affine hull A [303, §2], but r is notnecessarily equal to rank of X (1017).

For the particular example illustrated in Figure 113, P is the triangle inunion with its relative interior while its three vertices constitute the entirelist X . Affine hull A is the unique plane that contains the triangle, so affinedimension r = 2 in that example while rank of X is 3. Were there only twopoints in Figure 113, then the affine hull would instead be the unique linepassing through them; r would become 1 while rank would then be 2.

5.7.1 Determining affine dimension

Knowledge of affine dimension r becomes important because we lose anyabsolute offset common to all the generating xℓ in Rn when reconstructingconvex polyhedra given only distance information. (§5.5.1) To calculate r , wefirst remove any offset that serves to increase dimensionality of the subspacerequired to contain polyhedron P ; subtracting any α∈A in the affine hullfrom every list member will work,

X − α1T (999)

translating A to the origin:5.33

A− α = aff(X − α1T) = aff(X) − α (1000)

P − α = conv(X − α1T) = conv(X) − α (1001)

5.33The manipulation of hull functions aff and conv follows from their definitions.

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436 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Because (997) and (998) translate,

Rn ⊇ A−α = aff(X −α1T) = aff(P−α) ⊇ P−α ⊇ xℓ−α (1002)

where from the previous relations it is easily shown

aff(P − α) = aff(P) − α (1003)

Translating A neither changes its dimension or the dimension of theembedded polyhedron P ; (77)

r , dimA = dim(A− α) , dim(P − α) = dimP (1004)

For any α ∈ Rn, (1000)-(1004) remain true. [303, p.4, p.12] Yet whenα ∈ A , the affine set A− α becomes a unique subspace of Rn in whichthe xℓ − α and their convex hull P − α are embedded (1002), and whosedimension is more easily calculated.

5.7.1.0.1 Example. Translating first list-member to origin.Subtracting the first member α , x1 from every list member will translatetheir affine hull A and their convex hull P and, in particular, x1∈ P ⊆ A tothe origin in Rn ; videlicet,

X−x11T = X−Xe11

T = X(I−e11T) = X

[

0√

2VN]

∈ Rn×N (1005)

where VN is defined in (873), and e1 in (883). Applying (1002) to (1005),

Rn ⊇ R(XVN ) = A−x1 = aff(X−x11T) = aff(P−x1) ⊇ P−x1 ∋ 0

(1006)where XVN ∈ Rn×N−1. Hence

r = dimR(XVN ) (1007)

2

Since shifting the geometric center to the origin (§5.5.1.0.1) translates theaffine hull to the origin as well, then it must also be true

r = dimR(XV ) (1008)

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5.7. EMBEDDING IN AFFINE HULL 437

For any matrix whose range is R(V )=N (1T) we get the same result; e.g.,

r = dimR(XV †TN ) (1009)

because

R(XV ) = Xz | z∈N (1T) (1010)

and R(V ) = R(VN ) = R(V †TN ) (§E). These auxiliary matrices (§B.4.2) are

more closely related;

V = VNV †N (1621)

5.7.1.1 Affine dimension r versus rank

Now, suppose D is an EDM as defined by

D(X) = δ(XTX)1T + 1δ(XTX)T− 2XTX ∈ EDMN (867)

and we premultiply by −V TN and postmultiply by VN . Then because V T

N 1=0(874), it is always true that

−V TNDVN = 2V T

NXTXVN = 2V TN GVN ∈ SN−1 (1011)

where G is a Gram matrix. Similarly pre- and postmultiplying by V(confer (888))

−V DV = 2V XTXV = 2V GV ∈ SN (1012)

always holds because V 1=0 (1611). Likewise, multiplying inner-productform EDM definition (934), it always holds:

−V TNDVN = ΘTΘ ∈ SN−1 (938)

For any matrix A , rank ATA = rank A = rank AT. [199, §0.4]5.34 So, by(1010), affine dimension

r = rankXV = rankXVN = rank XV †TN = rank Θ

= rankV DV = rank V GV = rank V TNDVN = rankV T

N GVN(1013)

5.34For A∈Rm×n, N (ATA) = N (A). [328, §3.3]

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438 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

By conservation of dimension, (§A.7.3.0.1)

r + dimN (V TNDVN ) = N−1 (1014)

r + dimN (V DV ) = N (1015)

For D∈ EDMN

−V TNDVN ≻ 0 ⇔ r = N−1 (1016)

but −V DV ⊁ 0. The general fact5.35 (confer (899))

r ≤ minn , N−1 (1017)

is evident from (1005) but can be visualized in the example illustrated inFigure 113. There we imagine a vector from the origin to each point in thelist. Those three vectors are linearly independent in R3, but affine dimensionr is 2 because the three points lie in a plane. When that plane is translatedto the origin, it becomes the only subspace of dimension r=2 that cancontain the translated triangular polyhedron.

5.7.2 Precis

We collect expressions for affine dimension r : for list X∈ Rn×N and Grammatrix G∈ SN

+

r , dim(P − α) = dimP = dim conv X= dim(A− α) = dimA = dim aff X= rank(X − x11

T) = rank(X − αc1T)

= rank Θ (936)

= rankXVN = rank XV = rank XV †TN

= rankX , Xe1 = 0 or X1=0

= rankV TN GVN = rank V GV = rank V †

NGVN= rankG , Ge1 = 0 (884) or G1=0 (888)

= rankV TNDVN = rank V DV = rankV †

NDVN = rank VN (V TNDVN )V T

N= rank Λ (1104)

= N−1 − dimN([

0 1T

1 −D

])

= rank

[

0 1T

1 −D

]

− 2 (1025)

D ∈ EDMN

(1018)

5.35 rankX ≤ minn , N

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5.7. EMBEDDING IN AFFINE HULL 439

5.7.3 Eigenvalues of −V DV versus −V †NDVN

Suppose for D∈ EDMN we are given eigenvectors vi∈ RN of −V DV andcorresponding eigenvalues λ∈RN so that

−V DV vi = λi vi , i = 1 . . . N (1019)

From these we can determine the eigenvectors and eigenvalues of −V †NDVN :

Defineνi , V †

N vi , λi 6= 0 (1020)

Then we have:

−V DVNV †N vi = λi vi (1021)

−V †NV DVN νi = λiV

†N vi (1022)

−V †NDVN νi = λi νi (1023)

the eigenvectors of −V †NDVN are given by (1020) while its corresponding

nonzero eigenvalues are identical to those of −V DV although −V †NDVN

is not necessarily positive semidefinite. In contrast, −V TNDVN is positive

semidefinite but its nonzero eigenvalues are generally different.

5.7.3.0.1 Theorem. EDM rank versus affine dimension r .[159, §3] [180, §3] [158, §3] For D∈ EDMN (confer (1180))

1. r = rank(D) − 1 ⇔ 1TD†1 6= 0Points constituting a list X generating the polyhedron corresponding toD lie on the relative boundary of an r-dimensional circumhyperspherehaving

diameter =√

2(

1TD†1)−1/2

circumcenter = XD†1

1TD†1

(1024)

2. r = rank(D) − 2 ⇔ 1TD†1 = 0There can be no circumhypersphere whose relative boundary containsa generating list for the corresponding polyhedron.

3. In Cayley-Menger form [110, §6.2] [86, §3.3] [48, §40] (§5.11.2),

r = N−1 − dimN([

0 1T

1 −D

])

= rank

[

0 1T

1 −D

]

− 2 (1025)

Circumhyperspheres exist for r< rank(D)−2. [344, §7] ⋄

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440 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

For all practical purposes, (1017)

max0 , rank(D)− 2 ≤ r ≤ minn , N−1 (1026)

5.8 Euclidean metric versus matrix criteria

5.8.1 Nonnegativity property 1

When D=[dij] is an EDM (867), then it is apparent from (1011)

2V TNXTXVN = −V T

NDVN º 0 (1027)

because for any matrix A , ATAº0 .5.36 We claim nonnegativity of the dij

is enforced primarily by the matrix inequality (1027); id est,

−V TNDVN º 0

D ∈ SNh

⇒ dij ≥ 0 , i 6= j (1028)

(The matrix inequality to enforce strict positivity differs by a stroke of thepen. (1031))

We now support our claim: If any matrix A∈Rm×m is positivesemidefinite, then its main diagonal δ(A)∈ Rm must have all nonnegativeentries. [155, §4.2] Given D∈ SN

h

−V TNDVN =

d1212 (d12+d13−d23)

12 (d1,i+1+d1,j+1−di+1,j+1) · · · 1

2 (d12+d1N−d2N )12 (d12+d13−d23) d13

12 (d1,i+1+d1,j+1−di+1,j+1) · · · 1

2 (d13+d1N−d3N )

12 (d1,j+1+d1,i+1−dj+1,i+1)

12 (d1,j+1+d1,i+1−dj+1,i+1) d1,i+1

. . . 12 (d14+d1N−d4N )

......

. . .. . .

...12 (d12+d1N−d2N ) 1

2 (d13+d1N−d3N ) 12 (d14+d1N−d4N ) · · · d1N

= 12(1D1,2:N + D2:N,11

T− D2:N,2:N) ∈ SN−1 (1029)

5.36For A∈Rm×n, ATA º 0 ⇔ yTATAy = ‖Ay‖2 ≥ 0 for all ‖y‖ = 1. When A isfull-rank skinny-or-square, ATA ≻ 0.

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5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 441

where row,column indices i,j∈1 . . . N−1. [308] It follows:

−V TNDVN º 0

D ∈ SNh

⇒ δ(−V TNDVN ) =

d12

d13...

d1N

º 0 (1030)

Multiplication of VN by any permutation matrix Ξ has null effect on its rangeand nullspace. In other words, any permutation of the rows or columns of VNproduces a basis for N (1T); id est, R(ΞrVN )=R(VN Ξc)=R(VN )=N (1T).Hence, −V T

NDVN º 0 ⇔ −V TN ΞT

rDΞrVN º 0 (⇔ −ΞTc V T

NDVN Ξc º 0).Various permutation matrices5.37 will sift the remaining dij similarlyto (1030) thereby proving their nonnegativity. Hence −V T

NDVN º 0 isa sufficient test for the first property (§5.2) of the Euclidean metric,nonnegativity. ¨

When affine dimension r equals 1, in particular, nonnegativity symmetryand hollowness become necessary and sufficient criteria satisfying matrixinequality (1027). (§6.6.0.0.1)

5.8.1.1 Strict positivity

Should we require the points in Rn to be distinct, then entries of D off themain diagonal must be strictly positive dij > 0, i 6= j and only those entriesalong the main diagonal of D are 0. By similar argument, the strict matrixinequality is a sufficient test for strict positivity of Euclidean distance-square;

−V TNDVN ≻ 0

D ∈ SNh

⇒ dij > 0 , i 6= j (1031)

5.37The rule of thumb is: If Ξr(i , 1) = 1, then δ(−V TN ΞT

rDΞrVN )∈RN−1 is somepermutation of the ith row or column of D excepting the 0 entry from the main diagonal.

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442 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.8.2 Triangle inequality property 4

In light of Kreyszig’s observation [223, §1.1, prob.15] that properties 2through 4 of the Euclidean metric (§5.2) together imply nonnegativityproperty 1,

2√

djk =√

djk +√

dkj ≥√

djj = 0 , j 6=k (1032)

nonnegativity criterion (1028) suggests that matrix inequality −V TNDVN º 0

might somehow take on the role of triangle inequality; id est,

δ(D) = 0DT = D

−V TNDVN º 0

⇒√

dij ≤√

dik +√

dkj , i 6=j 6=k (1033)

We now show that is indeed the case: Let T be the leading principalsubmatrix in S2 of −V T

NDVN (upper left 2×2 submatrix from (1029));

T ,

[

d1212(d12+d13−d23)

12(d12+d13−d23) d13

]

(1034)

Submatrix T must be positive (semi)definite whenever −V TNDVN is.

(§A.3.1.0.4, §5.8.3) Now we have,

−V TNDVN º 0 ⇒ T º 0 ⇔ λ1 ≥ λ2 ≥ 0

−V TNDVN ≻ 0 ⇒ T ≻ 0 ⇔ λ1 > λ2 > 0

(1035)

where λ1 and λ2 are the eigenvalues of T , real due only to symmetry of T :

λ1 = 12

(

d12 + d13 +√

d 223 − 2(d12 + d13)d23 + 2(d 2

12 + d 213)

)

∈ R

λ2 = 12

(

d12 + d13 −√

d 223 − 2(d12 + d13)d23 + 2(d 2

12 + d 213)

)

∈ R(1036)

Nonnegativity of eigenvalue λ1 is guaranteed by only nonnegativity of the dij

which in turn is guaranteed by matrix inequality (1028). Inequality betweenthe eigenvalues in (1035) follows from only realness of the dij . Since λ1

always equals or exceeds λ2 , conditions for the positive (semi)definiteness ofsubmatrix T can be completely determined by examining λ2 the smaller ofits two eigenvalues. A triangle inequality is made apparent when we express

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5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 443

T eigenvalue nonnegativity in terms of D matrix entries; videlicet,

T º 0 ⇔ det T = λ1λ2 ≥ 0 , d12 , d13 ≥ 0 (c)⇔

λ2 ≥ 0 (b)⇔

|√

d12 −√

d23 | ≤√

d13 ≤√

d12 +√

d23 (a)

(1037)

Triangle inequality (1037a) (confer (932) (1049)), in terms of three rootedentries from D , is equivalent to metric property 4

d13 ≤√

d12 +√

d23√

d23 ≤√

d12 +√

d13√

d12 ≤√

d13 +√

d23

(1038)

for the corresponding points x1 , x2 , x3 from some length-N list.5.38

5.8.2.1 Comment

Given D whose dimension N equals or exceeds 3, there are N !/(3!(N− 3)!)distinct triangle inequalities in total like (932) that must be satisfied, of whicheach dij is involved in N−2, and each point xi is in (N−1)!/(2!(N−1 − 2)!).We have so far revealed only one of those triangle inequalities; namely,(1037a) that came from T (1034). Yet we claim if −V T

NDVN º 0 thenall triangle inequalities will be satisfied simultaneously;

|√

dik −√

dkj | ≤√

dij ≤√

dik +√

dkj , i<k<j (1039)

(There are no more.) To verify our claim, we must prove the matrix inequality−V T

NDVN º 0 to be a sufficient test of all the triangle inequalities; moreefficient, we mention, for larger N :

5.38Accounting for symmetry property 3, the fourth metric property demands threeinequalities be satisfied per one of type (1037a). The first of those inequalities in (1038)is self evident from (1037a), while the two remaining follow from the left-hand side of(1037a) and the fact for scalars, |a| ≤ b ⇔ a ≤ b and −a ≤ b .

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444 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.8.2.1.1 Shore. The columns of ΞrVN Ξc hold a basis for N (1T)when Ξr and Ξc are permutation matrices. In other words, any permutationof the rows or columns of VN leaves its range and nullspace unchanged;id est, R(ΞrVN Ξc)=R(VN )=N (1T) (874). Hence, two distinct matrixinequalities can be equivalent tests of the positive semidefiniteness of D onR(VN ) ; id est, −V T

NDVN º 0 ⇔ −(ΞrVN Ξc)TD(ΞrVN Ξc)º 0. By properly

choosing permutation matrices,5.39 the leading principal submatrix TΞ ∈ S2

of −(ΞrVN Ξc)TD(ΞrVN Ξc) may be loaded with the entries of D needed to

test any particular triangle inequality (similarly to (1029)-(1037)). Becauseall the triangle inequalities can be individually tested using a test equivalentto the lone matrix inequality −V T

NDVN º0, it logically follows that the lonematrix inequality tests all those triangle inequalities simultaneously. Weconclude that −V T

NDVN º 0 is a sufficient test for the fourth property of theEuclidean metric, triangle inequality. ¨

5.8.2.2 Strict triangle inequality

Without exception, all the inequalities in (1037) and (1038) can be madestrict while their corresponding implications remain true. The thenstrict inequality (1037a) or (1038) may be interpreted as a strict triangleinequality under which collinear arrangement of points is not allowed.[219, §24/6, p.322] Hence by similar reasoning, −V T

NDVN ≻ 0 is a sufficienttest of all the strict triangle inequalities; id est,

δ(D) = 0

DT = D−V T

NDVN ≻ 0

⇒√

dij <√

dik +√

dkj , i 6=j 6=k (1040)

5.8.3 −V TNDVN nesting

From (1034) observe that T =−V TNDVN |N←3 . In fact, for D∈ EDMN ,

the leading principal submatrices of −V TNDVN form a nested sequence (by

inclusion) whose members are individually positive semidefinite [155] [199][328] and have the same form as T ; videlicet,5.40

5.39To individually test triangle inequality |√

dik −√

dkj | ≤√

dij ≤√

dik +√

dkj forparticular i , k, j , set Ξr(i , 1)= Ξr(k, 2)= Ξr(j, 3)=1 and Ξc = I .5.40 −V DV |N←1 = 0 ∈ S0

+ (§B.4.1)

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5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 445

−V TNDVN |N←1 = [ ∅ ] (o)

−V TNDVN |N←2 = [d12] ∈ S+ (a)

−V TNDVN |N←3 =

[

d1212(d12+d13−d23)

12(d12+d13−d23) d13

]

= T ∈ S2

+ (b)

−V TNDVN |N←4 =

d1212(d12+d13−d23)

12(d12+d14−d24)

12(d12+d13−d23) d13

12(d13+d14−d34)

12(d12+d14−d24)

12(d13+d14−d34) d14

(c)

...

−V TNDVN |N← i =

−V TNDVN |N← i−1 ν(i)

ν(i)T d1i

∈ Si−1+ (d)

...

−V TNDVN =

−V TNDVN |N←N−1 ν(N)

ν(N)T d1N

∈ SN−1+ (e)

(1041)

where

ν(i) ,1

2

d12+d1i−d2i

d13+d1i−d3i...

d1,i−1+d1i−di−1,i

∈ Ri−2, i > 2 (1042)

Hence, the leading principal submatrices of EDM D must also be EDMs.5.41

Bordered symmetric matrices in the form (1041d) are known to haveintertwined [328, §6.4] (or interlaced [199, §4.3] [325, §IV.4.1]) eigenvalues;(confer §5.11.1) that means, for the particular submatrices (1041a) and(1041b),

5.41In fact, each and every principal submatrix of an EDM D is another EDM. [231, §4.1]

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446 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

λ2 ≤ d12 ≤ λ1 (1043)

where d12 is the eigenvalue of submatrix (1041a) and λ1 , λ2 are theeigenvalues of T (1041b) (1034). Intertwining in (1043) predicts that shouldd12 become 0, then λ2 must go to 0 .5.42 The eigenvalues are similarlyintertwined for submatrices (1041b) and (1041c);

γ3 ≤ λ2 ≤ γ2 ≤ λ1 ≤ γ1 (1044)

where γ1 , γ2 , γ3 are the eigenvalues of submatrix (1041c). Intertwininglikewise predicts that should λ2 become 0 (a possibility revealed in §5.8.3.1),then γ3 must go to 0. Combining results so far for N = 2, 3, 4: (1043) (1044)

γ3 ≤ λ2 ≤ d12 ≤ λ1 ≤ γ1 (1045)

The preceding logic extends by induction through the remaining membersof the sequence (1041).

5.8.3.1 Tightening the triangle inequality

Now we apply Schur complement from §A.4 to tighten the triangle inequalityfrom (1033) in case: cardinality N = 4. We find that the gains by doing soare modest. From (1041) we identify:

[

A BBT C

]

, −V TNDVN |N←4 (1046)

A , T = −V TNDVN |N←3 (1047)

both positive semidefinite by assumption, where B= ν(4) (1042), andC = d14 . Using nonstrict CC†-form (1484), Cº 0 by assumption (§5.8.1)and CC†= I . So by the positive semidefinite ordering of eigenvalues theorem(§A.3.1.0.1),

−V TNDVN |N←4 º 0 ⇔ T º d−1

14 ν(4)ν(4)T ⇒

λ1 ≥ d−114 ‖ν(4)‖2

λ2 ≥ 0(1048)

where d−114 ‖ν(4)‖2, 0 are the eigenvalues of d−1

14 ν(4)ν(4)T while λ1 , λ2 arethe eigenvalues of T .

5.42If d12 were 0, eigenvalue λ2 becomes 0 (1036) because d13 must then be equal tod23 ; id est, d12 = 0 ⇔ x1 = x2 . (§5.4)

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5.8.3.1.1 Example. Small completion problem, II.Applying the inequality for λ1 in (1048) to the small completion problem onpage 388 Figure 114, the lower bound on

√d14 (1.236 in (860)) is tightened

to 1.289 . The correct value of√

d14 to three significant figures is 1.414 .

2

5.8.4 Affine dimension reduction in two dimensions

(confer §5.14.4) The leading principal 2×2 submatrix T of −V TNDVN has

largest eigenvalue λ1 (1036) which is a convex function of D .5.43 λ1 cannever be 0 unless d12 = d13 = d23 = 0. Eigenvalue λ1 can never be negativewhile the dij are nonnegative. The remaining eigenvalue λ2 is a concavefunction of D that becomes 0 only at the upper and lower bounds of inequality(1037a) and its equivalent forms: (confer (1039))

|√

d12 −√

d23 | ≤√

d13 ≤√

d12 +√

d23 (a)⇔

|√

d12 −√

d13 | ≤√

d23 ≤√

d12 +√

d13 (b)⇔

|√

d13 −√

d23 | ≤√

d12 ≤√

d13 +√

d23 (c)

(1049)

In between those bounds, λ2 is strictly positive; otherwise, it would benegative but prevented by the condition T º 0.

When λ2 becomes 0, it means triangle 123 has collapsed to a linesegment; a potential reduction in affine dimension r . The same logic is validfor any particular principal 2×2 submatrix of −V T

NDVN , hence applicableto other triangles.

5.43The largest eigenvalue of any symmetric matrix is always a convex function of itsentries, while the smallest eigenvalue is always concave. [59, exmp.3.10] In our particular

case, say d ,

d12

d13

d23

∈ R3. Then the Hessian (1723) ∇2λ1(d)º 0 certifies convexity

whereas ∇2λ2(d)¹ 0 certifies concavity. Each Hessian has rank equal to 1. The respectivegradients ∇λ1(d) and ∇λ2(d) are nowhere 0.

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448 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.9 Bridge: Convex polyhedra to EDMs

The criteria for the existence of an EDM include, by definition (867) (934),the properties imposed upon its entries dij by the Euclidean metric. From§5.8.1 and §5.8.2, we know there is a relationship of matrix criteria to thoseproperties. Here is a snapshot of what we are sure: for i , j , k∈1 . . . N(confer §5.2)

dij ≥ 0 , i 6= j√

dij = 0 , i = j√

dij =√

dji√

dij ≤√

dik +√

dkj , i 6=j 6=k

⇐−V T

NDVN º 0δ(D) = 0DT = D

(1050)

all implied by D∈ EDMN . In words, these four Euclidean metric propertiesare necessary conditions for D to be a distance matrix. At the moment,we have no converse. As of concern in §5.3, we have yet to establishmetric requirements beyond the four Euclidean metric properties that wouldallow D to be certified an EDM or might facilitate polyhedron or listreconstruction from an incomplete EDM. We deal with this problem in §5.14.Our present goal is to establish ab initio the necessary and sufficient matrixcriteria that will subsume all the Euclidean metric properties and any furtherrequirements5.44 for all N >1 (§5.8.3); id est,

−V TNDVN º 0

D ∈ SNh

⇔ D ∈ EDMN (886)

or for EDM definition (943),

Ω º 0√

δ(d) º 0

⇔ D = D(Ω , d) ∈ EDMN (1051)

5.44In 1935, Schoenberg [308, (1)] first extolled matrix product −V TNDVN (1029)

(predicated on symmetry and self-distance) specifically incorporating VN , albeitalgebraically. He showed: nonnegativity −yTV T

NDVN y ≥ 0, for all y∈RN−1, is necessaryand sufficient for D to be an EDM. Gower [158, §3] remarks how surprising it is that sucha fundamental property of Euclidean geometry was obtained so late.

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Figure 128: Elliptope E3 in isometrically isomorphic R6 (projected on R3)is a convex body that appears to possess some kind of symmetry in thisdimension; it resembles a malformed pillow in the shape of a bulgingtetrahedron. Elliptope relative boundary is not smooth and comprises all setmembers (1052) having at least one 0 eigenvalue. [234, §2.1] This elliptopehas an infinity of vertices, but there are only four vertices corresponding toa rank-1 matrix. Those yyT, evident in the illustration, have binary vectory ∈ R3 with entries in ±1.

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450 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

0

Figure 129: Elliptope E2 in isometrically isomorphic R3 is a line segmentillustrated interior to positive semidefinite cone S2

+ (Figure 42).

5.9.1 Geometric arguments

5.9.1.0.1 Definition. Elliptope: [234] [231, §2.3] [110, §31.5]a unique bounded immutable convex Euclidean body in Sn ; intersection ofpositive semidefinite cone Sn

+ with that set of n hyperplanes defined by unitymain diagonal;

En , Sn+ ∩ Φ∈ Sn | δ(Φ)=1 (1052)

a.k.a, the set of all correlation matrices of dimension

dim En = n(n−1)/2 in Rn(n+1)/2 (1053)

An elliptope En is not a polyhedron, in general, but has some polyhedralfaces and an infinity of vertices.5.45 Of those, 2n−1 vertices (some extremepoints of the elliptope) are extreme directions yyT of the positive semidefinitecone, where the entries of vector y ∈ Rn belong to ±1 and exercise everycombination. Each of the remaining vertices has rank belonging to the setk>0 | k(k + 1)/2≤ n. Each and every face of an elliptope is exposed.

5.45Laurent defines vertex distinctly from the sense herein (§2.6.1.0.1); she defines vertexas a point with full-dimensional (nonempty interior) normal cone (§E.10.3.2.1). Herdefinition excludes point C in Figure 31, for example.

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5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 451

In fact, any positive semidefinite matrix whose entries belong to ±1 isa correlation matrix whose rank must be one:5.46

5.9.1.0.2 Theorem. Some elliptope vertices. [122, §2.1.1]For Y ∈ Sn, y∈ Rn, and all i, j∈1 . . . n (confer §2.3.1.0.2)

Y º 0 , Yij ∈ ±1 ⇔ Y = yyT, yi ∈ ±1 (1054)⋄

The elliptope for dimension n = 2 is a line segment in isometricallyisomorphic Rn(n+1)/2 (Figure 129). Obviously, cone(En) 6= Sn

+ . The elliptopefor dimension n = 3 is realized in Figure 128.

5.9.1.0.3 Lemma. Hypersphere. (confer bullet p.398) [17, §4]Matrix Ψ = [Ψij]∈ SN belongs to the elliptope in SN iff there exist N points pon the boundary of a hypersphere in RrankΨ having radius 1 such that

‖pi − pj‖2 = 2(1 − Ψij) , i, j =1 . . . N (1055)⋄

There is a similar theorem for Euclidean distance matrices:We derive matrix criteria for D to be an EDM, validating (886) using

simple geometry; distance to the polyhedron formed by the convex hull of alist of points (76) in Euclidean space Rn.

5.9.1.0.4 EDM assertion.D is a Euclidean distance matrix if and only if D∈ SN

h and distances-squarefrom the origin

‖p(y)‖2 = −yTV TNDVN y | y ∈ S − β (1056)

correspond to points p in some bounded convex polyhedron

P − α = p(y) | y ∈ S − β (1057)

having N or fewer vertices embedded in an r-dimensional subspace A− αof Rn, where α ∈ A = aff P and where the domain of linear surjection p(y)is the unit simplex S⊂RN−1

+ shifted such that its vertex at the origin istranslated to −β in RN−1. When β = 0, then α = x1 . ⋄5.46As there are few equivalent conditions for rank constraints, this device is ratherimportant for relaxing integer, combinatorial, or Boolean problems.

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452 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

In terms of VN , the unit simplex (272) in RN−1 has an equivalentrepresentation:

S = s ∈ RN−1 |√

2VN s º −e1 (1058)

where e1 is as in (883). Incidental to the EDM assertion, shifting theunit-simplex domain in RN−1 translates the polyhedron P in Rn. Indeed,there is a map from vertices of the unit simplex to members of the listgenerating P ;

p : RN−1

p

−βe1 − βe2 − β

...eN−1 − β

=

Rn

x1 − αx2 − αx3 − α

...xN − α

(1059)

5.9.1.0.5 Proof. EDM assertion.(⇒) We demonstrate that if D is an EDM, then each distance-square ‖p(y)‖2

described by (1056) corresponds to a point p in some embedded polyhedronP − α . Assume D is indeed an EDM; id est, D can be made from some listX of N unknown points in Euclidean space Rn ; D = D(X) for X∈ Rn×N

as in (867). Since D is translation invariant (§5.5.1), we may shift the affinehull A of those unknown points to the origin as in (999). Then take anypoint p in their convex hull (86);

P − α = p = (X − Xb1T)a | aT1 = 1, a º 0 (1060)

where α = Xb ∈ A ⇔ bT1 = 1. Solutions to aT1 = 1 are:5.47

a ∈

e1 +√

2VN s | s ∈ RN−1

(1061)

where e1 is as in (883). Similarly, b = e1 +√

2VN β .

P − α = p = X(I − (e1 +√

2VNβ)1T)(e1 +√

2VN s) |√

2VN s º −e1= p = X

√2VN (s − β) |

√2VN s º −e1 (1062)

5.47Since R(VN )=N (1T) and N (1T)⊥R(1) , then over all s∈RN−1, VN s is ahyperplane through the origin orthogonal to 1. Thus the solutions a constitute ahyperplane orthogonal to the vector 1, and offset from the origin in RN by any particularsolution; in this case, a = e1 .

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5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 453

that describes the domain of p(s) as the unit simplex

S = s |√

2VN s º −e1 ⊂ RN−1+ (1058)

Making the substitution s − β ← y

P − α = p = X√

2VN y | y ∈ S − β (1063)

Point p belongs to a convex polyhedron P−α embedded in an r-dimensionalsubspace of Rn because the convex hull of any list forms a polyhedron, andbecause the translated affine hull A− α contains the translated polyhedronP − α (1002) and the origin (when α ∈A), and because A has dimensionr by definition (1004). Now, any distance-square from the origin to thepolyhedron P − α can be formulated

pTp = ‖p‖2 = 2yTV TNXTXVN y | y ∈ S − β (1064)

Applying (1011) to (1064) we get (1056).(⇐) To validate the EDM assertion in the reverse direction, we prove: If eachdistance-square ‖p(y)‖2 (1056) on the shifted unit-simplex S−β⊂ RN−1

corresponds to a point p(y) in some embedded polyhedron P − α , thenD is an EDM. The r-dimensional subspace A− α ⊆ Rn is spanned by

p(S − β) = P − α (1065)

because A− α = aff(P − α) ⊇ P − α (1002). So, outside the domain S − βof linear surjection p(y) , the simplex complement \S − β ⊂ RN−1 mustcontain the domain of the distance-square ‖p(y)‖2 = p(y)Tp(y) to remainingpoints in the subspace A− α ; id est, to the polyhedron’s relative exterior\P − α . For ‖p(y)‖2 to be nonnegative on the entire subspace A− α ,−V T

NDVN must be positive semidefinite and is assumed symmetric;5.48

−V TNDVN , ΘT

p Θp (1066)

where5.49 Θp∈ Rm×N−1 for some m≥ r . Because p(S − β) is a convexpolyhedron, it is necessarily a set of linear combinations of points from some

5.48The antisymmetric part(

−V TNDVN − (−V T

NDVN )T)

/2 is annihilated by ‖p(y)‖2. Bythe same reasoning, any positive (semi)definite matrix A is generally assumed symmetricbecause only the symmetric part (A +AT)/2 survives the test yTAy ≥ 0. [199, §7.1]5.49 AT = A º 0 ⇔ A = RTR for some real matrix R . [328, §6.3]

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454 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

length-N list because every convex polyhedron having N or fewer verticescan be generated that way (§2.12.2). Equivalent to (1056) are

pTp | p ∈ P − α = pTp = yTΘTp Θp y | y ∈ S − β (1067)

Because p ∈ P − α may be found by factoring (1067), the list Θp is foundby factoring (1066). A unique EDM can be made from that list usinginner-product form definition D(Θ)|Θ=Θp (934). That EDM will be identicalto D if δ(D)=0, by injectivity of D (985). ¨

5.9.2 Necessity and sufficiency

From (1027) we learned that matrix inequality −V TNDVN º 0 is a necessary

test for D to be an EDM. In §5.9.1, the connection between convex polyhedraand EDMs was pronounced by the EDM assertion; the matrix inequalitytogether with D∈ SN

h became a sufficient test when the EDM assertiondemanded that every bounded convex polyhedron have a correspondingEDM. For all N >1 (§5.8.3), the matrix criteria for the existence of an EDMin (886), (1051), and (862) are therefore necessary and sufficient and subsumeall the Euclidean metric properties and further requirements.

5.9.3 Example revisited

Now we apply the necessary and sufficient EDM criteria (886) to an earlierproblem.

5.9.3.0.1 Example. Small completion problem, III. (confer §5.8.3.1.1)Continuing Example 5.3.0.0.2 pertaining to Figure 114 where N = 4,distance-square d14 is ascertainable from the matrix inequality −V T

NDVN º 0.Because all distances in (859) are known except

√d14 , we may simply

calculate the smallest eigenvalue of −V TNDVN over a range of d14 as in

Figure 130. We observe a unique value of d14 satisfying (886) where theabscissa axis is tangent to the hypograph of the smallest eigenvalue. Sincethe smallest eigenvalue of a symmetric matrix is known to be a concavefunction (§5.8.4), we calculate its second partial derivative with respect tod14 evaluated at 2 and find −1/3. We conclude there are no other satisfyingvalues of d14 . Further, that value of d14 does not meet an upper or lowerbound of a triangle inequality like (1039), so neither does it cause the collapse

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5.10. EDM-ENTRY COMPOSITION 455

of any triangle. Because the smallest eigenvalue is 0, affine dimension r ofany point list corresponding to D cannot exceed N−2. (§5.7.1.1) 2

5.10 EDM-entry composition

Laurent [231, §2.3] applies results from Schoenberg (1938) [309] to showcertain nonlinear compositions of individual EDM entries yield EDMs;in particular,

D ∈ EDMN ⇔ [1 − e−αdij ] ∈ EDMN ∀α > 0

⇔ [e−αdij ] ∈ EN ∀α > 0(1068)

where D = [dij] and EN is the elliptope (1052).

5.10.0.0.1 Proof. (Monique Laurent, 2003) [309] (confer [223])

Lemma 2.1. from A Tour d’Horizon . . . on Completion Problems.[231] For D=[dij , i, j =1 . . . N ]∈ SN

h and EN the elliptope in SN

(§5.9.1.0.1), the following assertions are equivalent:

(i) D ∈ EDMN

(ii) e−αD , [e−αdij ] ∈ EN for all α > 0

(iii) 11T− e−αD , [1 − e−αdij ] ∈ EDMN for all α > 0 ⋄

1) Equivalence of Lemma 2.1 (i) (ii) is stated in Schoenberg’s Theorem 1[309, p.527].

2) (ii) ⇒ (iii) can be seen from the statement in the beginning of section 3,saying that a distance space embeds in L2 iff some associated matrixis PSD. We reformulate it:

Let d =(dij)i,j=0,1...N be a distance space on N+1 points(i.e., symmetric hollow matrix of order N+1) and letp =(pij)i,j=1...N be the symmetric matrix of order N related by:

(A) 2pij = d0i + d0j − dij for i, j = 1 . . . Nor equivalently

(B) d0i = pii , dij = pii + pjj − 2pij for i, j = 1 . . . N

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456 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

1 2 3 4

-0.6

-0.4

-0.2

smallest eigenvalue

d14

Figure 130: Smallest eigenvalue of −V TNDVN makes it a PSD matrix for

only one value of d14 : 2.

0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

1.2

1.4

0.5 1 1.5 2

1

2

3

4

d1/αij

log2(1+d1/αij )

1− e−αdij

dij

α

(a)

(b)d

1/αij

log2(1+d1/αij )

1− e−αdij

Figure 131: Some entrywise EDM compositions: (a) α = 2. Concavenondecreasing in dij . (b) Trajectory convergence in α for dij = 2.

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5.10. EDM-ENTRY COMPOSITION 457

Then d embeds in L2 iff p is a positive semidefinite matrix iff dis of negative type (second half page 525/top of page 526 in [309]).

For the implication from (ii) to (iii), set: p = e−αd and define d ′

from p using (B) above. Then d ′ is a distance space on N+1points that embeds in L2 . Thus its subspace of N points alsoembeds in L2 and is precisely 1− e−αd.

Note that (iii) ⇒ (ii) cannot be read immediately from this argumentsince (iii) involves the subdistance of d ′ on N points (and not the fulld ′ on N+1 points).

3) Show (iii) ⇒ (i) by using the series expansion of the function 1− e−αd :the constant term cancels, α factors out; there remains a summationof d plus a multiple of α . Letting α go to 0 gives the result.

This is not explicitly written in Schoenberg, but he also uses suchan argument; expansion of the exponential function then α→ 0 (firstproof on [309, p.526]). ¨

Schoenberg’s results [309, §6, thm.5] (confer [223, p.108 -109]) alsosuggest certain finite positive roots of EDM entries produce EDMs;specifically,

D ∈ EDMN ⇔ [d1/αij ] ∈ EDMN ∀α > 1 (1069)

The special case α = 2 is of interest because it corresponds to absolutedistance; e.g.,

D∈ EDMN ⇒ √

D ∈ EDMN (1070)

Assuming that points constituting a corresponding list X are distinct(1031), then it follows: for D∈ SN

h

limα→∞

[d1/αij ] = lim

α→∞[1 − e−αdij ] = −E , 11T− I (1071)

Negative elementary matrix −E (§B.3) is relatively interior to the EDMcone (§6.6) and terminal to the respective trajectories (1068) and (1069)as functions of α . Both trajectories are confined to the EDM cone; inengineering terms, the EDM cone is an invariant set [306] to either trajectory.Further, if D is not an EDM but for some particular αp it becomes an EDM,then for all greater values of α it remains an EDM.

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458 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.10.0.0.2 Exercise. Concave nondecreasing EDM-entry composition.Given EDM D = [dij] , empirical evidence suggests that the composition

[log2(1 + d1/αij )] is also an EDM for each fixed α≥ 1 [sic] . Its concavity

in dij is illustrated in Figure 131 together with functions from (1068) and(1069). Prove whether it holds more generally: Any concave nondecreasingcomposition of individual EDM entries dij on R+ produces another EDM.

H

5.10.0.0.3 Exercise. Taxicab distance matrix as EDM.Determine whether taxicab distance matrices (D1(X) in Example 3.8.0.0.2)are all numerically equivalent to EDMs. Explain why or why not. H

5.10.1 EDM by elliptope

(confer (893)) For some κ∈R+ and C∈ SN+ in the elliptope EN (§5.9.1.0.1),

Alfakih asserts any given EDM D is expressible [9] [110, §31.5]

D = κ(11T− C) ∈ EDMN (1072)

This expression exhibits nonlinear combination of variables κ and C . Wetherefore propose a different expression requiring redefinition of the elliptope(1052) by scalar parametrization;

Ent , Sn

+ ∩ Φ∈ Sn | δ(Φ)= t1 (1073)

where, of course, En = En1 . Then any given EDM D is expressible

D = t11T− E ∈ EDMN (1074)

which is linear in variables t∈R+ and E∈ENt .

5.11 EDM indefiniteness

By known result (§A.7.2) regarding a 0-valued entry on the main diagonal ofa symmetric positive semidefinite matrix, there can be no positive or negativesemidefinite EDM except the 0 matrix because EDMN⊆ SN

h (866) and

SNh ∩ SN

+ = 0 (1075)

the origin. So when D∈ EDMN , there can be no factorization D =ATAor −D =ATA . [328, §6.3] Hence eigenvalues of an EDM are neither allnonnegative or all nonpositive; an EDM is indefinite and possibly invertible.

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5.11. EDM INDEFINITENESS 459

5.11.1 EDM eigenvalues, congruence transformation

For any symmetric −D , we can characterize its eigenvalues by congruencetransformation: [328, §6.3]

−WTDW = −[

V TN

1T

]

D [ VN 1 ] = −[

V TNDVN V T

ND1

1TDVN 1TD1

]

∈ SN (1076)

Because

W , [ VN 1 ] ∈ RN×N (1077)

is full-rank, then (1489)

inertia(−D) = inertia(

−WTDW)

(1078)

the congruence (1076) has the same number of positive, zero, and negativeeigenvalues as −D . Further, if we denote by γi , i=1 . . . N−1 theeigenvalues of −V T

NDVN and denote eigenvalues of the congruence −WTDWby ζi , i=1 . . . N and if we arrange each respective set of eigenvalues innonincreasing order, then by theory of interlacing eigenvalues for borderedsymmetric matrices [199, §4.3] [328, §6.4] [325, §IV.4.1]

ζN ≤ γN−1 ≤ ζN−1 ≤ γN−2 ≤ · · · ≤ γ2 ≤ ζ2 ≤ γ1 ≤ ζ1 (1079)

When D∈ EDMN , then γi ≥ 0 ∀ i (1426) because −V TNDVN º 0 as we

know. That means the congruence must have N−1 nonnegative eigenvalues;ζi ≥ 0, i=1 . . . N−1. The remaining eigenvalue ζN cannot be nonnegativebecause then −D would be positive semidefinite, an impossibility; so ζN < 0.By congruence, nontrivial −D must therefore have exactly one negativeeigenvalue;5.50 [110, §2.4.5]

5.50All the entries of the corresponding eigenvector must have the same sign with respectto each other [87, p.116] because that eigenvector is the Perron vector corresponding tothe spectral radius; [199, §8.2.6] the predominant characteristic of negative [sic ] matrices.

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460 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

D ∈ EDMN ⇒

λ(−D)i ≥ 0 , i=1 . . . N−1(

N∑

i=1

λ(−D)i = 0

)

D ∈ SNh ∩ RN×N

+

(1080)

where the λ(−D)i are nonincreasingly ordered eigenvalues of −D whosesum must be 0 only because tr D = 0 [328, §5.1]. The eigenvalue summationcondition, therefore, can be considered redundant. Even so, all theseconditions are insufficient to determine whether some given H∈ SN

h is anEDM, as shown by counter-example.5.51

5.11.1.0.1 Exercise. Spectral inequality.Prove whether it holds: for D=[dij]∈ EDMN

λ(−D)1 ≥ dij ≥ λ(−D)N−1 ∀ i 6= j (1081)

H

Terminology: a spectral cone is a convex cone containing all eigenspectra[223, p.365] [325, p.26] corresponding to some set of matrices. Anypositive semidefinite matrix, for example, possesses a vector of nonnegativeeigenvalues corresponding to an eigenspectrum contained in a spectral conethat is a nonnegative orthant.

5.11.2 Spectral cones

Denoting the eigenvalues of Cayley-Menger matrix

[

0 1T

1 −D

]

∈ SN+1 by

λ

([

0 1T

1 −D

])

∈ RN+1 (1082)

5.51When N = 3, for example, the symmetric hollow matrix

H =

0 1 11 0 51 5 0

∈ SNh ∩ RN×N

+

is not an EDM, although λ(−H) = [5 0.3723 −5.3723]T conforms to (1080).

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5.11. EDM INDEFINITENESS 461

we have the Cayley-Menger form (§5.7.3.0.1) of necessary and sufficientconditions for D∈ EDMN from the literature: [180, §3]5.52 [72, §3] [110, §6.2](confer (886) (862))

D ∈ EDMN ⇔

λ

([

0 1T

1 −D

])

i

≥ 0 , i = 1 . . . N

D ∈ SNh

−V TNDVN º 0

D ∈ SNh

(1083)

These conditions say the Cayley-Menger form has one and only one negative

eigenvalue. When D is an EDM, eigenvalues λ

([

0 1T

1 −D

])

belong to that

particular orthant in RN+1 having the N+1th coordinate as sole negativecoordinate5.53:

[

RN+

R−

]

= cone e1 , e2 , · · · eN , −eN+1 (1084)

5.11.2.1 Cayley-Menger versus Schoenberg

Connection to the Schoenberg criterion (886) is made when theCayley-Menger form is further partitioned:

[

0 1T

1 −D

]

=

[

0 11 0

] [

1T

−D1,2:N

]

[1 −D2:N,1 ] −D2:N,2:N

(1085)

Matrix D∈ SNh is an EDM if and only if the Schur complement of

[

0 11 0

]

(§A.4) in this partition is positive semidefinite; [17, §1] [213, §3] id est,

5.52Recall: for D∈ SNh , −V T

NDVN º 0 subsumes nonnegativity property 1 (§5.8.1).5.53Empirically, all except one entry of the corresponding eigenvector have the same signwith respect to each other.

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462 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

(confer (1029))

D ∈ EDMN

−D2:N,2:N − [1 −D2:N,1 ]

[

0 11 0

] [

1T

−D1,2:N

]

= −2V TNDVN º 0

and

D ∈ SNh

(1086)

Positive semidefiniteness of that Schur complement insures nonnegativity(D ∈ RN×N

+ , §5.8.1), whereas complementary inertia (1491) insures existenceof that lone negative eigenvalue of the Cayley-Menger form.

Now we apply results from chapter 2 with regard to polyhedral cones andtheir duals.

5.11.2.2 Ordered eigenspectra

Conditions (1083) specify eigenvalue membership to the smallest pointed

polyhedral spectral cone for

[

0 1T

1 −EDMN

]

:

Kλ , ζ∈ RN+1 | ζ1 ≥ ζ2 ≥ · · · ≥ ζN ≥ 0 ≥ ζN+1 , 1Tζ = 0

= KM ∩[

RN+

R−

]

∩ ∂H

= λ

([

0 1T

1 −EDMN

])

(1087)

where∂H , ζ∈ RN+1 | 1Tζ = 0 (1088)

is a hyperplane through the origin, and KM is the monotone cone(§2.13.9.4.3, implying ordered eigenspectra) which is full-dimensional but isnot pointed;

KM = ζ∈ RN+1 | ζ1 ≥ ζ2 ≥ · · · ≥ ζN+1 (414)

K∗M = [ e1− e2 e2− e3 · · · eN− eN+1 ] a | a º 0 ⊂ RN+1 (415)

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5.11. EDM INDEFINITENESS 463

So because of the hyperplane,

dim aff Kλ = dim ∂H = N (1089)

indicating Kλ is not full-dimensional. Defining

A ,

eT1 − eT

2

eT2 − eT

3...

eTN − eT

N+1

∈ RN×N+1 , B ,

eT1

eT2...

eTN

−eTN+1

∈ RN+1×N+1

(1090)we have the halfspace-description:

Kλ = ζ∈ RN+1 | Aζ º 0 , Bζ º 0 , 1Tζ = 0 (1091)

From this and (422) we get a vertex-description for a pointed spectral conethat is not full-dimensional:

Kλ =

VN

([

A

B

]

VN

)†b | b º 0

(1092)

where VN ∈ RN+1×N , and where [sic]

B = eTN ∈ R1×N+1 (1093)

and

A =

eT1 − eT

2

eT2 − eT

3...

eTN−1− eT

N

∈ RN−1×N+1 (1094)

hold those rows of A and B corresponding to conically independent rows

(§2.10) in

[

AB

]

VN .

Conditions (1083) can be equivalently restated in terms of a spectral conefor Euclidean distance matrices:

D ∈ EDMN ⇔

λ

([

0 1T

1 −D

])

∈ KM ∩[

RN+

R−

]

∩ ∂H

D ∈ SNh

(1095)

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464 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Vertex-description of the dual spectral cone is, (292)

K∗λ = K∗

M +

[

RN+

R−

]∗+ ∂H∗ ⊆ RN+1

= [

AT BT 1 −1]

b | b º 0

= [

AT BT 1 −1]

a | a º 0

(1096)From (1092) and (423) we get a halfspace-description:

K∗λ = y ∈ RN+1 | (V T

N [ AT BT ])†V TN y º 0 (1097)

This polyhedral dual spectral cone K∗λ is closed, convex, full-dimensional

because Kλ is pointed, but is not pointed because Kλ is not full-dimensional.

5.11.2.3 Unordered eigenspectra

Spectral cones are not unique; eigenspectra ordering can be rendered benignwithin a cone by presorting a vector of eigenvalues into nonincreasingorder.5.54 Then things simplify: Conditions (1083) now specify eigenvaluemembership to the spectral cone

λ

([

0 1T

1 −EDMN

])

=

[

RN+

R−

]

∩ ∂H

= ζ∈ RN+1 | Bζ º 0 , 1Tζ = 0(1098)

where B is defined in (1090), and ∂H in (1088). From (422) we get avertex-description for a pointed spectral cone not full-dimensional:

λ

([

0 1T

1 −EDMN

])

=

VN (BVN )† b | b º 0

=

[

I−1T

]

b | b º 0

(1099)

where VN ∈ RN+1×N and

B ,

eT1

eT2...

eTN

∈ RN×N+1 (1100)

5.54Eigenspectra ordering (represented by a cone having monotone description such as(1087)) becomes benign in (1308), for example, where projection of a given presorted vectoron the nonnegative orthant in a subspace is equivalent to its projection on the monotonenonnegative cone in that same subspace; equivalence is a consequence of presorting.

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5.11. EDM INDEFINITENESS 465

holds only those rows of B corresponding to conically independent rowsin BVN .

For presorted eigenvalues, (1083) can be equivalently restated

D ∈ EDMN ⇔

λ

([

0 1T

1 −D

])

∈[

RN+

R−

]

∩ ∂H

D ∈ SNh

(1101)

Vertex-description of the dual spectral cone is, (292)

λ

([

0 1T

1 −EDMN

])∗=

[

RN+

R−

]

+ ∂H∗ ⊆ RN+1

= [

BT 1 −1]

b | b º 0

=[

BT 1 −1]

a | a º 0

(1102)From (423) we get a halfspace-description:

λ

([

0 1T

1 −EDMN

])∗= y ∈ RN+1 | (V T

N BT)†V TN y º 0

= y ∈ RN+1 | [ I −1 ] y º 0(1103)

This polyhedral dual spectral cone is closed, convex, full-dimensional but isnot pointed. (Notice that any nonincreasingly ordered eigenspectrum belongsto this dual spectral cone.)

5.11.2.4 Dual cone versus dual spectral cone

An open question regards the relationship of convex cones and their duals tothe corresponding spectral cones and their duals. A positive semidefinitecone, for example, is selfdual. Both the nonnegative orthant and themonotone nonnegative cone are spectral cones for it. When we considerthe nonnegative orthant, then that spectral cone for the selfdual positivesemidefinite cone is also selfdual.

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466 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.12 List reconstruction

The term metric multidimensional scaling 5.55 [253] [103] [350] [101] [256][87] refers to any reconstruction of a list X∈ Rn×N in Euclidean spacefrom interpoint distance information, possibly incomplete (§6.4), ordinal(§5.13.2), or specified perhaps only by bounding-constraints (§5.4.2.3.9)[348]. Techniques for reconstruction are essentially methods for optimallyembedding an unknown list of points, corresponding to given Euclideandistance data, in an affine subset of desired or minimum dimension.The oldest known precursor is called principal component analysis [161]which analyzes the correlation matrix (§5.9.1.0.1); [51, §22] a.k.a,

Karhunen-Loeve transform in the digital signal processing literature.Isometric reconstruction (§5.5.3) of point list X is best performed by

eigenvalue decomposition of a Gram matrix; for then, numerical errors offactorization are easily spotted in the eigenvalues: Now we consider howrotation/reflection and translation invariance factor into a reconstruction.

5.12.1 x1 at the origin. VN

At the stage of reconstruction, we have D∈ EDMN and wish to find agenerating list (§2.3.2) for polyhedron P − α by factoring Gram matrix−V T

NDVN (884) as in (1066). One way to factor −V TNDVN is via

diagonalization of symmetric matrices; [328, §5.6] [199] (§A.5.1, §A.3)

−V TNDVN , QΛQT (1104)

QΛQT º 0 ⇔ Λ º 0 (1105)

where Q∈ RN−1×N−1 is an orthogonal matrix containing eigenvectorswhile Λ∈ SN−1 is a diagonal matrix containing corresponding nonnegativeeigenvalues ordered by nonincreasing value. From the diagonalization,identify the list using (1011);

−V TNDVN = 2V T

NXTXVN , Q√

Λ QTpQp

√Λ QT (1106)

5.55Scaling [346] means making a scale, i.e., a numerical representation of qualitative data.If the scale is multidimensional, it’s multidimensional scaling. −Jan de LeeuwA goal of multidimensional scaling is to find a low-dimensional representation of listX so that distances between its elements best fit a given set of measured pairwisedissimilarities. When data comprises measurable distances, then reconstruction is termedmetric multidimensional scaling. In one dimension, N coordinates in X define the scale.

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5.12. LIST RECONSTRUCTION 467

where√

Λ QTpQp

√Λ , Λ =

√Λ√

Λ and where Qp ∈ Rn×N−1 is unknown as isits dimension n . Rotation/reflection is accounted for by Qp yet only itsfirst r columns are necessarily orthonormal.5.56 Assuming membership tothe unit simplex y∈S (1063), then point p = X

√2VN y = Qp

√Λ QTy in Rn

belongs to the translated polyhedron

P − x1 (1107)

whose generating list constitutes the columns of (1005)

[

0 X√

2VN]

=[

0 Qp

√Λ QT

]

∈ Rn×N

= [0 x2−x1 x3−x1 · · · xN −x1 ](1108)

The scaled auxiliary matrix VN represents that translation. A simple choicefor Qp has n set to N − 1; id est, Qp = I . Ideally, each member of thegenerating list has at most r nonzero entries; r being, affine dimension

rank V TNDVN = rankQp

√Λ QT = rank Λ = r (1109)

Each member then has at least N−1 − r zeros in its higher-dimensionalcoordinates because r ≤ N−1. (1017) To truncate those zeros, choose nequal to affine dimension which is the smallest n possible because XVN hasrank r ≤ n (1013).5.57 In that case, the simplest choice for Qp is [ I 0 ]having dimensions r×N−1.

We may wish to verify the list (1108) found from the diagonalization of−V T

NDVN . Because of rotation/reflection and translation invariance (§5.5),EDM D can be uniquely made from that list by calculating: (867)

5.56Recall r signifies affine dimension. Qp is not necessarily an orthogonal matrix. Qp isconstrained such that only its first r columns are necessarily orthonormal because thereare only r nonzero eigenvalues in Λ when −V T

NDVN has rank r (§5.7.1.1). Remainingcolumns of Qp are arbitrary.

5.57 If we write QT =

qT1...

qTN−1

as rowwise eigenvectors, Λ =

λ1 0. . .

λr0 . . .

0 0

in terms of eigenvalues, and Qp =[

qp1· · · qpN−1

]

as column vectors, then

Qp

√Λ QT =

r∑

i=1

λi qpiqTi is a sum of r linearly independent rank-one matrices (§B.1.1).

Hence the summation has rank r .

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468 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

D(X) = D(X[0√

2VN ]) = D(Qp[0√

Λ QT ]) = D([0√

Λ QT ]) (1110)

This suggests a way to find EDM D given −V TNDVN (confer (989))

D =

[

0δ(

−V TNDVN

)

]

1T+ 1[

0 δ(

−V TNDVN

)T]

− 2

[

0 0T

0 −V TNDVN

]

(985)

5.12.2 0 geometric center. V

Alternatively we may perform reconstruction using auxiliary matrix V(§B.4.1) and Gram matrix −V DV 1

2(888) instead; to find a generating list

for polyhedron

P − αc (1111)

whose geometric center has been translated to the origin. Redimensioningdiagonalization factors Q, Λ∈RN×N and unknown Qp ∈ Rn×N , (1012)

−V DV = 2V XTXV , Q√

Λ QTpQp

√Λ QT , QΛQT (1112)

where the geometrically centered generating list constitutes (confer (1108))

XV = 1√2Qp

√Λ QT ∈ Rn×N

= [ x1− 1N

X1 x2− 1N

X1 x3− 1N

X1 · · · xN − 1N

X1 ](1113)

where αc = 1N

X1. (§5.5.1.0.1) The simplest choice for Qp is [ I 0 ]∈ Rr×N .

Now EDM D can be uniquely made from the list found: (867)

D(X) = D(XV ) = D(1√2

Qp

√Λ QT) = D(

√Λ QT)

1

2(1114)

This EDM is, of course, identical to (1110). Similarly to (985), from −V DVwe can find EDM D (confer (976))

D = δ(−V DV 12)1T + 1δ(−V DV 1

2)T − 2(−V DV 1

2) (975)

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5.12. LIST RECONSTRUCTION 469

(a)

(b)

(c)

(d)

(f) (e)

Figure 132: Map of United States of America showing some state boundariesand the Great Lakes. All plots made by connecting 5020 points. Anydifference in scale in (a) through (d) is artifact of plotting routine.(a) Shows original map made from decimated (latitude, longitude) data.(b) Original map data rotated (freehand) to highlight curvature of Earth.(c) Map isometrically reconstructed from an EDM (from distance only).(d) Same reconstructed map illustrating curvature.(e)(f) Two views of one isotonic reconstruction (from comparative distance);problem (1123) with no sort constraint Π d (and no hidden line removal).

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470 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.13 Reconstruction examples

5.13.1 Isometric reconstruction

5.13.1.0.1 Example. Cartography.The most fundamental application of EDMs is to reconstruct relative pointposition given only interpoint distance information. Drawing a map of theUnited States is a good illustration of isometric reconstruction from completedistance data. We obtained latitude and longitude information for the coast,border, states, and Great Lakes from the usalo atlas data file within Matlab

Mapping Toolbox; conversion to Cartesian coordinates (x, y , z) via:

φ , π/2 − latitude

θ , longitudex = sin(φ) cos(θ)y = sin(φ) sin(θ)z = cos(φ)

(1115)

We used 64% of the available map data to calculate EDM D from N = 5020points. The original (decimated) data and its isometric reconstruction via(1106) are shown in Figure 132a-d. Matlab code is on Wıκımization. Theeigenvalues computed for (1104) are

λ(−V TNDVN ) = [199.8 152.3 2.465 0 0 0 · · · ]T (1116)

The 0 eigenvalues have absolute numerical error on the order of 2E-13 ;meaning, the EDM data indicates three dimensions (r = 3) are required forreconstruction to nearly machine precision. 2

5.13.2 Isotonic reconstruction

Sometimes only comparative information about distance is known (Earth iscloser to the Moon than it is to the Sun). Suppose, for example, EDM D forthree points is unknown:

D = [dij] =

0 d12 d13

d12 0 d23

d13 d23 0

∈ S3

h (856)

but comparative distance data is available:

d13 ≥ d23 ≥ d12 (1117)

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5.13. RECONSTRUCTION EXAMPLES 471

With vectorization d = [d12 d13 d23]T∈ R3, we express the comparative data

as the nonincreasing sorting

Π d =

0 1 00 0 11 0 0

d12

d13

d23

=

d13

d23

d12

∈ KM+ (1118)

where Π is a given permutation matrix expressing known sorting action onthe entries of unknown EDM D , and KM+ is the monotone nonnegativecone (§2.13.9.4.2)

KM+ = z | z1 ≥ z2 ≥ · · · ≥ zN(N−1)/2 ≥ 0 ⊆ RN(N−1)/2+ (407)

where N(N−1)/2 = 3 for the present example. From sorted vectorization(1118) we create the sort-index matrix

O =

0 12 32

12 0 22

32 22 0

∈ S3

h ∩ R3×3

+ (1119)

generally defined

Oij , k2 | dij = (Ξ Π d)k , j 6= i (1120)

where Ξ is a permutation matrix (1691) completely reversing order of vectorentries.

Replacing EDM data with indices-square of a nonincreasing sorting likethis is, of course, a heuristic we invented and may be regarded as a nonlinearintroduction of much noise into the Euclidean distance matrix. For largedata sets, this heuristic makes an otherwise intense problem computationallytractable; we see an example in relaxed problem (1124).

Any process of reconstruction that leaves comparative distanceinformation intact is called ordinal multidimensional scaling or isotonicreconstruction. Beyond rotation, reflection, and translation error, (§5.5)list reconstruction by isotonic reconstruction is subject to error in absolutescale (dilation) and distance ratio. Yet Borg & Groenen argue: [51, §2.2]reconstruction from complete comparative distance information for a largenumber of points is as highly constrained as reconstruction from an EDM;the larger the number, the better.

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472 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

700

800

900

j

λ(−V TN OVN )j

Figure 133: Largest ten eigenvalues, of −V TN OVN for map of USA, sorted by

nonincreasing value.

5.13.2.1 Isotonic cartography

To test Borg & Groenen’s conjecture, suppose we make a complete sort-indexmatrix O∈ SN

h ∩ RN×N+ for the map of the USA and then substitute O in place

of EDM D in the reconstruction process of §5.12. Whereas EDM D returnedonly three significant eigenvalues (1116), the sort-index matrix O is generallynot an EDM (certainly not an EDM with corresponding affine dimension 3)so returns many more. The eigenvalues, calculated with absolute numericalerror approximately 5E-7 , are plotted in Figure 133: (In the code onWıκımization, matrix O is normalized by (N(N−1)/2)2.)

λ(−V TN OVN ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ]T

(1121)

The extra eigenvalues indicate that affine dimension corresponding to anEDM near O is likely to exceed 3. To realize the map, we must simultaneouslyreduce that dimensionality and find an EDM D closest to O in somesense5.58 while maintaining the known comparative distance relationship.For example: given permutation matrix Π expressing the known sortingaction like (1118) on entries

5.58 a problem explored more in §7.

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5.13. RECONSTRUCTION EXAMPLES 473

d ,1√2

dvec D =

d12

d13

d23

d14

d24

d34...dN−1,N

∈ RN(N−1)/2 (1122)

of unknown D∈ SNh , we can make sort-index matrix O input to the

optimization problem

minimizeD

‖−V TN (D − O)VN‖F

subject to rank V TNDVN ≤ 3

Π d ∈ KM+

D ∈ EDMN

(1123)

that finds the EDM D (corresponding to affine dimension not exceeding 3 inisomorphic dvec EDMN∩ ΠTKM+) closest to O in the sense of Schoenberg(886).

Analytical solution to this problem, ignoring the sort constraintΠ d ∈KM+ , is known [350]: we get the convex optimization [sic] (§7.1)

minimizeD

‖−V TN (D − O)VN‖F

subject to rank V TNDVN ≤ 3

D ∈ EDMN

(1124)

Only the three largest nonnegative eigenvalues in (1121) need be retained tomake list (1108); the rest are discarded. The reconstruction from EDM Dfound in this manner is plotted in Figure 132e-f. Matlab code is onWıκımization. From these plots it becomes obvious that inclusion of thesort constraint is necessary for isotonic reconstruction.

That sort constraint demands: any optimal solution D⋆ must possess theknown comparative distance relationship that produces the original ordinaldistance data O (1120). Ignoring the sort constraint, apparently, violates it.Yet even more remarkable is how much the map reconstructed using onlyordinal data still resembles the original map of the USA after suffering themany violations produced by solving relaxed problem (1124). This suggeststhe simple reconstruction techniques of §5.12 are robust to a significantamount of noise.

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474 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.13.2.2 Isotonic solution with sort constraint

Because problems involving rank are generally difficult, we will partition(1123) into two problems we know how to solve and then alternate theirsolution until convergence:

minimizeD

‖−V TN (D − O)VN‖F

subject to rank V TNDVN ≤ 3

D ∈ EDMN

(a)

minimizeσ

‖σ − Π d‖subject to σ ∈ KM+

(b)

(1125)

where sort-index matrix O (a given constant in (a)) becomes an implicitvector variable o i solving the ith instance of (1125b)

1√2

dvec Oi = o i , ΠTσ⋆ ∈ RN(N−1)/2 , i∈1, 2, 3 . . . (1126)

As mentioned in discussion of relaxed problem (1124), a closed-formsolution to problem (1125a) exists. Only the first iteration of (1125a)sees the original sort-index matrix O whose entries are nonnegative wholenumbers; id est, O0 =O∈ SN

h ∩ RN×N+ (1120). Subsequent iterations i take

the previous solution of (1125b) as input

Oi = dvec−1(√

2 o i ) ∈ SN (1127)

real successors, estimating distance-square not order, to the sort-indexmatrix O .

New convex problem (1125b) finds the unique minimum-distanceprojection of Π d on the monotone nonnegative cone KM+ . By defining

Y †T = [e1− e2 e2− e3 e3− e4 · · · em] ∈ Rm×m (408)

where m,N(N−1)/2, we may rewrite (1125b) as an equivalent quadraticprogram; a convex problem in terms of the halfspace-description of KM+ :

minimizeσ

(σ − Π d)T(σ − Π d)

subject to Y †σ º 0(1128)

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5.13. RECONSTRUCTION EXAMPLES 475

This quadratic program can be converted to a semidefinite program viaSchur-form (§3.6.2); we get the equivalent problem

minimizet∈R , σ

t

subject to

[

tI σ − Π d(σ − Π d)T 1

]

º 0

Y †σ º 0

(1129)

5.13.2.3 Convergence

In §E.10 we discuss convergence of alternating projection on intersectingconvex sets in a Euclidean vector space; convergence to a point in theirintersection. Here the situation is different for two reasons:

Firstly, sets of positive semidefinite matrices having an upper bound onrank are generally not convex. Yet in §7.1.4.0.1 we prove (1125a) is equivalentto a projection of nonincreasingly ordered eigenvalues on a subset of thenonnegative orthant:

minimizeD

‖−V TN (D − O)VN‖F

subject to rank V TNDVN ≤ 3

D ∈ EDMN

≡minimize

Υ‖Υ − Λ‖F

subject to δ(Υ)∈[

R3

+

0

]

(1130)

where −V TNDVN ,UΥUT∈ SN−1 and −V T

N OVN ,QΛQT∈ SN−1 areordered diagonalizations (§A.5). It so happens: optimal orthogonal U⋆

always equals Q given. Linear operator T (A) = U⋆TAU⋆, acting on squarematrix A , is a bijective isometry because Frobenius’ norm is orthogonallyinvariant (48). This isometric isomorphism T thus maps a nonconvexproblem to a convex one that preserves distance.

Secondly, the second half (1125b) of the alternation takes place in adifferent vector space; SN

h (versus SN−1). From §5.6 we know these twovector spaces are related by an isomorphism, SN−1 =VN (SN

h ) (994), but notby an isometry.

We have, therefore, no guarantee from theory of alternating projectionthat the alternation (1125) converges to a point, in the set of allEDMs corresponding to affine dimension not in excess of 3, belonging todvec EDMN∩ ΠTKM+ .

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5.13.2.4 Interlude

Map reconstruction from comparative distance data, isotonic reconstruction,would also prove invaluable to stellar cartography where absolute interstellardistance is difficult to acquire. But we have not yet implemented the secondhalf (1128) of alternation (1125) for USA map data because memory-demandsexceed capability of our computer.

5.13.2.4.1 Exercise. Convergence of isotonic solution by alternation.Empirically demonstrate convergence, discussed in §5.13.2.3, on a smallerdata set. H

It would be remiss not to mention another method of solution to thisisotonic reconstruction problem: Once again we assume only comparativedistance data like (1117) is available. Given known set of indices I

minimizeD

rank V DV

subject to dij ≤ dkl ≤ dmn ∀(i, j, k , l ,m, n)∈ ID ∈ EDMN

(1131)

this problem minimizes affine dimension while finding an EDM whoseentries satisfy known comparative relationships. Suitable rank heuristicsare discussed in §4.4.1 and §7.2.2 that will transform this to a convexoptimization problem.

Using contemporary computers, even with a rank heuristic in place of theobjective function, this problem formulation is more difficult to compute thanthe relaxed counterpart problem (1124). That is because there exist efficientalgorithms to compute a selected few eigenvalues and eigenvectors from avery large matrix. Regardless, it is important to recognize: the optimalsolution set for this problem (1131) is practically always different from theoptimal solution set for its counterpart, problem (1123).

5.14 Fifth property of Euclidean metric

We continue now with the question raised in §5.3 regarding the necessityfor at least one requirement more than the four properties of the Euclideanmetric (§5.2) to certify realizability of a bounded convex polyhedron or to

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5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 477

reconstruct a generating list for it from incomplete distance information.There we saw that the four Euclidean metric properties are necessary forD∈ EDMN in the case N = 3, but become insufficient when cardinality Nexceeds 3 (regardless of affine dimension).

5.14.1 Recapitulate

In the particular case N = 3, −V TNDVN º 0 (1035) and D∈ S3

h are necessaryand sufficient conditions for D to be an EDM. By (1037), triangle inequality isthen the only Euclidean condition bounding the necessarily nonnegative dij ;and those bounds are tight. That means the first four properties of theEuclidean metric are necessary and sufficient conditions for D to be an EDMin the case N = 3 ; for i, j∈1, 2, 3

dij ≥ 0 , i 6= j√

dij = 0 , i = j√

dij =√

dji√

dij ≤√

dik +√

dkj , i 6=j 6=k

⇔ −V TNDVN º 0

D ∈ S3

h

⇔ D ∈ EDM3

(1132)

Yet those four properties become insufficient when N > 3.

5.14.2 Derivation of the fifth

Correspondence between the triangle inequality and the EDM was developedin §5.8.2 where a triangle inequality (1037a) was revealed within theleading principal 2×2 submatrix of −V T

NDVN when positive semidefinite.Our choice of the leading principal submatrix was arbitrary; actually,a unique triangle inequality like (932) corresponds to any one of the(N−1)!/(2!(N−1 − 2)!) principal 2×2 submatrices.5.59 Assuming D∈ S4

h

and −V TNDVN ∈ S3, then by the positive (semi)definite principal submatrices

theorem (§A.3.1.0.4) it is sufficient to prove all dij are nonnegative, alltriangle inequalities are satisfied, and det(−V T

NDVN ) is nonnegative. WhenN = 4, in other words, that nonnegative determinant becomes the fifth andlast Euclidean metric requirement for D∈ EDMN . We now endeavor toascribe geometric meaning to it.

5.59There are fewer principal 2×2 submatrices in −V TNDVN than there are triangles made

by four or more points because there are N !/(3!(N− 3)!) triangles made by point triples.The triangles corresponding to those submatrices all have vertex x1 . (confer §5.8.2.1)

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478 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.14.2.1 Nonnegative determinant

By (938) when D∈EDM4, −V TNDVN is equal to inner product (933),

ΘTΘ =

d12

d12d13 cos θ213

d12d14 cos θ214√

d12d13 cos θ213 d13

d13d14 cos θ314√

d12d14 cos θ214

d13d14 cos θ314 d14

(1133)

Because Euclidean space is an inner-product space, the more conciseinner-product form of the determinant is admitted;

det(ΘTΘ) = −d12d13d14

(

cos(θ213)2+cos(θ214)

2+cos(θ314)2 − 2 cos θ213 cos θ214 cos θ314 − 1

)

(1134)The determinant is nonnegative if and only if

cos θ214 cos θ314 −√

sin(θ214)2 sin(θ314)2 ≤ cos θ213 ≤ cos θ214 cos θ314 +√

sin(θ214)2 sin(θ314)2

⇔cos θ213 cos θ314 −

sin(θ213)2 sin(θ314)2 ≤ cos θ214 ≤ cos θ213 cos θ314 +√

sin(θ213)2 sin(θ314)2

⇔cos θ213 cos θ214 −

sin(θ213)2 sin(θ214)2 ≤ cos θ314 ≤ cos θ213 cos θ214 +√

sin(θ213)2 sin(θ214)2

(1135)

which simplifies, for 0 ≤ θi1ℓ , θℓ1j , θi1j ≤ π and all i 6=j 6=ℓ∈2, 3, 4 , to

cos(θi1ℓ + θℓ1j) ≤ cos θi1j ≤ cos(θi1ℓ − θℓ1j) (1136)

Analogously to triangle inequality (1049), the determinant is 0 upon equalityon either side of (1136) which is tight. Inequality (1136) can be equivalentlywritten linearly as a triangle inequality between relative angles [390, §1.4];

|θi1ℓ − θℓ1j| ≤ θi1j ≤ θi1ℓ + θℓ1j

θi1ℓ + θℓ1j + θi1j ≤ 2π

0 ≤ θi1ℓ , θℓ1j , θi1j ≤ π

(1137)

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5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 479

θ213

-θ214

-θ314

π

Figure 134: The relative-angle inequality tetrahedron (1138) bounding EDM4

is regular; drawn in entirety. Each angle θ (930) must belong to this solidto be realizable.

Generalizing this:

5.14.2.1.1 Fifth property of Euclidean metric - restatement.Relative-angle inequality. (confer §5.3.1.0.1) [47] [48, p.17, p.107] [231, §3.1]Augmenting the four fundamental Euclidean metric properties in Rn, for alli, j, ℓ 6= k∈1 . . . N , i<j<ℓ , and for N ≥ 4 distinct points xk , theinequalities

|θikℓ − θℓkj| ≤ θikj ≤ θikℓ + θℓkj (a)

θikℓ + θℓkj + θikj ≤ 2π (b)

0 ≤ θikℓ , θℓkj , θikj ≤ π (c)

(1138)

where θikj = θjki is the angle between vectors at vertex xk (as defined in(930) and illustrated in Figure 115), must be satisfied at each point xk

regardless of affine dimension. ⋄

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480 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

Because point labelling is arbitrary, this fifth Euclidean metricrequirement must apply to each of the N points as though each were in turnlabelled x1 ; hence the new index k in (1138). Just as the triangle inequalityis the ultimate test for realizability of only three points, the relative-angleinequality is the ultimate test for only four. For four distinct points, thetriangle inequality remains a necessary although penultimate test; (§5.4.3)

Four Euclidean metric properties (§5.2).Angle θ inequality (861) or (1138).

⇔ −V TNDVN º 0

D ∈ S4

h

⇔ D = D(Θ) ∈ EDM4

(1139)

The relative-angle inequality, for this case, is illustrated in Figure 134.

5.14.2.2 Beyond the fifth metric property

When cardinality N exceeds 4, the first four properties of the Euclideanmetric and the relative-angle inequality together become insufficientconditions for realizability. In other words, the four Euclidean metricproperties and relative-angle inequality remain necessary but become asufficient test only for positive semidefiniteness of all the principal 3 × 3submatrices [sic] in −V T

NDVN . Relative-angle inequality can be consideredthe ultimate test only for realizability at each vertex xk of each and everypurported tetrahedron constituting a hyperdimensional body.

When N = 5 in particular, relative-angle inequality becomes thepenultimate Euclidean metric requirement while nonnegativity of thenunwieldy det(ΘTΘ) corresponds (by the positive (semi)definite principalsubmatrices theorem in §A.3.1.0.4) to the sixth and last Euclidean metricrequirement. Together these six tests become necessary and sufficient, andso on.

Yet for all values of N , only assuming nonnegative dij , relative-anglematrix inequality in (1051) is necessary and sufficient to certify realizability;(§5.4.3.1)

Euclidean metric property 1 (§5.2).Angle matrix inequality Ω º 0 (939).

⇔ −V TNDVN º 0

D ∈ SNh

⇔ D = D(Ω , d) ∈ EDMN

(1140)

Like matrix criteria (862), (886), and (1051), the relative-angle matrixinequality and nonnegativity property subsume all the Euclidean metricproperties and further requirements.

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5.14.3 Path not followed

As a means to test for realizability of four or more points, an intuitivelyappealing way to augment the four Euclidean metric properties is torecognize generalizations of the triangle inequality: In the case ofcardinality N = 4, the three-dimensional analogue to triangle & distance istetrahedron & facet-area, while in case N = 5 the four-dimensional analogueis polychoron & facet-volume, ad infinitum. For N points, N + 1 metricproperties are required.

5.14.3.1 N = 4

Each of the four facets of a general tetrahedron is a triangle and its relativeinterior. Suppose we identify each facet of the tetrahedron by its area-square:c1 , c2 , c3 , c4 . Then analogous to metric property 4, we may write a tight5.60

area inequality for the facets

√ci ≤ √

cj +√

ck +√

cℓ , i 6=j 6=k 6=ℓ∈1, 2, 3, 4 (1141)

which is a generalized “triangle” inequality [223, §1.1] that follows from

√ci =

√cj cos ϕij +

√ck cos ϕik +

√cℓ cos ϕiℓ (1142)

[238] [370, Law of Cosines ] where ϕij is the dihedral angle at the commonedge between triangular facets i and j .

If D is the EDM corresponding to the whole tetrahedron, then area-squareof the ith triangular facet has a convenient formula in terms of Di∈EDMN−1

the EDM corresponding to that particular facet: From the Cayley-Mengerdeterminant5.61 for simplices, [370] [128] [158, §4] [86, §3.3] the ith facetarea-square for i∈1 . . . N is (§A.4.1)

5.60The upper bound is met when all angles in (1142) are simultaneously 0 ; that occurs,for example, if one point is relatively interior to the convex hull of the three remaining.5.61 whose foremost characteristic is: the determinant vanishes if and only if affine

dimension does not equal penultimate cardinality; id est, det

[

0 1T

1 −D

]

= 0 ⇔ r < N−1

where D is any EDM (§5.7.3.0.1). Otherwise, the determinant is negative.

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482 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

ci =−1

2N−2(N−2)!2det

[

0 1T

1 −Di

]

(1143)

=(−1)N

2N−2(N−2)!2det Di

(

1TD−1i 1

)

(1144)

=(−1)N

2N−2(N−2)!21Tcof(Di)

T1 (1145)

where Di is the ith principal N−1×N−1 submatrix5.62 of D∈ EDMN ,and cof(Di) is the N−1×N−1 matrix of cofactors [328, §4] correspondingto Di . The number of principal 3 × 3 submatrices in D is, of course, equalto the number of triangular facets in the tetrahedron; four (N !/(3!(N−3)!))when N = 4.

5.14.3.1.1 Exercise. Sufficiency conditions for an EDM of four points.Triangle inequality (property 4) and area inequality (1141) are conditionsnecessary for D to be an EDM. Prove their sufficiency in conjunction withthe remaining three Euclidean metric properties. H

5.14.3.2 N = 5

Moving to the next level, we might encounter a Euclidean body calledpolychoron: a bounded polyhedron in four dimensions.5.63 Our polychoronhas five (N !/(4!(N−4)!)) facets, each of them a general tetrahedron whosevolume-square ci is calculated using the same formula; (1143) whereD is the EDM corresponding to the polychoron, and Di is the EDMcorresponding to the ith facet (the principal 4 × 4 submatrix of D∈ EDMN

corresponding to the ith tetrahedron). The analogue to triangle & distanceis now polychoron & facet-volume. We could then write another generalized“triangle” inequality like (1141) but in terms of facet volume; [376, §IV]

√ci ≤ √

cj +√

ck +√

cℓ +√

cm , i 6=j 6=k 6=ℓ 6=m∈1 . . . 5 (1146)

5.62Every principal submatrix of an EDM remains an EDM. [231, §4.1]5.63The simplest polychoron is called a pentatope [370]; a regular simplex hence convex.(A pentahedron is a three-dimensional body having five vertices.)

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5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 483

.

Figure 135: Length of one-dimensional face a equals height h=a=1 of thisconvex nonsimplicial pyramid in R3 with square base inscribed in a circle ofradius R centered at the origin. [370, Pyramid ]

5.14.3.2.1 Exercise. Sufficiency for an EDM of five points.For N = 5, triangle (distance) inequality (§5.2), area inequality (1141), andvolume inequality (1146) are conditions necessary for D to be an EDM. Provetheir sufficiency. H

5.14.3.3 Volume of simplices

There is no known formula for the volume of a bounded general convexpolyhedron expressed either by halfspace or vertex-description. [388, §2.1][278, p.173] [229] [169] [170] Volume is a concept germane to R3 ; in higherdimensions it is called content. Applying the EDM assertion (§5.9.1.0.4)and a result from [59, p.407], a general nonempty simplex (§2.12.3) in RN−1

corresponding to an EDM D∈ SNh has content

√c = content(S)

det(−V TNDVN ) (1147)

where content-square of the unit simplex S⊂ RN−1 is proportional to itsCayley-Menger determinant;

content(S)2 =−1

2N−1(N−1)!2det

[

0 1T

1 −D([0 e1 e2 · · · eN−1 ])

]

(1148)

where ei∈ RN−1 and the EDM operator used is D(X) (867).

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484 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

5.14.3.3.1 Example. Pyramid.A formula for volume of a pyramid is known;5.64 it is 1

3the product of its

base area with its height. [219] The pyramid in Figure 135 has volume 13.

To find its volume using EDMs, we must first decompose the pyramid intosimplicial parts. Slicing it in half along the plane containing the line segmentscorresponding to radius R and height h we find the vertices of one simplex,

X =

1/2 1/2 −1/2 01/2 −1/2 −1/2 00 0 0 1

∈ Rn×N (1149)

where N = n + 1 for any nonempty simplex in Rn. The volume ofthis simplex is half that of the entire pyramid; id est,

√c = 1

6found by

evaluating (1147). 2

With that, we conclude digression of path.

5.14.4 Affine dimension reduction in three dimensions

(confer §5.8.4) The determinant of any M×M matrix is equal to the productof its M eigenvalues. [328, §5.1] When N = 4 and det(ΘTΘ) is 0, thatmeans one or more eigenvalues of ΘTΘ∈R3×3 are 0. The determinant will goto 0 whenever equality is attained on either side of (861), (1138a), or (1138b),meaning that a tetrahedron has collapsed to a lower affine dimension; id est,r = rank ΘTΘ = rank Θ is reduced below N−1 exactly by the number of0 eigenvalues (§5.7.1.1).

In solving completion problems of any size N where one or more entriesof an EDM are unknown, therefore, dimension r of the affine hull requiredto contain the unknown points is potentially reduced by selecting distancesto attain equality in (861) or (1138a) or (1138b).

5.64Pyramid volume is independent of the paramount vertex position as long as its heightremains constant.

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5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 485

5.14.4.1 Exemplum redux

We now apply the fifth Euclidean metric property to an earlier problem:

5.14.4.1.1 Example. Small completion problem, IV. (confer §5.9.3.0.1)Returning again to Example 5.3.0.0.2 that pertains to Figure 114 whereN =4, distance-square d14 is ascertainable from the fifth Euclidean metricproperty. Because all distances in (859) are known except

√d14 , then

cos θ123 =0 and θ324 =0 result from identity (930). Applying (861),

cos(θ123 + θ324) ≤ cos θ124 ≤ cos(θ123 − θ324)0 ≤ cos θ124 ≤ 0

(1150)

It follows again from (930) that d14 can only be 2. As explained in thissubsection, affine dimension r cannot exceed N−2 because equality isattained in (1150). 2

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486 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX

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Chapter 6

Cone of distance matrices

For N > 3, the cone of EDMs is no longer a circular cone andthe geometry becomes complicated. . .

−Hayden, Wells, Liu, & Tarazaga (1991) [181, §3]

In the subspace of symmetric matrices SN , we know that the convex cone ofEuclidean distance matrices EDMN (the EDM cone) does not intersect thepositive semidefinite cone SN

+ (PSD cone) except at the origin, their onlyvertex; there can be no positive or negative semidefinite EDM. (1075) [231]

EDMN ∩ SN+ = 0 (1151)

Even so, the two convex cones can be related. In §6.8.1 we prove the equality

EDMN = SNh ∩

(

SN⊥c − SN

+

)

(1244)

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

487

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488 CHAPTER 6. CONE OF DISTANCE MATRICES

a resemblance to EDM definition (867) where

SNh =

A ∈ SN | δ(A) = 0

(66)

is the symmetric hollow subspace (§2.2.3) and where

SN⊥c = u1T + 1uT | u∈RN (1962)

is the orthogonal complement of the geometric center subspace (§E.7.2.0.2)

SNc = Y ∈ SN | Y 1 = 0 (1960)

6.0.1 gravity

Equality (1244) is equally important as the known isomorphisms (983) (984)(995) (996) relating the EDM cone EDMN to positive semidefinite cone SN−1

+

(§5.6.2.1) or to an N(N−1)/2-dimensional face of SN+ (§5.6.1.1).6.1 But

those isomorphisms have never led to this equality relating whole conesEDMN and SN

+ .

Equality (1244) is not obvious from the various EDM definitions such as(867) or (1170) because inclusion must be proved algebraically in order toestablish equality; EDMN ⊇ SN

h ∩ (SN⊥c − SN

+ ). We will instead prove (1244)using purely geometric methods.

6.0.2 highlight

In §6.8.1.7 we show: the Schoenberg criterion for discriminating Euclideandistance matrices

D ∈ EDMN ⇔

−V TNDVN ∈ SN−1

+

D ∈ SNh

(886)

is a discretized membership relation (§2.13.4, dual generalized inequalities)between the EDM cone and its ordinary dual.

6.1Because both positive semidefinite cones are frequently in play, dimension is explicit.

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6.1. DEFINING EDM CONE 489

6.1 Defining EDM cone

We invoke a popular matrix criterion to illustrate correspondence between theEDM and PSD cones belonging to the ambient space of symmetric matrices:

D ∈ EDMN ⇔

−V DV ∈ SN+

D ∈ SNh

(890)

where V ∈ SN is the geometric centering matrix (§B.4). The set of all EDMsof dimension N×N forms a closed convex cone EDMN because any pair ofEDMs satisfies the definition of a convex cone (166); videlicet, for each andevery ζ1 , ζ2 ≥ 0 (§A.3.1.0.2)

ζ1 V D1V + ζ2 V D2V º 0ζ1 D1 + ζ2 D2 ∈ SN

h

⇐ V D1V º 0 , V D2V º 0D1 ∈ SN

h , D2 ∈ SNh

(1152)

and convex cones are invariant to inverse linear transformation [303, p.22].

6.1.0.0.1 Definition. Cone of Euclidean distance matrices.In the subspace of symmetric matrices, the set of all Euclidean distancematrices forms a unique immutable pointed closed convex cone called theEDM cone: for N > 0

EDMN ,

D ∈ SNh | −V DV ∈ SN

+

=⋂

z∈N (1T)

D ∈ SN | 〈zzT,−D〉≥ 0 , δ(D)=0 (1153)

The EDM cone in isomorphic RN(N+1)/2 [sic] is the intersection of an infinitenumber (when N >2) of halfspaces about the origin and a finite numberof hyperplanes through the origin in vectorized variable D = [dij] . HenceEDMN is not full-dimensional with respect to SN because it is confined to thesymmetric hollow subspace SN

h . The EDM cone relative interior comprises

rel int EDMN =⋂

z∈N (1T)

D ∈ SN | 〈zzT,−D〉> 0 , δ(D)=0

=

D ∈ EDMN | rank(V DV ) = N−1

(1154)

while its relative boundary comprises

rel ∂EDMN =

D ∈ EDMN | 〈zzT,−D〉 = 0 for some z∈N (1T)

=

D ∈ EDMN | rank(V DV ) < N−1

(1155)

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490 CHAPTER 6. CONE OF DISTANCE MATRICES

0

0.2

0.4

0.6

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0

0.2

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0

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1

0

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0.8

(a) (d)

(c)(b)

d12d13

d23

√d12

√d13

√d23

d12d13

d23

dvec rel ∂EDM3

Figure 136: Relative boundary (tiled) of EDM cone EDM3 drawn truncatedin isometrically isomorphic subspace R3. (a) EDM cone drawn in usualdistance-square coordinates dij . View is from interior toward origin. Unlikepositive semidefinite cone, EDM cone is not selfdual; neither is it properin ambient symmetric subspace (dual EDM cone for this example belongsto isomorphic R6). (b) Drawn in its natural coordinates

dij (absolutedistance), cone remains convex (confer §5.10); intersection of three halfspaces(1038) whose partial boundaries each contain origin. Cone geometry becomesnonconvex (nonpolyhedral) in higher dimension. (§6.3) (c) Two coordinatesystems artificially superimposed. Coordinate transformation from dij to√

dij appears a topological contraction. (d) Sitting on its vertex 0, pointedEDM3 is a circular cone having axis of revolution dvec(−E)= dvec(11T− I )(1071) (73). (Rounded vertex is plot artifact.)

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6.2. POLYHEDRAL BOUNDS 491

This cone is more easily visualized in the isomorphic vector subspaceRN(N−1)/2 corresponding to SN

h :

In the case N = 1 point, the EDM cone is the origin in R0.

In the case N = 2, the EDM cone is the nonnegative real line in R ; ahalfline in a subspace of the realization in Figure 144.

The EDM cone in the case N = 3 is a circular cone in R3 illustrated inFigure 136(a)(d); rather, the set of all matrices

D =

0 d12 d13

d12 0 d23

d13 d23 0

∈ EDM3 (1156)

makes a circular cone in this dimension. In this case, the first four Euclideanmetric properties are necessary and sufficient tests to certify realizabilityof triangles; (1132). Thus triangle inequality property 4 describes threehalfspaces (1038) whose intersection makes a polyhedral cone in R3 ofrealizable

dij (absolute distance); an isomorphic subspace representationof the set of all EDMs D in the natural coordinates

D ,

0√

d12

√d13√

d12 0√

d23√d13

√d23 0

(1157)

illustrated in Figure 136b.

6.2 Polyhedral bounds

The convex cone of EDMs is nonpolyhedral in dij for N > 2 ; e.g.,Figure 136a. Still we found necessary and sufficient bounding polyhedralrelations consistent with EDM cones for cardinality N = 1, 2, 3, 4:

N = 3. Transforming distance-square coordinates dij by taking their positivesquare root provides the polyhedral cone in Figure 136b; polyhedralbecause an intersection of three halfspaces in natural coordinates√

dij is provided by triangle inequalities (1038). This polyhedralcone implicitly encompasses necessary and sufficient metric properties:nonnegativity, self-distance, symmetry, and triangle inequality.

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492 CHAPTER 6. CONE OF DISTANCE MATRICES

∂H

dvec rel ∂EDM3

0

(b) (c)

(a)

Figure 137: (a) In isometrically isomorphic subspace R3, intersection ofEDM3 with hyperplane ∂H representing one fixed symmetric entry d23 =κ(both drawn truncated, rounded vertex is artifact of plot). EDMs in thisdimension corresponding to affine dimension 1 comprise relative boundary ofEDM cone (§6.6). Since intersection illustrated includes a nontrivial subsetof cone’s relative boundary, then it is apparent there exist infinitely manyEDM completions corresponding to affine dimension 1. In this dimension it isimpossible to represent a unique nonzero completion corresponding to affinedimension 1, for example, using a single hyperplane because any hyperplanesupporting relative boundary at a particular point Γ contains an entire rayζΓ | ζ≥0 belonging to rel ∂EDM3 by Lemma 2.8.0.0.1. (b) d13 =κ .(c) d12 =κ .

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EDM CONE IS NOT CONVEX 493

N = 4. Relative-angle inequality (1138) together with four Euclidean metricproperties are necessary and sufficient tests for realizability oftetrahedra. (1139) Albeit relative angles θikj (930) are nonlinearfunctions of the dij , relative-angle inequality provides a regulartetrahedron in R3 [sic] (Figure 134) bounding angles θikj at vertexxk consistently with EDM4 .6.2

Yet were we to employ the procedure outlined in §5.14.3 for makinggeneralized triangle inequalities, then we would find all the necessary andsufficient dij -transformations for generating bounding polyhedra consistentwith EDMs of any higher dimension (N > 3).

6.3√

EDM cone is not convex

For some applications, like a molecular conformation problem (Figure 3,Figure 125) or multidimensional scaling [100] [351], absolute distance

dij

is the preferred variable. Taking square root of the entries in all EDMs Dof dimension N , we get another cone but not a convex cone when N > 3(Figure 136b): [87, §4.5.2]

EDMN , √

D | D∈ EDMN (1158)

where √

D is defined like (1157). It is a cone simply because any coneis completely constituted by rays emanating from the origin: (§2.7) Anygiven ray ζΓ∈ RN(N−1)/2 | ζ≥0 remains a ray under entrywise square root:

ζΓ∈ RN(N−1)/2 | ζ≥0. It is already established that

D∈ EDMN ⇒ √

D ∈ EDMN (1070)

But because of how√

EDMN is defined, it is obvious that (confer §5.10)

D∈ EDMN ⇔ √

D∈√

EDMN (1159)

Were√

EDMN convex, then given √

D1 , √

D2 ∈√

EDMN we would

expect their conic combination √

D1 + √

D2 to be a member of√

EDMN .

6.2Still, property-4 triangle inequalities (1038) corresponding to each principal 3×3submatrix of −V T

NDVN demand that the corresponding√

dij belong to a polyhedralcone like that in Figure 136b.

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494 CHAPTER 6. CONE OF DISTANCE MATRICES

That is easily proven false by counter-example via (1159), for then( √

D1 + √

D2 ) ( √

D1 + √

D2 ) would need to be a member of EDMN .Notwithstanding,

EDMN ⊆ EDMN (1160)

by (1070) (Figure 136), and we learn how to transform a nonconvexproximity problem in the natural coordinates

dij to a convex optimizationin §7.2.1.

6.4 A geometry of completion

It is not known how to proceed if one wishes to restrict thedimension of the Euclidean space in which the configuration ofpoints may be constructed.

−Michael W. Trosset (2000) [349, §1]

Given an incomplete noiseless EDM, intriguing is the question of whether alist in X∈ Rn×N (76) may be reconstructed and under what circumstancesreconstruction is unique. [2] [4] [5] [6] [8] [17] [65] [202] [213] [230] [231] [232]

If one or more entries of a particular EDM are fixed, then geometricinterpretation of the feasible set of completions is the intersection of the EDMcone EDMN in isomorphic subspace RN(N−1)/2 with as many hyperplanes asthere are fixed symmetric entries.6.3 Assuming a nonempty intersection, thenthe number of completions is generally infinite, and those corresponding toparticular affine dimension r<N− 1 belong to some generally nonconvexsubset of that intersection (confer §2.9.2.6.2) that can be as small as a point.

6.4.0.0.1 Example. Maximum variance unfolding. [369]A process minimizing affine dimension (§2.1.5) of certain kinds of Euclideanmanifold by topological transformation can be posed as a completionproblem (confer §E.10.2.1.2). Weinberger & Saul, who originated thetechnique, specify an applicable manifold in three dimensions by analogy toan ordinary sheet of paper (confer §2.1.6); imagine, we find it deformed fromflatness in some way introducing neither holes, tears, or self-intersections.[369, §2.2] The physical process is intuitively described as unfurling,

6.3Depicted in Figure 137a is an intersection of the EDM cone EDM3 with a singlehyperplane representing the set of all EDMs having one fixed symmetric entry.

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6.4. A GEOMETRY OF COMPLETION 495

(a)

(b)

2 nearest neighbors

3 nearest neighbors

Figure 138: Neighborhood graph (dashed) with one dimensionless EDMsubgraph completion (solid) superimposed (but not obscuring dashed). Localview of a few dense samples # from relative interior of some arbitraryEuclidean manifold whose affine dimension appears two-dimensional in thisneighborhood. All line segments measure absolute distance. Dashed linesegments help visually locate nearest neighbors; suggesting, best number ofnearest neighbors can be greater than value of embedding dimension aftertopological transformation. (confer [208, §2]) Solid line segments representcompletion of EDM subgraph from available distance data for an arbitrarilychosen sample and its nearest neighbors. Each distance from EDM subgraphbecomes distance-square in corresponding EDM submatrix.

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496 CHAPTER 6. CONE OF DISTANCE MATRICES

unfolding, diffusing, decompacting, or unraveling. In particular instances,the process is a sort of flattening by stretching until taut (but not bycrushing); e.g., unfurling a three-dimensional Euclidean body resembling abillowy national flag reduces that manifold’s affine dimension to r=2.

Data input to the proposed process originates from distances betweenneighboring relatively dense samples of a given manifold. Figure 138 realizesa densely sampled neighborhood; called, neighborhood graph. Essentially,the algorithmic process preserves local isometry between nearest neighborsallowing distant neighbors to excurse expansively by “maximizing variance”(Figure 5). The common number of nearest neighbors to each sample isa data-dependent algorithmic parameter whose minimum value connectsthe graph. The dimensionless EDM subgraph between each sample andits nearest neighbors is completed from available data and included asinput; one such EDM subgraph completion is drawn superimposed upon theneighborhood graph in Figure 138 .6.4 The consequent dimensionless EDMgraph comprising all the subgraphs is incomplete, in general, because theneighbor number is relatively small; incomplete even though it is a supersetof the neighborhood graph. Remaining distances (those not graphed at all)are squared then made variables within the algorithm; it is this variabilitythat admits unfurling.

To demonstrate, consider untying the trefoil knot drawn in Figure 139a.A corresponding Euclidean distance matrix D = [dij , i, j =1 . . . N ]employing only 2 nearest neighbors is banded having the incomplete form

D =

0 d12 d13 ? · · · ? d1,N−1 d1N

d12 0 d23 d24. . . ? ? d2N

d13 d23 0 d34. . . ? ? ?

? d24 d34 0. . . . . . ? ?

.... . . . . . . . . . . . . . . . . . ?

? ? ?. . . . . . 0 dN−2,N−1 dN−2,N

d1,N−1 ? ? ?. . . dN−2,N−1 0 dN−1,N

d1N d2N ? ? ? dN−2,N dN−1,N 0

(1161)

where dij denotes a given fixed distance-square. The unfurling algorithm can

6.4Local reconstruction of point position, from the EDM submatrix corresponding to acomplete dimensionless EDM subgraph, is unique to within an isometry (§5.6, §5.12).

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6.4. A GEOMETRY OF COMPLETION 497

(a)

(b)

Figure 139: (a) Trefoil knot in R3 from Weinberger & Saul [369].(b) Topological transformation algorithm employing 4 nearest neighbors andN = 539 samples reduces affine dimension of knot to r=2. Choosing instead2 nearest neighbors would make this embedding more circular.

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498 CHAPTER 6. CONE OF DISTANCE MATRICES

be expressed as an optimization problem; constrained total distance-squaremaximization:

maximizeD

〈−V , D〉subject to 〈D , eie

Tj + eje

Ti 〉1

2= dij ∀(i, j)∈ I

rank(V DV ) = 2

D ∈ EDMN

(1162)

where ei∈ RN is the ith member of the standard basis, where set I indexesthe given distance-square data like that in (1161), where V ∈ RN×N is thegeometric centering matrix (§B.4.1), and where

〈−V , D〉 = tr(−V DV ) = 2 tr G =1

N

i,j

dij (891)

where G is the Gram matrix producing D assuming G1 = 0.If the (rank) constraint on affine dimension is ignored, then problem

(1162) becomes convex, a corresponding solution D⋆ can be found, and anearest rank-2 solution is then had by ordered eigenvalue decomposition

of −V D⋆V followed by spectral projection (§7.1.3) on

[

R2

+

0

]

⊂ RN . This

two-step process is necessarily suboptimal. Yet because the decompositionfor the trefoil knot reveals only two dominant eigenvalues, the spectralprojection is nearly benign. Such a reconstruction of point position (§5.12)utilizing 4 nearest neighbors is drawn in Figure 139b; a low-dimensionalembedding of the trefoil knot.

This problem (1162) can, of course, be written equivalently in terms ofGram matrix G , facilitated by (897); videlicet, for Φij as in (865)

maximizeG∈SN

c

〈I , G〉subject to 〈G , Φij〉 = dij ∀(i, j)∈ I

rank G = 2

G º 0

(1163)

The advantage to converting EDM to Gram is: Gram matrix G is a bridgebetween point list X and EDM D ; constraints on any or all of thesethree variables may now be introduced. (Example 5.4.2.3.5) Confinementto the geometric center subspace SN

c (implicit constraint G1 = 0) keeps G

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6.4. A GEOMETRY OF COMPLETION 499

Figure 140: Trefoil ribbon; courtesy, Kilian Weinberger. Same topologicaltransformation algorithm with 5 nearest neighbors and N = 1617 samples.

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500 CHAPTER 6. CONE OF DISTANCE MATRICES

independent of its translation-invariant subspace SN⊥c (§5.5.1.1, Figure 146)

so as not to become numerically unbounded.A problem dual to maximum variance unfolding problem (1163) (less the

Gram rank constraint) has been called the fastest mixing Markov process.That dual has simple interpretations in graph and circuit theory and inmechanical and thermal systems, explored in [337], and has direct applicationto quick calculation of PageRank by search engines [227, §4]. Optimal Gramrank turns out to be tightly bounded above by minimum multiplicity of thesecond smallest eigenvalue of a dual optimal variable. 2

6.5 EDM definition in 11T

Any EDM D corresponding to affine dimension r has representation

D(VX ) , δ(VXV TX )1T + 1δ(VXV T

X )T − 2VXV TX ∈ EDMN (1164)

where R(VX ∈ RN×r)⊆ N (1T) = 1⊥

V TX VX = δ2(V T

X VX ) and VX is full-rank with orthogonal columns.(1165)

Equation (1164) is simply the standard EDM definition (867) with acentered list X as in (951); Gram matrix XTX has been replaced withthe subcompact singular value decomposition (§A.6.2)6.5

VXV TX ≡ V TXTXV ∈ SN

c ∩ SN+ (1166)

This means: inner product V TX VX is an r×r diagonal matrix Σ of nonzero

singular values.Vector δ(VXV T

X ) may me decomposed into complementary parts byprojecting it on orthogonal subspaces 1⊥ and R(1) : namely,

P1⊥

(

δ(VXV TX )

)

= V δ(VXV TX ) (1167)

P1

(

δ(VXV TX )

)

=1

N11Tδ(VXV T

X ) (1168)

Of course

δ(VXV TX ) = V δ(VXV T

X ) +1

N11Tδ(VXV T

X ) (1169)

6.5Subcompact SVD: VXV TX , Q

√Σ√

ΣQT≡ V TXTXV . So V TX is not necessarily XV

(§5.5.1.0.1), although affine dimension r = rank(V TX ) = rank(XV ). (1008)

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6.5. EDM DEFINITION IN 11T 501

by (889). Substituting this into EDM definition (1164), we get theHayden, Wells, Liu, & Tarazaga EDM formula [181, §2]

D(VX , y) , y1T + 1yT +λ

N11T − 2VXV T

X ∈ EDMN (1170)

where

λ , 2‖VX‖2F = 1Tδ(VXV T

X )2 and y , δ(VXV TX )− λ

2N1 = V δ(VXV T

X )

(1171)

and y=0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomesan eigenvalue when corresponding eigenvector 1 exists.6.6

Then the particular dyad sum from (1170)

y1T + 1yT +λ

N11T ∈ SN⊥

c (1172)

must belong to the orthogonal complement of the geometric center subspace(p.719), whereas VXV T

X ∈ SNc ∩ SN

+ (1166) belongs to the positive semidefinitecone in the geometric center subspace.

Proof. We validate eigenvector 1 and eigenvalue λ .(⇒) Suppose 1 is an eigenvector of EDM D . Then because

V TX 1 = 0 (1173)

it follows

D1 = δ(VXV TX )1T1 + 1δ(VXV T

X )T1 = N δ(VXV TX ) + ‖VX‖2

F1

⇒ δ(VXV TX ) ∝ 1

(1174)

For some κ∈R+

δ(VXV TX )T1 = N κ = tr(V T

X VX ) = ‖VX‖2F ⇒ δ(VXV T

X ) =1

N‖VX‖2

F1 (1175)

so y=0.

(⇐) Now suppose δ(VXV TX )=

λ

2N1 ; id est, y=0. Then

D =λ

N11T− 2VXV T

X ∈ EDMN (1176)

1 is an eigenvector with corresponding eigenvalue λ . ¨

6.6 e.g., when X = I in EDM definition (867).

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502 CHAPTER 6. CONE OF DISTANCE MATRICES

VX

N (1T)

1

δ(VXV TX )

Figure 141: Example of VX selection to make an EDM corresponding tocardinality N = 3 and affine dimension r = 1 ; VX is a vector in nullspaceN (1T)⊂ R3. Nullspace of 1T is hyperplane in R3 (drawn truncated) havingnormal 1. Vector δ(VXV T

X ) may or may not be in plane spanned by 1 , VX ,but belongs to nonnegative orthant which is strictly supported by N (1T).

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6.5. EDM DEFINITION IN 11T 503

6.5.1 Range of EDM D

From §B.1.1 pertaining to linear independence of dyad sums: If the transposehalves of all the dyads in the sum (1164)6.7 make a linearly independent set,then the nontranspose halves constitute a basis for the range of EDM D .Saying this mathematically: For D∈ EDMN

R(D)=R([ δ(VXV TX ) 1 VX ]) ⇐ rank([ δ(VXV T

X ) 1 VX ])= 2 + r

R(D)=R([1 VX ]) ⇐ otherwise (1177)

6.5.1.0.1 Proof. We need that condition under which the rankequality is satisfied: We know R(VX )⊥1, but what is the relativegeometric orientation of δ(VXV T

X ) ? δ(VXV TX )º 0 because VXV T

X º 0,and δ(VXV T

X )∝1 remains possible (1174); this means δ(VXV TX ) /∈ N (1T)

simply because it has no negative entries. (Figure 141) If the projection ofδ(VXV T

X ) on N (1T) does not belong to R(VX ) , then that is a necessary andsufficient condition for linear independence (l.i.) of δ(VXV T

X ) with respect toR([1 VX ]) ; id est,

V δ(VXV TX ) 6= VX a for any a∈Rr

(I − 1N11T)δ(VXV T

X ) 6= VX a

δ(VXV TX ) − 1

N‖VX‖2

F1 6= VX a

δ(VXV TX ) − λ

2N1 = y 6= VX a ⇔ 1 , δ(VXV T

X ) , VX is l.i.

(1178)

On the other hand when this condition is violated (when (1171) y=VX ap

for some particular a∈Rr), then from (1170) we have

R(

D = y1T + 1yT + λN11T− 2VXV T

X)

= R(

(VX ap + λN1)1T + (1aT

p − 2VX )V TX

)

= R([ VX ap + λN1 1aT

p − 2VX ])

= R([1 VX ]) (1179)

An example of such a violation is (1176) where, in particular, ap = 0. ¨

6.7Identifying columns VX , [ v1 · · · vr ] , then VXV TX =

i

vivTi is also a sum of dyads.

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504 CHAPTER 6. CONE OF DISTANCE MATRICES

Then a statement parallel to (1177) is, for D∈ EDMN (Theorem 5.7.3.0.1)

rank(D) = r + 2 ⇔ y /∈R(VX )(

⇔ 1TD†1 = 0)

rank(D) = r + 1 ⇔ y∈R(VX )(

⇔ 1TD†1 6= 0) (1180)

6.5.2 Boundary constituents of EDM cone

Expression (1164) has utility in forming the set of all EDMs correspondingto affine dimension r :

D∈ EDMN | rank(V DV )= r

=

D(VX ) | VX ∈ RN×r, rank VX = r , V TX VX = δ2(V T

X VX ) , R(VX )⊆ N (1T)

(1181)

whereas D∈ EDMN | rank(V DV )≤ r is the closure of this same set;

D∈ EDMN | rank(V DV )≤ r

=

D∈ EDMN | rank(V DV )= r

(1182)

For example,

rel ∂EDMN =

D∈ EDMN | rank(V DV )< N−1

=N−2⋃

r=0

D∈ EDMN | rank(V DV )= r (1183)

None of these are necessarily convex sets, although

EDMN =N−1⋃

r=0

D∈ EDMN | rank(V DV )= r

=

D∈ EDMN | rank(V DV )= N−1

rel int EDMN =

D∈ EDMN | rank(V DV )= N−1

(1184)

are pointed convex cones.When cardinality N = 3 and affine dimension r = 2, for example, the

relative interior rel int EDM3 is realized via (1181). (§6.6)When N = 3 and r = 1, the relative boundary of the EDM cone

dvec rel ∂EDM3 is realized in isomorphic R3 as in Figure 136d. This figurecould be constructed via (1182) by spiraling vector VX tightly about theorigin in N (1T) ; as can be imagined with aid of Figure 141. Vectors closeto the origin in N (1T) are correspondingly close to the origin in EDMN .As vector VX orbits the origin in N (1T) , the corresponding EDM orbitsthe axis of revolution while remaining on the boundary of the circular conedvec rel ∂EDM3. (Figure 142)

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6.5. EDM DEFINITION IN 11T 505

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

VX ∈ R3×1տ

(a)

(b)

dvecD(VX ) ⊂ dvec rel ∂EDM3←−

Figure 142: (a) Vector VX from Figure 141 spirals in N (1T)⊂ R3

decaying toward origin. (Spiral is two-dimensional in vector space R3.)(b) Corresponding trajectory D(VX ) on EDM cone relative boundary createsa vortex also decaying toward origin. There are two complete orbits on EDMcone boundary about axis of revolution for every single revolution of VXabout origin. (Vortex is three-dimensional in isometrically isomorphic R3.)

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506 CHAPTER 6. CONE OF DISTANCE MATRICES

6.5.3 Faces of EDM cone

Like the positive semidefinite cone, EDM cone faces are EDM cones.

6.5.3.0.1 Exercise. Isomorphic faces.Prove that in high cardinality N , any set of EDMs made via (1181) or (1182)with particular affine dimension r is isomorphic with any set admitting thesame affine dimension but made in lower cardinality. H

6.5.3.1 smallest face containing an EDM

Now suppose we are given a particular EDM D(VXp)∈ EDMN correspondingto affine dimension r and parametrized by VXp in (1164). The EDM cone’ssmallest face that contains D(VXp) is

F(

EDMN ∋D(VXp))

=

D(VX ) | VX ∈ RN×r, rank VX = r , V TX VX = δ2(V T

X VX ) , R(VX )⊆R(VXp)

≃ EDMr+1 (1185)

which is isomorphic6.8 with convex cone EDMr+1, hence of dimension

dimF(

EDMN ∋D(VXp))

= (r + 1)r/2 (1186)

in isomorphic RN(N−1)/2. Not all dimensions are represented; e.g., the EDMcone has no two-dimensional faces.

When cardinality N = 4 and affine dimension r=2 so that R(VXp) is anytwo-dimensional subspace of three-dimensional N (1T) in R4, for example,then the corresponding face of EDM4 is isometrically isomorphic with: (1182)

EDM3 = D∈ EDM3 | rank(V DV )≤ 2 ≃ F(EDM4∋D(VXp)) (1187)

Each two-dimensional subspace of N (1T) corresponds to anotherthree-dimensional face.

Because each and every principal submatrix of an EDM in EDMN

(§5.14.3) is another EDM [231, §4.1], for example, then each principalsubmatrix belongs to a particular face of EDMN .

6.8The fact that the smallest face is isomorphic with another EDM cone (perhaps smallerthan EDMN) is implicit in [181, §2].

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6.5. EDM DEFINITION IN 11T 507

6.5.3.2 extreme directions of EDM cone

In particular, extreme directions (§2.8.1) of EDMN correspond to affinedimension r = 1 and are simply represented: for any particular cardinalityN ≥ 2 (§2.8.2) and each and every nonzero vector z in N (1T)

Γ , (z z)1T + 1(z z)T − 2zzT ∈ EDMN

= δ(zzT)1T + 1δ(zzT)T − 2zzT(1188)

is an extreme direction corresponding to a one-dimensional face of the EDMcone EDMN that is a ray in isomorphic subspace RN(N−1)/2.

Proving this would exercise the fundamental definition (177) of extremedirection. Here is a sketch: Any EDM may be represented

D(VX ) = δ(VXV TX )1T + 1δ(VXV T

X )T − 2VXV TX ∈ EDMN (1164)

where matrix VX (1165) has orthogonal columns. For the same reason (1443)that zzT is an extreme direction of the positive semidefinite cone (§2.9.2.4)for any particular nonzero vector z , there is no conic combination of distinctEDMs (each conically independent of Γ (§2.10)) equal to Γ . ¥

6.5.3.2.1 Example. Biorthogonal expansion of an EDM.(confer §2.13.7.1.1) When matrix D belongs to the EDM cone, nonnegativecoordinates for biorthogonal expansion are the eigenvalues λ∈RN of−V DV 1

2: For any D∈ SN

h it holds

D = δ(

−V DV 12

)

1T + 1δ(

−V DV 12

)T − 2(

−V DV 12

)

(975)

By diagonalization −V DV 12, QΛQT∈ SN

c (§A.5.1) we may write

D = δ

(

N∑

i=1

λi qiqTi

)

1T + 1δ

(

N∑

i=1

λi qiqTi

)T

− 2N∑

i=1

λi qiqTi

=N∑

i=1

λi

(

δ(qiqTi )1T + 1δ(qiq

Ti )T − 2qiq

Ti

)

(1189)

where qi is the ith eigenvector of −V DV 12

arranged columnar in orthogonalmatrix

Q = [ q1 q2 · · · qN ] ∈ RN×N (378)

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508 CHAPTER 6. CONE OF DISTANCE MATRICES

and where δ(qiqTi )1T + 1δ(qiq

Ti )T− 2qiq

Ti , i=1 . . . N are extreme

directions of some pointed polyhedral cone K⊂ SNh and extreme directions

of EDMN . Invertibility of (1189)

−V DV 12

= −VN∑

i=1

λi

(

δ(qiqTi )1T + 1δ(qiq

Ti )T− 2qiq

Ti

)

V 12

=N∑

i=1

λi qiqTi

(1190)

implies linear independence of those extreme directions. Then thebiorthogonal expansion is expressed

dvec D = Y Y † dvec D = Y λ(

−V DV 12

)

(1191)

where

Y ,[

dvec(

δ(qiqTi )1T + 1δ(qiq

Ti )T− 2qiq

Ti

)

, i = 1 . . . N]

∈ RN(N−1)/2×N (1192)

When D belongs to the EDM cone in the subspace of symmetric hollowmatrices, unique coordinates Y † dvec D for this biorthogonal expansionmust be the nonnegative eigenvalues λ of −V DV 1

2. This means D

simultaneously belongs to the EDM cone and to the pointed polyhedral conedvecK= cone(Y ). 2

6.5.3.3 open question

Result (1186) is analogous to that for the positive semidefinite cone (213),although the question remains open whether all faces of EDMN (whosedimension is less than dimension of the cone) are exposed like they are forthe positive semidefinite cone.6.9 (§2.9.2.3) [344]

6.6 Correspondence to PSD cone SN−1+

Hayden, Wells, Liu, & Tarazaga [181, §2] assert one-to-one correspondenceof EDMs with positive semidefinite matrices in the symmetric subspace.Because rank(V DV )≤N−1 (§5.7.1.1), that PSD cone corresponding to

6.9Elementary example of face not exposed is given by the closed convex set in Figure 31and in Figure 41.

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6.6. CORRESPONDENCE TO PSD CONE SN−1+ 509

the EDM cone can only be SN−1+ . [8, §18.2.1] To clearly demonstrate this

correspondence, we invoke inner-product form EDM definition

D(Φ) =

[

0δ(Φ)

]

1T + 1[

0 δ(Φ)T]

− 2

[

0 0T

0 Φ

]

∈ EDMN

⇔Φ º 0

(993)

Then the EDM cone may be expressed

EDMN =

D(Φ) | Φ ∈ SN−1+

(1193)

Hayden & Wells’ assertion can therefore be equivalently stated in terms ofan inner-product form EDM operator

D(SN−1+ ) = EDMN (995)

VN (EDMN) = SN−1+ (996)

identity (996) holding because R(VN )=N (1T) (874), linear functions D(Φ)and VN (D)=−V T

NDVN (§5.6.2.1) being mutually inverse.

In terms of affine dimension r , Hayden & Wells claim particularcorrespondence between PSD and EDM cones:

r = N−1: Symmetric hollow matrices −D positive definite on N (1T) correspondto points relatively interior to the EDM cone.

r < N−1: Symmetric hollow matrices −D positive semidefinite on N (1T) , where−V T

NDVN has at least one 0 eigenvalue, correspond to points on therelative boundary of the EDM cone.

r = 1: Symmetric hollow nonnegative matrices rank-one on N (1T) correspondto extreme directions (1188) of the EDM cone; id est, for some nonzerovector u (§A.3.1.0.7)

rank V TNDVN =1

D ∈ SNh ∩ RN×N

+

⇔ D ∈ EDMN

D is an extreme direction⇔

−V TNDVN ≡ uuT

D ∈ SNh

(1194)

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510 CHAPTER 6. CONE OF DISTANCE MATRICES

6.6.0.0.1 Proof. Case r = 1 is easily proved: From the nonnegativitydevelopment in §5.8.1, extreme direction (1188), and Schoenberg criterion(886), we need show only sufficiency; id est, prove

rank V TNDVN =1

D ∈ SNh ∩ RN×N

+

⇒ D ∈ EDMN

D is an extreme direction

Any symmetric matrix D satisfying the rank condition must have the form,for z,q∈RN and nonzero z∈N (1T) ,

D = ±(1qT + q1T− 2zzT) (1195)

because (§5.6.2.1, confer §E.7.2.0.2)

N (VN (D)) = 1qT + q1T | q∈RN ⊆ SN (1196)

Hollowness demands q = δ(zzT) while nonnegativity demands choice ofpositive sign in (1195). Matrix D thus takes the form of an extremedirection (1188) of the EDM cone. ¨

The foregoing proof is not extensible in rank: An EDMwith corresponding affine dimension r has the general form, forzi∈N (1T) , i=1 . . . r an independent set,

D = 1δ

(

r∑

i=1

zizTi

)T

+ δ

(

r∑

i=1

zizTi

)

1T− 2r

i=1

zizTi ∈ EDMN (1197)

The EDM so defined relies principally on the sum∑

zizTi having positive

summand coefficients (⇔ −V TNDVN º 0)6.10. Then it is easy to find a

sum incorporating negative coefficients while meeting rank, nonnegativity,and symmetric hollowness conditions but not positive semidefiniteness onsubspace R(VN ) ; e.g., from page 460,

−V

0 1 11 0 51 5 0

V1

2= z1z

T1 − z2z

T2 (1198)

6.10 (⇐) For ai∈RN−1, let zi =V †TN ai .

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6.6. CORRESPONDENCE TO PSD CONE SN−1+ 511

6.6.0.0.2 Example. Extreme rays versus rays on the boundary.

The EDM D =

0 1 41 0 14 1 0

is an extreme direction of EDM3 where

u =

[

12

]

in (1194). Because −V TNDVN has eigenvalues 0, 5 , the ray

whose direction is D also lies on the relative boundary of EDM3.

In exception, EDM D = κ

[

0 11 0

]

, for any particular κ > 0, is an

extreme direction of EDM2 but −V TNDVN has only one eigenvalue: κ.

Because EDM2 is a ray whose relative boundary (§2.6.1.3.1) is the origin,this conventional boundary does not include D which belongs to the relativeinterior in this dimension. (§2.7.0.0.1) 2

6.6.1 Gram-form correspondence to SN−1+

With respect to D(G)= δ(G)1T + 1δ(G)T− 2G (879) the linear Gram-formEDM operator, results in §5.6.1 provide [2, §2.6]

EDMN = D(

V(EDMN))

≡ D(

VN SN−1+ V T

N)

(1199)

VN SN−1+ V T

N ≡ V(

D(

VN SN−1+ V T

N))

= V(EDMN) , −V EDMN V 12

= SNc ∩ SN

+

(1200)a one-to-one correspondence between EDMN and SN−1

+ .

6.6.2 EDM cone by elliptope

Having defined the elliptope parametrized by scalar t>0

ENt = SN

+ ∩ Φ∈ SN | δ(Φ)= t1 (1073)

then following Alfakih [9] we have

EDMN = cone11T− EN1 = t(11T− EN

1 ) | t ≥ 0 (1201)

Identification EN = EN1 equates the standard elliptope (§5.9.1.0.1,

Figure 128) to our parametrized elliptope.

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512 CHAPTER 6. CONE OF DISTANCE MATRICES

dvec rel ∂ EDM3

dvec(11T− E3)

EDMN = cone11T− EN = t(11T− EN) | t ≥ 0 (1201)

Figure 143: Three views of translated negated elliptope 11T− E3

1

(confer Figure 128) shrouded by truncated EDM cone. Fractal on EDMcone relative boundary is numerical artifact belonging to intersection withelliptope relative boundary. The fractal is trying to convey existence of aneighborhood about the origin where the translated elliptope boundary andEDM cone boundary intersect.

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6.6. CORRESPONDENCE TO PSD CONE SN−1+ 513

6.6.2.0.1 Expository. Normal cone, tangent cone, elliptope.Define TE(11T) to be the tangent cone to the elliptope E at point 11T ;id est,

TE(11T) , t(E − 11T) | t≥ 0 (1202)

The normal cone K⊥E (11T) to the elliptope at 11T is a closed convex cone

defined (§E.10.3.2.1, Figure 172)

K⊥E (11T) , B | 〈B , Φ − 11T〉 ≤ 0 , Φ∈E (1203)

The polar cone of any set K is the closed convex cone (confer (276))

K, B | 〈B , A〉≤ 0 , for all A∈K (1204)

The normal cone is well known to be the polar of the tangent cone,

K⊥E (11T) = TE(11T)

(1205)

and vice versa; [196, §A.5.2.4]

K⊥E (11T)

= TE(11T) (1206)

From Deza & Laurent [110, p.535] we have the EDM cone

EDM = −TE(11T) (1207)

The polar EDM cone is also expressible in terms of the elliptope. From (1205)we have

EDM

= −K⊥E (11T) (1208)

In §5.10.1 we proposed the expression for EDM D

D = t11T− E ∈ EDMN (1074)

where t∈R+ and E belongs to the parametrized elliptope ENt . We further

propose, for any particular t>0

EDMN = conet11T− ENt (1209)

Proof. Pending. ¨

6.6.2.0.2 Exercise. EDM cone from elliptope.Relationship of the translated negated elliptope with the EDM cone isillustrated in Figure 143. Prove whether it holds that

EDMN = limt→∞

t11T− ENt (1210)

H

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514 CHAPTER 6. CONE OF DISTANCE MATRICES

6.7 Vectorization & projection interpretation

In §E.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection[6, §2] of vectorized −D∈ SN

h on the subspace of geometrically centeredsymmetric matrices

SNc = G∈ SN | G1 = 0 (1960)

= G∈ SN | N (G) ⊇ 1 = G∈ SN | R(G) ⊆ N (1T)= V Y V | Y ∈ SN ⊂ SN (1961)

≡ VNAV TN | A ∈ SN−1

(966)

because elementary auxiliary matrix V is an orthogonal projector (§B.4.1).Yet there is another useful projection interpretation:

Revising the fundamental matrix criterion for membership to the EDMcone (862),6.11

〈zzT,−D〉 ≥ 0 ∀ zzT | 11TzzT = 0

D ∈ SNh

⇔ D ∈ EDMN (1211)

this is equivalent, of course, to the Schoenberg criterion

−V TNDVN º 0

D ∈ SNh

⇔ D ∈ EDMN (886)

because N (11T) = R(VN ). When D∈ EDMN , correspondence (1211)means −zTDz is proportional to a nonnegative coefficient of orthogonalprojection (§E.6.4.2, Figure 145) of −D in isometrically isomorphicRN(N+1)/2 on the range of each and every vectorized (§2.2.2.1) symmetricdyad (§B.1) in the nullspace of 11T ; id est, on each and every member of

T ,

svec(zzT) | z ∈N (11T)=R(VN )

⊂ svec ∂ SN+

=

svec(VN υυTV TN ) | υ∈RN−1

(1212)

whose dimension is

dim T = N(N − 1)/2 (1213)

6.11 N (11T)=N (1T) and R(zzT)=R(z)

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6.7. VECTORIZATION & PROJECTION INTERPRETATION 515

The set of all symmetric dyads zzT | z∈RN constitute the extremedirections of the positive semidefinite cone (§2.8.1, §2.9) SN

+ , hence lie onits boundary. Yet only those dyads in R(VN ) are included in the test (1211),thus only a subset T of all vectorized extreme directions of SN

+ is observed.In the particularly simple case D∈ EDM2 = D∈ S2

h | d12 ≥ 0 , forexample, only one extreme direction of the PSD cone is involved:

zzT =

[

1 −1−1 1

]

(1214)

Any nonnegative scaling of vectorized zzT belongs to the set T illustratedin Figure 144 and Figure 145.

6.7.1 Face of PSD cone SN+ containing V

In any case, set T (1212) constitutes the vectorized extreme directions ofan N(N−1)/2-dimensional face of the PSD cone SN

+ containing auxiliarymatrix V ; a face isomorphic with SN−1

+ = Srank V+ (§2.9.2.3).

To show this, we must first find the smallest face that contains auxiliarymatrix V and then determine its extreme directions. From (212),

F(

SN+ ∋V

)

= W ∈ SN+ | N (W ) ⊇ N (V ) = W ∈ SN

+ | N (W ) ⊇ 1= V Y V º 0 | Y ∈ SN ≡ VNBV T

N | B∈ SN−1+

≃ Srank V+ = −V T

N EDMN VN (1215)

where the equivalence ≡ is from §5.6.1 while isomorphic equality ≃ withtransformed EDM cone is from (996). Projector V belongs to F

(

SN+ ∋V

)

because VNV †NV †T

N V TN = V . (§B.4.3) Each and every rank-one matrix

belonging to this face is therefore of the form:

VN υυTV TN | υ∈RN−1 (1216)

Because F(

SN+ ∋V

)

is isomorphic with a positive semidefinite cone SN−1+ ,

then T constitutes the vectorized extreme directions of F , the originconstitutes the extreme points of F , and auxiliary matrix V is some convexcombination of those extreme points and directions by the extremes theorem(§2.8.1.1.1). ¨

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516 CHAPTER 6. CONE OF DISTANCE MATRICES

d11

√2d12

d22svec EDM2

0

−T

svec ∂ S2

+

[

d11 d12

d12 d22

]

T ,

svec(zzT) | z ∈N (11T)= κ

[

1−1

]

, κ∈R

⊂ svec ∂ S2

+

Figure 144: Truncated boundary of positive semidefinite cone S2

+ inisometrically isomorphic R3 (via svec (56)) is, in this dimension, constitutedsolely by its extreme directions. Truncated cone of Euclidean distancematrices EDM2 in isometrically isomorphic subspace R . Relativeboundary of EDM cone is constituted solely by matrix 0. HalflineT = κ2[ 1 −

√2 1 ]T | κ∈R on PSD cone boundary depicts that lone

extreme ray (1214) on which orthogonal projection of −D must be positivesemidefinite if D is to belong to EDM2. aff cone T = svec S2

c . (1219) DualEDM cone is halfspace in R3 whose bounding hyperplane has inward-normalsvec EDM2.

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6.7. VECTORIZATION & PROJECTION INTERPRETATION 517

d11

√2d12

d22svec S2

h

0

−T

svec ∂ S2

+

[

d11 d12

d12 d22

]

D

−D

Projection of vectorized −D on range of vectorized zzT :

Psvec zzT(svec(−D)) =〈zzT, −D〉〈zzT, zzT〉 zzT

D ∈ EDMN ⇔

〈zzT,−D〉 ≥ 0 ∀ zzT | 11TzzT = 0

D ∈ SNh

(1211)

Figure 145: Given-matrix D is assumed to belong to symmetric hollowsubspace S2

h ; a line in this dimension. Negating D , we find its polar alongS2

h . Set T (1212) has only one ray member in this dimension; not orthogonalto S2

h . Orthogonal projection of −D on T (indicated by half dot) hasnonnegative projection coefficient. Matrix D must therefore be an EDM.

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518 CHAPTER 6. CONE OF DISTANCE MATRICES

In fact the smallest face, that contains auxiliary matrix V , of the PSDcone SN

+ is the intersection with the geometric center subspace (1960) (1961);

F(

SN+ ∋V

)

= cone

VN υυTV TN | υ∈RN−1

= SNc ∩ SN

+

≡ Xº 0 | 〈X , 11T〉 = 0 (1559)

(1217)

In isometrically isomorphic RN(N+1)/2

svecF(

SN+ ∋V

)

= cone T (1218)

related to SNc by

aff cone T = svec SNc (1219)

6.7.2 EDM criteria in 11T

(confer §6.5, (893)) Laurent specifies an elliptope trajectory condition forEDM cone membership: [231, §2.3]

D ∈ EDMN ⇔ [1 − e−αdij ] ∈ EDMN ∀α > 0 (1068)

From the parametrized elliptope ENt in §6.6.2 and §5.10.1 we propose

D∈ EDMN ⇔ ∃ t∈R+

E∈ENt

Ä D = t11T− E (1220)

Chabrillac & Crouzeix [72, §4] prove a different criterion they attributeto Finsler (1937) [140]. We apply it to EDMs: for D∈ SN

h (1016)

−V TNDVN ≻ 0 ⇔ ∃κ>0 Ä −D + κ11T ≻ 0

⇔D∈ EDMN with corresponding affine dimension r=N−1

(1221)

This Finsler criterion has geometric interpretation in terms of thevectorization & projection already discussed in connection with (1211). Withreference to Figure 144, the offset 11T is simply a direction orthogonal toT in isomorphic R3. Intuitively, translation of −D in direction 11T is likeorthogonal projection on T in so far as similar information can be obtained.

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6.8. DUAL EDM CONE 519

When the Finsler criterion (1221) is applied despite lower affinedimension, the constant κ can go to infinity making the test −D+κ11Tº 0impractical for numerical computation. Chabrillac & Crouzeix invent acriterion for the semidefinite case, but is no more practical: for D∈ SN

h

D∈ EDMN

⇔∃κp > 0 Ä ∀κ≥κp , −D − κ11T [sic] has exactly one negative eigenvalue

(1222)

6.8 Dual EDM cone

6.8.1 Ambient SN

We consider finding the ordinary dual EDM cone in ambient space SN whereEDMN is pointed, closed, convex, but not full-dimensional. The set of allEDMs in SN is a closed convex cone because it is the intersection of halfspacesabout the origin in vectorized variable D (§2.4.1.1.1, §2.7.2):

EDMN =⋂

z∈N (1T)i=1...N

D ∈ SN | 〈eieTi , D〉 ≥ 0 , 〈eie

Ti , D〉 ≤ 0 , 〈zzT,−D〉 ≥ 0

(1223)

By definition (276), dual cone K∗comprises each and every vector

inward-normal to a hyperplane supporting convex cone K . The dual EDMcone in the ambient space of symmetric matrices is therefore expressible asthe aggregate of every conic combination of inward-normals from (1223):

EDMN∗= coneeie

Ti , −eje

Tj | i , j =1 . . . N − conezzT | 11TzzT=0

= N∑

i=1

ζieieTi −

N∑

j=1

ξj ejeTj | ζi , ξj ≥ 0 − conezzT | 11TzzT=0

= δ(u) | u∈RN − cone

VN υυTV TN | υ∈RN−1, (‖v‖= 1)

⊂ SN

= δ2(Y ) − VNΨV TN | Y ∈ SN , Ψ∈ SN−1

+ (1224)

The EDM cone is not selfdual in ambient SN because its affine hull belongsto a proper subspace

aff EDMN = SNh (1225)

The ordinary dual EDM cone cannot, therefore, be pointed. (§2.13.1.1)

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520 CHAPTER 6. CONE OF DISTANCE MATRICES

When N = 1, the EDM cone is the point at the origin in R . Auxiliarymatrix VN is empty [ ∅ ] , and dual cone EDM∗ is the real line.

When N = 2, the EDM cone is a nonnegative real line in isometricallyisomorphic R3 ; there S2

h is a real line containing the EDM cone. Dual cone

EDM2∗ is the particular halfspace in R3 whose boundary has inward-normalEDM2. Diagonal matrices δ(u) in (1224) are represented by a hyperplanethrough the origin d | [ 0 1 0 ]d = 0 while the term coneVN υυTV T

N is represented by the halfline T in Figure 144 belonging to the positivesemidefinite cone boundary. The dual EDM cone is formed by translatingthe hyperplane along the negative semidefinite halfline −T ; the union ofeach and every translation. (confer §2.10.2.0.1)

When cardinality N exceeds 2, the dual EDM cone can no longer bepolyhedral simply because the EDM cone cannot. (§2.13.1.1)

6.8.1.1 EDM cone and its dual in ambient SN

Consider the two convex cones

K1 , SNh

K2 ,⋂

y∈N (1T)

A ∈ SN | 〈yyT, −A〉 ≥ 0

=

A ∈ SN | −zTV AV z ≥ 0 ∀ zzT(º 0)

=

A ∈ SN | −V AV º 0

(1226)

so

K1 ∩ K2 = EDMN (1227)

Dual cone K∗1 = SN⊥

h ⊆ SN (72) is the subspace of diagonal matrices.From (1224) via (292),

K∗2 = − cone

VN υυTV TN | υ∈RN−1

⊂ SN (1228)

Gaffke & Mathar [144, §5.3] observe that projection on K1 and K2 havesimple closed forms: Projection on subspace K1 is easily performed bysymmetrization and zeroing the main diagonal or vice versa, while projectionof H∈ SN on K2 is

PK2H = H − PSN+(V H V ) (1229)

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6.8. DUAL EDM CONE 521

Proof. First, we observe membership of H−PSN+(V H V ) to K2 because

PSN+(V H V ) − H =

(

PSN+(V H V ) − V H V

)

+ (V H V − H) (1230)

The term PSN+(V H V ) − V H V necessarily belongs to the (dual) positive

semidefinite cone by Theorem E.9.2.0.1. V 2 = V , hence

−V(

H−PSN+(V H V )

)

V º 0 (1231)

by Corollary A.3.1.0.5.Next, we require

〈PK2H−H , PK2H 〉 = 0 (1232)

Expanding,

〈−PSN+(V H V ) , H−PSN

+(V H V )〉 = 0 (1233)

〈PSN+(V H V ) , (PSN

+(V H V ) − V H V ) + (V H V − H)〉 = 0 (1234)

〈PSN+(V H V ) , (V H V − H)〉 = 0 (1235)

Product V H V belongs to the geometric center subspace; (§E.7.2.0.2)

V H V ∈ SNc = Y ∈ SN | N (Y )⊇1 (1236)

Diagonalize V H V ,QΛQT (§A.5) whose nullspace is spanned bythe eigenvectors corresponding to 0 eigenvalues by Theorem A.7.3.0.1.Projection of V H V on the PSD cone (§7.1) simply zeroes negativeeigenvalues in diagonal matrix Λ . Then

N (PSN+(V H V )) ⊇ N (V H V ) (⊇ N (V ) ) (1237)

from which it follows:PSN

+(V H V ) ∈ SN

c (1238)

so PSN+(V H V ) ⊥ (V H V −H) because V H V −H∈ SN⊥

c .

Finally, we must have PK2H−H =−PSN+(V H V )∈K∗

2 . From §6.7.1

we know dual cone K∗2 =−F

(

SN+ ∋V

)

is the negative of the positivesemidefinite cone’s smallest face that contains auxiliary matrix V . ThusPSN

+(V H V )∈F

(

SN+ ∋V

)

⇔ N (PSN+(V H V ))⊇N (V ) (§2.9.2.3) which was

already established in (1237). ¨

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522 CHAPTER 6. CONE OF DISTANCE MATRICES

svec ∂S2

+

svec S2⊥c

0

svec S2

c

svec S2

h

EDM2 = S2

h ∩(

S2⊥c − S2

+

)

Figure 146: Orthogonal complement S2⊥c (1962) (§B.2) of geometric center

subspace (a plane in isometrically isomorphic R3 ; drawn is a tiled fragment)apparently supporting positive semidefinite cone. (Rounded vertex is artifactof plot.) Line svec S2

c = aff cone T (1219) intersects svec ∂S2

+ , also drawn inFigure 144; it runs along PSD cone boundary. (confer Figure 127)

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6.8. DUAL EDM CONE 523

− svec ∂S2

+

svec S2⊥c

0

svec S2

c

svec S2

h

EDM2 = S2

h ∩(

S2⊥c − S2

+

)

Figure 147: EDM cone construction in isometrically isomorphic R3 by addingpolar PSD cone to svec S2⊥

c . Difference svec(

S2⊥c − S2

+

)

is halfspace partially

bounded by svec S2⊥c . EDM cone is nonnegative halfline along svec S2

h inthis dimension.

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524 CHAPTER 6. CONE OF DISTANCE MATRICES

From the results in §E.7.2.0.2, we know matrix product V H V is theorthogonal projection of H∈ SN on the geometric center subspace SN

c . Thusthe projection product

PK2H = H − PSN+PSN

cH (1239)

6.8.1.1.1 Lemma. Projection on PSD cone ∩ geometric center subspace.

PSN+∩ SN

c= PSN

+PSN

c(1240)

Proof. For each and every H∈ SN , projection of PSNcH on the positive

semidefinite cone remains in the geometric center subspace

SNc = G∈ SN | G1 = 0 (1960)

= G∈ SN | N (G) ⊇ 1 = G∈ SN | R(G) ⊆ N (1T)= V Y V | Y ∈ SN ⊂ SN (1961)

(966)

That is because: eigenvectors of PSNcH corresponding to 0 eigenvalues

span its nullspace N (PSNcH ). (§A.7.3.0.1) To project PSN

cH on the positive

semidefinite cone, its negative eigenvalues are zeroed. (§7.1.2) The nullspaceis thereby expanded while eigenvectors originally spanning N (PSN

cH )

remain intact. Because the geometric center subspace is invariant toprojection on the PSD cone, then the rule for projection on a convex setin a subspace governs (§E.9.5, projectors do not commute) and statement(1240) follows directly. ¨

From the lemma it follows

PSN+PSN

cH | H∈ SN = PSN

+∩ SNcH | H∈ SN (1241)

Then from (1986)

−(

SNc ∩ SN

+

)∗= H − PSN

+PSN

cH | H∈ SN (1242)

From (292) we get closure of a vector sum

K2 = −(

SNc ∩ SN

+

)∗= SN⊥

c − SN+ (1243)

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6.8. DUAL EDM CONE 525

therefore the equality

EDMN = K1 ∩ K2 = SNh ∩

(

SN⊥c − SN

+

)

(1244)

whose veracity is intuitively evident, in hindsight, [87, p.109] from the mostfundamental EDM definition (867). Formula (1244) is not a matrix criterionfor membership to the EDM cone, it is not an EDM definition, and it isnot an equivalence between EDM operators or an isomorphism. Rather, it isa recipe for constructing the EDM cone whole from large Euclidean bodies:the positive semidefinite cone, orthogonal complement of the geometric centersubspace, and symmetric hollow subspace. A realization of this constructionin low dimension is illustrated in Figure 146 and Figure 147.

The dual EDM cone follows directly from (1244) by standard propertiesof cones (§2.13.1.1):

EDMN∗= K∗

1 + K∗2 = SN⊥

h − SNc ∩ SN

+ (1245)

which bears strong resemblance to (1224).

6.8.1.2 nonnegative orthant contains EDMN

That EDMN is a proper subset of the nonnegative orthant is not obviousfrom (1244). We wish to verify

EDMN = SNh ∩

(

SN⊥c − SN

+

)

⊂ RN×N+ (1246)

While there are many ways to prove this, it is sufficient to show that allentries of the extreme directions of EDMN must be nonnegative; id est, forany particular nonzero vector z = [zi , i=1 . . . N ]∈ N (1T) (§6.5.3.2),

δ(zzT)1T + 1δ(zzT)T− 2zzT ≥ 0 (1247)

where the inequality denotes entrywise comparison. The inequality holdsbecause the i, j th entry of an extreme direction is squared: (zi− zj)

2.We observe that the dyad 2zzT∈ SN

+ belongs to the positive semidefinitecone, the doublet

δ(zzT)1T + 1δ(zzT)T ∈ SN⊥c (1248)

to the orthogonal complement (1962) of the geometric center subspace, whiletheir difference is a member of the symmetric hollow subspace SN

h .

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526 CHAPTER 6. CONE OF DISTANCE MATRICES

Here is an algebraic method to prove nonnegativity: Suppose we aregiven A∈ SN⊥

c and B = [Bij]∈ SN+ and A−B∈ SN

h . Then we have, forsome vector u , A = u1T + 1uT = [Aij] = [ui + uj] and δ(B)= δ(A)= 2u .Positive semidefiniteness of B requires nonnegativity A−B≥ 0 because

(ei−ej)TB(ei−ej) = (Bii−Bij)−(Bji−Bjj) = 2(ui+uj)−2Bij ≥ 0 (1249)

6.8.1.3 Dual Euclidean distance matrix criterion

Conditions necessary for membership of a matrix D∗∈ SN to the dualEDM cone EDMN∗

may be derived from (1224): D∗∈ EDMN∗ ⇒D∗= δ(y)− VNAV T

N for some vector y and positive semidefinite matrixA∈ SN−1

+ . This in turn implies δ(D∗1)= δ(y). Then, for D∗∈ SN

D∗∈ EDMN∗ ⇔ δ(D∗1) − D∗ º 0 (1250)

where, for any symmetric matrix D∗

δ(D∗1) − D∗ ∈ SNc (1251)

To show sufficiency of the matrix criterion in (1250), recall Gram-formEDM operator

D(G) = δ(G)1T + 1δ(G)T− 2G (879)

where Gram matrix G is positive semidefinite by definition, and recall theself-adjointness property of the main-diagonal linear operator δ (§A.1):

〈D , D∗〉 =⟨

δ(G)1T + 1δ(G)T− 2G , D∗⟩ = 〈G , δ(D∗1) − D∗〉 2 (897)

Assuming 〈G , δ(D∗1) − D∗〉≥ 0 (1458), then we have known membershiprelation (§2.13.2.0.1)

D∗∈ EDMN∗ ⇔ 〈D , D∗〉 ≥ 0 ∀D∈ EDMN (1252)

¨

Elegance of this matrix criterion (1250) for membership to the dual EDMcone is lack of any other assumptions except D∗ be symmetric.6.12

6.12Recall: Schoenberg criterion (886) for membership to the EDM cone requiresmembership to the symmetric hollow subspace.

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6.8. DUAL EDM CONE 527

Linear Gram-form EDM operator (879) has adjoint, for Y ∈ SN

DT(Y ) , (δ(Y 1) − Y ) 2 (1253)

Then we have: [87, p.111]

EDMN∗= Y ∈ SN | δ(Y 1) − Y º 0 (1254)

the dual EDM cone expressed in terms of the adjoint operator. A dual EDMcone determined this way is illustrated in Figure 149.

6.8.1.3.1 Exercise. Dual EDM spectral cone.Find a spectral cone as in §5.11.2 corresponding to EDMN∗

. H

6.8.1.4 Nonorthogonal components of dual EDM

Now we tie construct (1245) for the dual EDM cone together with the matrixcriterion (1250) for dual EDM cone membership. For any D∗∈ SN it isobvious:

δ(D∗1) ∈ SN⊥h (1255)

any diagonal matrix belongs to the subspace of diagonal matrices (67). We

know when D∗∈ EDMN∗

δ(D∗1) − D∗ ∈ SNc ∩ SN

+ (1256)

this adjoint expression (1253) belongs to that face (1217) of the positivesemidefinite cone SN

+ in the geometric center subspace. Any nonzerodual EDM

D∗ = δ(D∗1) − (δ(D∗1) − D∗) ∈ SN⊥h ⊖ SN

c ∩ SN+ = EDMN∗

(1257)

can therefore be expressed as the difference of two linearly independent (whenvectorized) nonorthogonal components (Figure 127, Figure 148).

6.8.1.5 Affine dimension complementarity

From §6.8.1.3 we have, for some A∈ SN−1+ (confer (1256))

δ(D∗1) − D∗ = VNAV TN ∈ SN

c ∩ SN+ (1258)

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528 CHAPTER 6. CONE OF DISTANCE MATRICES

D = δ(D1) + (D− δ(D1)) ∈ SN⊥h ⊕ SN

c ∩ SN+ = EDMN

EDMN

EDMN

D− δ(D1)δ(D1)

D

0

SNc ∩ SN

+

SN⊥h

Figure 148: Hand-drawn abstraction of polar EDM cone (drawn truncated).Any member D of polar EDM cone can be decomposed into two linearlyindependent nonorthogonal components: δ(D1) and D− δ(D1).

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6.8. DUAL EDM CONE 529

if and only if D∗ belongs to the dual EDM cone. Call rank(VNAV TN ) dual

affine dimension. Empirically, we find a complementary relationship in affinedimension between the projection of some arbitrary symmetric matrix H onthe polar EDM cone, EDMN

= −EDMN∗, and its projection on the EDM

cone; id est, the optimal solution of 6.13

minimizeD∈ SN

‖D − H‖F

subject to D − δ(D1) º 0(1259)

has dual affine dimension complementary to affine dimension correspondingto the optimal solution of

minimizeD∈SN

h

‖D − H‖F

subject to −V TNDVN º 0

(1260)

Precisely,rank(D⋆− δ(D⋆ 1)) + rank(V T

ND⋆VN ) = N−1 (1261)

and rank(D⋆− δ(D⋆ 1))≤N−1 because vector 1 is always in the nullspaceof rank’s argument. This is similar to the known result for projection on theselfdual positive semidefinite cone and its polar:

rank P−SN+H + rank PSN

+H = N (1262)

When low affine dimension is a desirable result of projection on theEDM cone, projection on the polar EDM cone should be performed instead.Convex polar problem (1259) can be solved for D⋆ by transforming to anequivalent Schur-form semidefinite program (§3.6.2). Interior-point methodsfor numerically solving semidefinite programs tend to produce high-rank

6.13This polar projection can be solved quickly (without semidefinite programming) viaLemma 6.8.1.1.1; rewriting,

minimizeD∈ SN

‖(D− δ(D 1)) − (H − δ(D 1))‖F

subject to D− δ(D 1) º 0

which is the projection of affinely transformed optimal solution H − δ(D⋆ 1) on SNc ∩ SN

+ ;

D⋆− δ(D⋆ 1) = PSN+

PSNc(H − δ(D⋆ 1))

Foreknowledge of an optimal solution D⋆ as argument to projection suggests recursion.

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530 CHAPTER 6. CONE OF DISTANCE MATRICES

solutions. (§4.1.1.1) Then D⋆ = H − D⋆∈ EDMN by Corollary E.9.2.2.1,and D⋆ will tend to have low affine dimension. This approach breaks whenattempting projection on a cone subset discriminated by affine dimension orrank, because then we have no complementarity relation like (1261) or (1262)(§7.1.4.1).

6.8.1.6 EDM cone is not selfdual

In §5.6.1.1, via Gram-form EDM operator

D(G) = δ(G)1T + 1δ(G)T− 2G ∈ EDMN ⇐ G º 0 (879)

we established clear connection between the EDM cone and that face (1217)of positive semidefinite cone SN

+ in the geometric center subspace:

EDMN = D(SNc ∩ SN

+ ) (983)

V(EDMN) = SNc ∩ SN

+ (984)

whereV(D) = −V DV 1

2(972)

In §5.6.1 we established

SNc ∩ SN

+ = VN SN−1+ V T

N (970)

Then from (1250), (1258), and (1224) we can deduce

δ(EDMN∗1) − EDMN∗

= VN SN−1+ V T

N = SNc ∩ SN

+ (1263)

which, by (983) and (984), means the EDM cone can be related to the dualEDM cone by an equality:

EDMN = D(

δ(EDMN∗1) − EDMN∗)

(1264)

V(EDMN) = δ(EDMN∗1) − EDMN∗

(1265)

This means projection −V(EDMN) of the EDM cone on the geometriccenter subspace SN

c (§E.7.2.0.2) is a linear transformation of the dual EDM

cone: EDMN∗− δ(EDMN∗1). Secondarily, it means the EDM cone is not

selfdual in SN .

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6.8. DUAL EDM CONE 531

6.8.1.7 Schoenberg criterion is discretized membership relation

We show the Schoenberg criterion

−V TNDVN ∈ SN−1

+

D ∈ SNh

⇔ D ∈ EDMN (886)

to be a discretized membership relation (§2.13.4) between a closed convexcone K and its dual K∗

like

〈y , x〉 ≥ 0 for all y ∈ G(K∗) ⇔ x ∈ K (344)

where G(K∗) is any set of generators whose conic hull constructs closed

convex dual cone K∗:

The Schoenberg criterion is the same as

〈zzT,−D〉 ≥ 0 ∀ zzT | 11TzzT = 0

D ∈ SNh

⇔ D ∈ EDMN (1211)

which, by (1212), is the same as

〈zzT,−D〉 ≥ 0 ∀ zzT∈

VN υυTV TN | υ∈RN−1

D ∈ SNh

⇔ D ∈ EDMN (1266)

where the zzT constitute a set of generators G for the positive semidefinitecone’s smallest face F

(

SN+ ∋V

)

(§6.7.1) that contains auxiliary matrix V .From the aggregate in (1224) we get the ordinary membership relation,assuming only D∈ SN [196, p.58]

〈D∗, D〉 ≥ 0 ∀D∗∈ EDMN∗ ⇔ D ∈ EDMN

〈D∗, D〉 ≥ 0 ∀D∗∈ δ(u) | u∈RN − cone

VN υυTV TN | υ∈RN−1

⇔ D ∈ EDMN

(1267)

Discretization (344) yields:

〈D∗, D〉 ≥ 0 ∀D∗∈ eieTi , −eje

Tj , −VN υυTV T

N | i , j =1 . . . N , υ∈RN−1 ⇔ D ∈ EDMN

(1268)

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532 CHAPTER 6. CONE OF DISTANCE MATRICES

Because⟨

δ(u) | u∈RN , D⟩

≥ 0 ⇔ D∈ SNh , we can restrict observation

to the symmetric hollow subspace without loss of generality. Then for D∈ SNh

〈D∗, D〉 ≥ 0 ∀D∗ ∈

−VN υυTV TN | υ∈RN−1

⇔ D ∈ EDMN (1269)

this discretized membership relation becomes (1266); identical to theSchoenberg criterion.

Hitherto a correspondence between the EDM cone and a face of a PSDcone, the Schoenberg criterion is now accurately interpreted as a discretizedmembership relation between the EDM cone and its ordinary dual.

6.8.2 Ambient SNh

When instead we consider the ambient space of symmetric hollow matrices(1225), then still we find the EDM cone is not selfdual for N > 2. Thesimplest way to prove this is as follows:

Given a set of generators G= Γ (1188) for the pointed closed convexEDM cone, the discretized membership theorem in §2.13.4.2.1 asserts thatmembers of the dual EDM cone in the ambient space of symmetric hollowmatrices can be discerned via discretized membership relation:

EDMN∗∩ SNh , D∗∈ SN

h | 〈Γ , D∗〉 ≥ 0 ∀Γ ∈ G(EDMN)= D∗∈ SN

h | 〈δ(zzT)1T + 1δ(zzT)T− 2zzT , D∗〉 ≥ 0 ∀ z∈N (1T)= D∗∈ SN

h | 〈1δ(zzT)T− zzT , D∗〉 ≥ 0 ∀ z∈N (1T)

(1270)

By comparison

EDMN = D ∈ SNh | 〈−zzT , D〉 ≥ 0 ∀ z∈N (1T) (1271)

the term δ(zzT)TD∗1 foils any hope of selfdualness in ambient SNh . ¨

To find the dual EDM cone in ambient SNh per §2.13.9.4 we prune the

aggregate in (1224) describing the ordinary dual EDM cone, removing anymember having nonzero main diagonal:

EDMN∗∩ SNh = cone

δ2(VN υυTV TN ) − VN υυTV T

N | υ∈RN−1

= δ2(VNΨV TN ) − VNΨV T

N | Ψ∈ SN−1+

(1272)

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6.8. DUAL EDM CONE 533

dvec rel ∂EDM3

dvec rel ∂(EDM3∗∩ S3

h)

0

D∗∈ EDMN∗ ⇔ δ(D∗1) − D∗ º 0 (1250)

Figure 149: Ordinary dual EDM cone projected on S3

h shrouds EDM3 ;drawn tiled in isometrically isomorphic R3. (It so happens: intersection

EDMN∗∩ SNh (§2.13.9.3) is identical to projection of dual EDM cone on SN

h .)

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534 CHAPTER 6. CONE OF DISTANCE MATRICES

When N = 1, the EDM cone and its dual in ambient Sh each comprisethe origin in isomorphic R0 ; thus, selfdual in this dimension. (confer (103))

When N = 2, the EDM cone is the nonnegative real line in isomorphic R .(Figure 144) EDM2∗ in S2

h is identical, thus selfdual in this dimension.This result is in agreement with (1270), verified directly: for all κ∈R ,

z = κ

[

1−1

]

and δ(zzT) = κ2

[

11

]

⇒ d∗12≥ 0.

The first case adverse to selfdualness N = 3 may be deduced fromFigure 136; the EDM cone is a circular cone in isomorphic R3 correspondingto no rotation of Lorentz cone (169) (the selfdual circular cone). Figure 149illustrates the EDM cone and its dual in ambient S3

h ; no longer selfdual.

6.9 Theorem of the alternative

In §2.13.2.1.1 we showed how alternative systems of generalized inequalitycan be derived from closed convex cones and their duals. This section is,therefore, a fitting postscript to the discussion of the dual EDM cone.

6.9.0.0.1 Theorem. EDM alternative. [159, §1]Given D ∈ SN

h

D ∈ EDMN

or in the alternative

∃ z such that

1Tz = 1

Dz = 0

(1273)

In words, either N (D) intersects hyperplane z | 1Tz =1 or D is an EDM;the alternatives are incompatible. ⋄

When D is an EDM [255, §2]

N (D) ⊂ N (1T) = z | 1Tz = 0 (1274)

Because [159, §2] (§E.0.1)DD†1 = 1

1TD†D = 1T (1275)

thenR(1) ⊂ R(D) (1276)

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6.10. POSTSCRIPT 535

6.10 Postscript

We provided an equality (1244) relating the convex cone of Euclidean distancematrices to the convex cone of positive semidefinite matrices. Projection ona positive semidefinite cone, constrained by an upper bound on rank, is easyand well known; [126] simply, a matter of truncating a list of eigenvalues.Projection on a positive semidefinite cone with such a rank constraint is, infact, a convex optimization problem. (§7.1.4)

In the past, it was difficult to project on the EDM cone under aconstraint on rank or affine dimension. A surrogate method was to invokethe Schoenberg criterion (886) and then project on a positive semidefinitecone under a rank constraint bounding affine dimension from above. But asolution acquired that way is necessarily suboptimal.

In §7.3.3 we present a method for projecting directly on the EDM coneunder a constraint on rank or affine dimension.

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536 CHAPTER 6. CONE OF DISTANCE MATRICES

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Chapter 7

Proximity problems

In the “extremely large-scale case” (N of order of tens andhundreds of thousands), [iteration cost O(N3)] rules out alladvanced convex optimization techniques, including all knownpolynomial time algorithms.

−Arkadi Nemirovski (2004)

A problem common to various sciences is to find the Euclidean distancematrix (EDM) D∈ EDMN closest in some sense to a given complete matrixof measurements H under a constraint on affine dimension 0≤ r≤N−1(§2.3.1, §5.7.1.1); rather, r is bounded above by desired affine dimension ρ .

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

537

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538 CHAPTER 7. PROXIMITY PROBLEMS

7.0.1 Measurement matrix H

Ideally, we want a given matrix of measurements H∈ RN×N to conformwith the first three Euclidean metric properties (§5.2); to belong to theintersection of the orthant of nonnegative matrices RN×N

+ with the symmetrichollow subspace SN

h (§2.2.3.0.1). Geometrically, we want H to belong to thepolyhedral cone (§2.12.1.0.1)

K , SNh ∩ RN×N

+ (1277)

Yet in practice, H can possess significant measurement uncertainty (noise).

Sometimes realization of an optimization problem demands that itsinput, the given matrix H , possess some particular characteristics; perhapssymmetry and hollowness or nonnegativity. When that H given does nothave the desired properties, then we must impose them upon H prior tooptimization: When measurement matrix H is not symmetric or hollow, taking its

symmetric hollow part is equivalent to orthogonal projection on thesymmetric hollow subspace SN

h . When measurements of distance in H are negative, zeroing negativeentries effects unique minimum-distance projection on the orthant ofnonnegative matrices RN×N

+ in isomorphic RN 2

(§E.9.2.2.3).

7.0.1.1 Order of imposition

Since convex cone K (1277) is the intersection of an orthant with a subspace,we want to project on that subset of the orthant belonging to the subspace;on the nonnegative orthant in the symmetric hollow subspace that is, in fact,the intersection. For that reason alone, unique minimum-distance projectionof H on K (that member of K closest to H in isomorphic RN 2

in the Euclideansense) can be attained by first taking its symmetric hollow part, and onlythen clipping negative entries of the result to 0 ; id est, there is only onecorrect order of projection, in general, on an orthant intersecting a subspace:

Page 539: Convex Optimization & Euclidean Distance Geometry

539 project on the subspace, then project the result on the orthant in thatsubspace. (confer §E.9.5)

In contrast, order of projection on an intersection of subspaces is arbitrary.

That order-of-projection rule applies more generally, of course, tothe intersection of any convex set C with any subspace. Consider theproximity problem7.1 over convex feasible set SN

h ∩ C given nonsymmetricnonhollow H∈ RN×N :

minimizeB∈SN

h

‖B − H‖2F

subject to B ∈ C(1278)

a convex optimization problem. Because the symmetric hollow subspace SNh

is orthogonal to the antisymmetric antihollow subspace RN×N⊥h (§2.2.3), then

for B∈ SNh

tr

(

BT

(

1

2(H−HT) + δ2(H)

))

= 0 (1279)

so the objective function is equivalent to

‖B − H‖2F ≡

B −(

1

2(H +HT) − δ2(H)

)∥

2

F

+

1

2(H−HT) + δ2(H)

2

F

(1280)

This means the antisymmetric antihollow part of given matrix H wouldbe ignored by minimization with respect to symmetric hollow variable Bunder Frobenius’ norm; id est, minimization proceeds as though given thesymmetric hollow part of H .

This action of Frobenius’ norm (1280) is effectively a Euclidean projection(minimum-distance projection) of H on the symmetric hollow subspace SN

h

prior to minimization. Thus minimization proceeds inherently following thecorrect order for projection on SN

h ∩ C . Therefore we may either assumeH∈ SN

h , or take its symmetric hollow part prior to optimization.

7.1There are two equivalent interpretations of projection (§E.9): one finds a set normal,the other, minimum distance between a point and a set. Here we realize the latter view.

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540 CHAPTER 7. PROXIMITY PROBLEMS

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bb

bb

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bb

bb

bb

bb

bb

bb

bb

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bb

bb

bb

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K = SNh ∩ RN×N

+

EDMN

SNh

SN

RN×N

0 ©©©©©©©©©©©

cc

cc

cc

cc

cc

cc

hhhhhhhhhhhhhhhhhhh

Figure 150: Pseudo-Venn diagram: The EDM cone belongs to theintersection of the symmetric hollow subspace with the nonnegative orthant;EDMN ⊆ K (866). EDMN cannot exist outside SN

h , but RN×N+ does.

7.0.1.2 Egregious input error under nonnegativity demand

More pertinent to the optimization problems presented herein where

C , EDMN ⊆ K = SNh ∩ RN×N

+ (1281)

then should some particular realization of a proximity problem demandinput H be nonnegative, and were we only to zero negative entries of anonsymmetric nonhollow input H prior to optimization, then the ensuingprojection on EDMN would be guaranteed incorrect (out of order).

Now comes a surprising fact: Even were we to correctly follow theorder-of-projection rule and provide H ∈ K prior to optimization, then theensuing projection on EDMN will be incorrect whenever input H has negativeentries and some proximity problem demands nonnegative input H .

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541

H SNh

K = SNh ∩ RN×N

+

0

EDMN

Figure 151: Pseudo-Venn diagram from Figure 150 showing elbow placedin path of projection of H on EDMN ⊂ SN

h by an optimization problemdemanding nonnegative input matrix H . The first two line segmentsleading away from H result from correct order-of-projection required toprovide nonnegative H prior to optimization. Were H nonnegative, then itsprojection on SN

h would instead belong to K ; making the elbow disappear.(confer Figure 163)

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542 CHAPTER 7. PROXIMITY PROBLEMS

This is best understood referring to Figure 150: Suppose nonnegativeinput H is demanded, and then the problem realization correctly projectsits input first on SN

h and then directly on C= EDMN . That demandfor nonnegativity effectively requires imposition of K on input H prior tooptimization so as to obtain correct order of projection (on SN

h first). Yetsuch an imposition prior to projection on EDMN generally introduces anelbow into the path of projection (illustrated in Figure 151) caused bythe technique itself; that being, a particular proximity problem realizationrequiring nonnegative input.

Any procedure for imposition of nonnegativity on input H can only beincorrect in this circumstance. There is no resolution unless input H isguaranteed nonnegative with no tinkering. Otherwise, we have no choice butto employ a different problem realization; one not demanding nonnegativeinput.

7.0.2 Lower bound

Most of the problems we encounter in this chapter have the general form:

minimizeB

‖B − A‖F

subject to B ∈ C(1282)

where A∈ Rm×n is given data. This particular objective denotes Euclideanprojection (§E) of vectorized matrix A on the set C which may or may not beconvex. When C is convex, then the projection is unique minimum-distancebecause Frobenius’ norm when squared is a strictly convex function ofvariable B and because the optimal solution is the same regardless of thesquare (491). When C is a subspace, then the direction of projection isorthogonal to C .

Denoting by A,UAΣAQTA and B,UBΣBQT

B their full singular valuedecompositions (whose singular values are always nonincreasingly ordered(§A.6)), there exists a tight lower bound on the objective over the manifoldof orthogonal matrices;

‖ΣB − ΣA‖F ≤ infUA ,UB ,QA ,QB

‖B − A‖F (1283)

This least lower bound holds more generally for any orthogonally invariantnorm on Rm×n (§2.2.1) including the Frobenius and spectral norm [325, §II.3].[199, §7.4.51]

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543

7.0.3 Problem approach

Problems traditionally posed in terms of point position xi∈ Rn, i=1 . . . Nsuch as

minimizexi

i , j ∈ I(‖xi − xj‖ − hij)

2 (1284)

or

minimizexi

i , j ∈ I(‖xi − xj‖2 − hij)

2 (1285)

(where I is an abstract set of indices and hij is given data) are everywhereconverted herein to the distance-square variable D or to Gram matrix G ;the Gram matrix acting as bridge between position and distance. (Thatconversion is performed regardless of whether known data is complete.)Then the techniques of chapter 5 or chapter 6 are applied to find relativeor absolute position. This approach is taken because we prefer introductionof rank constraints into convex problems rather than searching a googol oflocal minima in nonconvex problems like (1284) [102] (§3.7.4.0.2, §7.2.2.7.1)or (1285).

7.0.4 Three prevalent proximity problems

There are three statements of the closest-EDM problem prevalent in theliterature, the multiplicity due primarily to choice of projection on theEDM versus positive semidefinite (PSD) cone and vacillation between thedistance-square variable dij versus absolute distance

dij . In their mostfundamental form, the three prevalent proximity problems are (1286.1),(1286.2), and (1286.3): [339] for D , [dij] and

√D , [

dij ]

(1)

minimizeD

‖−V (D − H)V ‖2F

subject to rank V DV ≤ ρ

D ∈ EDMN

minimize√

D‖ √

D − H‖2F

subject to rank V DV ≤ ρ√

D ∈√

EDMN

(2)

(3)

minimizeD

‖D − H‖2F

subject to rank V DV ≤ ρ

D ∈ EDMN

minimize√

D‖−V (

√D − H)V ‖2

F

subject to rank V DV ≤ ρ√

D ∈√

EDMN

(4)

(1286)

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544 CHAPTER 7. PROXIMITY PROBLEMS

where we have made explicit an imposed upper bound ρ on affine dimension

r = rank V TNDVN = rank V DV (1018)

that is benign when ρ =N−1 or H were realizable with r≤ ρ . Problems(1286.2) and (1286.3) are Euclidean projections of a vectorized matrix H onan EDM cone (§6.3), whereas problems (1286.1) and (1286.4) are Euclideanprojections of a vectorized matrix −VHV on a PSD cone.7.2 Problem(1286.4) is not posed in the literature because it has limited theoreticalfoundation.7.3

Analytical solution to (1286.1) is known in closed form for any bound ρalthough, as the problem is stated, it is a convex optimization only in the caseρ =N−1. We show in §7.1.4 how (1286.1) becomes a convex optimizationproblem for any ρ when transformed to the spectral domain. When expressedas a function of point list in a matrix X as in (1284), problem (1286.2)becomes a variant of what is known in statistics literature as a stress problem.[51, p.34] [100] [351] Problems (1286.2) and (1286.3) are convex optimizationproblems in D for the case ρ =N−1 wherein (1286.3) becomes equivalentto (1285). Even with the rank constraint removed from (1286.2), we will seethat the convex problem remaining inherently minimizes affine dimension.

Generally speaking, each problem in (1286) produces a different resultbecause there is no isometry relating them. Of the various auxiliaryV -matrices (§B.4), the geometric centering matrix V (889) appears in theliterature most often although VN (873) is the auxiliary matrix naturallyconsequent to Schoenberg’s seminal exposition [308]. Substitution of anyauxiliary matrix or its pseudoinverse into these problems produces anothervalid problem.

Substitution of V TN for left-hand V in (1286.1), in particular, produces a

different result because

minimizeD

‖−V (D − H)V ‖2F

subject to D ∈ EDMN(1287)

7.2Because −VHV is orthogonal projection of −H on the geometric center subspace SNc

(§E.7.2.0.2), problems (1286.1) and (1286.4) may be interpreted as oblique (nonminimumdistance) projections of −H on a positive semidefinite cone.7.3 D∈EDMN ⇒

√D∈EDMN , −V

√DV ∈ SN

+ (§5.10)

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7.1. FIRST PREVALENT PROBLEM: 545

finds D to attain Euclidean distance of vectorized −VHV to the positivesemidefinite cone in ambient isometrically isomorphic RN(N+1)/2, whereas

minimizeD

‖−V TN (D − H)VN‖2

F

subject to D ∈ EDMN(1288)

attains Euclidean distance of vectorized −V TNHVN to the positive

semidefinite cone in isometrically isomorphic subspace RN(N−1)/2 ; quitedifferent projections7.4 regardless of whether affine dimension is constrained.But substitution of auxiliary matrix V T

W (§B.4.3) or V †N yields the same

result as (1286.1) because V = VWV TW = VNV †

N ; id est,

‖−V (D − H)V ‖2F = ‖−VWV T

W(D − H)VWV TW‖2

F = ‖−V TW(D − H)VW‖2

F

= ‖−VNV †N (D − H)VNV †

N‖2F = ‖−V †

N (D − H)VN‖2F

(1289)

We see no compelling reason to prefer one particular auxiliary V -matrixover another. Each has its own coherent interpretations; e.g., §5.4.2, §6.7,§B.4.5. Neither can we say any particular problem formulation producesgenerally better results than another.7.5

7.1 First prevalent problem:

Projection on PSD cone

This first problem

minimizeD

‖−V TN (D − H)VN‖2

F

subject to rank V TNDVN ≤ ρ

D ∈ EDMN

Problem 1 (1290)

7.4The isomorphism T (Y )=V †TN Y V †

N onto SNc = V X V | X∈ SN relates the map in

(1288) to that in (1287), but is not an isometry.7.5All four problem formulations (1286) produce identical results when affine

dimension r , implicit to a realizable measurement matrix H , does not exceed desiredaffine dimension ρ ; because, the optimal objective value will vanish (‖ · ‖ = 0).

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546 CHAPTER 7. PROXIMITY PROBLEMS

poses a Euclidean projection of −V TNHVN in subspace SN−1 on a generally

nonconvex subset (when ρ <N−1) of the positive semidefinite coneboundary ∂SN−1

+ whose elemental matrices have rank no greater than desiredaffine dimension ρ (§5.7.1.1). Problem 1 finds the closest EDM D inthe sense of Schoenberg. (886) [308] As it is stated, this optimizationproblem is convex only when desired affine dimension is largest ρ =N−1although its analytical solution is known [253, thm.14.4.2] for all nonnegativeρ≤N−1 .7.6

We assume only that the given measurement matrix H is symmetric;7.7

H ∈ SN (1291)

Arranging the eigenvalues λi of −V TNHVN in nonincreasing order for all i ,

λi ≥ λi+1 with vi the corresponding ith eigenvector, then an optimal solutionto Problem 1 is [350, §2]

−V TND⋆VN =

ρ∑

i=1

max0, λi vivTi (1292)

where

−V TNHVN ,

N−1∑

i=1

λi vivTi ∈ SN−1 (1293)

is an eigenvalue decomposition and

D⋆ ∈ EDMN (1294)

is an optimal Euclidean distance matrix.

In §7.1.4 we show how to transform Problem 1 to a convex optimizationproblem for any ρ .

7.6 being first pronounced in the context of multidimensional scaling by Mardia [252] in1978 who attributes the generic result (§7.1.2) to Eckart & Young, 1936 [126].7.7Projection in Problem 1 is on a rank ρ subset of the positive semidefinite cone SN−1

+

(§2.9.2.1) in the subspace of symmetric matrices SN−1. It is wrong here to zero the maindiagonal of given H because first projecting H on the symmetric hollow subspace placesan elbow in the path of projection in Problem 1. (Figure 151)

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7.1. FIRST PREVALENT PROBLEM: 547

7.1.1 Closest-EDM Problem 1, convex case

7.1.1.0.1 Proof. Solution (1292), convex case.When desired affine dimension is unconstrained, ρ =N−1, the rank functiondisappears from (1290) leaving a convex optimization problem; a simpleunique minimum-distance projection on the positive semidefinite cone SN−1

+ :videlicet

minimizeD∈ SN

h

‖−V TN (D − H)VN‖2

F

subject to −V TNDVN º 0

(1295)

by (886). Because

SN−1 = −V TN SN

h VN (987)

then the necessary and sufficient conditions for projection in isometricallyisomorphic RN(N−1)/2 on the selfdual (355) positive semidefinite cone SN−1

+

are:7.8 (§E.9.2.0.1) (1556) (confer (1993))

−V TND⋆VN º 0

−V TND⋆VN

(

−V TND⋆VN + V T

NHVN)

= 0−V T

ND⋆VN + V TNHVN º 0

(1296)

Symmetric −V TNHVN is diagonalizable hence decomposable in terms of its

eigenvectors v and eigenvalues λ as in (1293). Therefore (confer (1292))

−V TND⋆VN =

N−1∑

i=1

max0, λivivTi (1297)

satisfies (1296), optimally solving (1295). To see that, recall: theseeigenvectors constitute an orthogonal set and

−V TND⋆VN + V T

NHVN = −N−1∑

i=1

min0, λivivTi (1298)

¨

7.8The Karush-Kuhn-Tucker (KKT) optimality conditions [276, p.328] [59, §5.5.3] forproblem (1295) are identical to these conditions for projection on a convex cone.

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548 CHAPTER 7. PROXIMITY PROBLEMS

7.1.2 Generic problem

Prior to determination of D⋆, analytical solution (1292) to Problem 1 isequivalent to solution of a generic rank-constrained projection problem:Given desired affine dimension ρ and

A , −V TNHVN =

N−1∑

i=1

λi vivTi ∈ SN−1 (1293)

Euclidean projection on a rank ρ subset of a PSD cone (on a generallynonconvex subset of the PSD cone boundary ∂SN−1

+ when ρ <N−1)

minimizeB∈SN−1

‖B − A‖2F

subject to rank B ≤ ρ

B º 0

Generic 1 (1299)

has well known optimal solution (Eckart & Young) [126]

B⋆ , −V TND⋆VN =

ρ∑

i=1

max0, λi vivTi ∈ SN−1 (1292)

Once optimal B⋆ is found, the technique of §5.12 can be used to determinea uniquely corresponding optimal Euclidean distance matrix D⋆ ; a uniquecorrespondence by injectivity arguments in §5.6.2.

7.1.2.1 Projection on rank ρ subset of PSD cone

Because Problem 1 is the same as

minimizeD∈ SN

h

‖−V TN (D − H)VN‖2

F

subject to rank V TNDVN ≤ ρ

−V TNDVN º 0

(1300)

and because (987) provides invertible mapping to the generic problem,then Problem 1 is truly a Euclidean projection of vectorized −V T

NHVN onthat generally nonconvex subset of symmetric matrices (belonging to thepositive semidefinite cone SN−1

+ ) having rank no greater than desired affine

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7.1. FIRST PREVALENT PROBLEM: 549

dimension ρ ; 7.9 called rank ρ subset: (240)

SN−1+ \SN−1

+ (ρ + 1) = X∈ SN−1+ | rank X ≤ ρ (207)

7.1.3 Choice of spectral cone

Spectral projection substitutes projection on a polyhedral cone, containinga complete set of eigenspectra, in place of projection on a convex set ofdiagonalizable matrices; e.g., (1312). In this section we develop a method ofspectral projection for constraining rank of positive semidefinite matrices ina proximity problem like (1299). We will see why an orthant turns out to bethe best choice of spectral cone, and why presorting is critical.

Define a nonlinear permutation-operator π(x) : Rn→ Rn that sorts itsvector argument x into nonincreasing order.

7.1.3.0.1 Definition. Spectral projection.Let R be an orthogonal matrix and Λ a nonincreasingly ordered diagonalmatrix of eigenvalues. Spectral projection means unique minimum-distanceprojection of a rotated (R , §B.5.4) nonincreasingly ordered (π) vector (δ)of eigenvalues

π(

δ(RTΛR))

(1301)

on a polyhedral cone containing all eigenspectra corresponding to a rank ρsubset of a positive semidefinite cone (§2.9.2.1) or the EDM cone (inCayley-Menger form, §5.11.2.3).

In the simplest and most common case, projection on a positivesemidefinite cone, orthogonal matrix R equals I (§7.1.4.0.1) and diagonalmatrix Λ is ordered during diagonalization (§A.5.1). Then spectralprojection simply means projection of δ(Λ) on a subset of the nonnegativeorthant, as we shall now ascertain:

It is curious how nonconvex Problem 1 has such a simple analyticalsolution (1292). Although solution to generic problem (1299) is well knownsince 1936 [126], its equivalence was observed in 1997 [350, §2] to projection

7.9Recall: affine dimension is a lower bound on embedding (§2.3.1), equal to dimensionof the smallest affine set in which points from a list X corresponding to an EDM D canbe embedded.

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550 CHAPTER 7. PROXIMITY PROBLEMS

of an ordered vector of eigenvalues (in diagonal matrix Λ) on a subset of themonotone nonnegative cone (§2.13.9.4.2)

KM+ = υ | υ1 ≥ υ2 ≥ · · · ≥ υN−1 ≥ 0 ⊆ RN−1+ (407)

Of interest, momentarily, is only the smallest convex subset of themonotone nonnegative cone KM+ containing every nonincreasingly orderedeigenspectrum corresponding to a rank ρ subset of the positive semidefinitecone SN−1

+ ; id est,

KρM+ , υ∈Rρ | υ1 ≥ υ2 ≥ · · · ≥ υρ ≥ 0 ⊆ Rρ

+ (1302)

a pointed polyhedral cone, a ρ-dimensional convex subset of themonotone nonnegative cone KM+⊆ RN−1

+ having property, for λ denotingeigenspectra,

[

KρM+

0

]

= π(λ(rank ρ subset)) ⊆ KN−1M+ , KM+ (1303)

For each and every elemental eigenspectrum

γ ∈ λ(rank ρ subset)⊆ RN−1+ (1304)

of the rank ρ subset (ordered or unordered in λ), there is a nonlinearsurjection π(γ) onto Kρ

M+ .

7.1.3.0.2 Exercise. Smallest spectral cone.Prove that there is no convex subset of KM+ smaller than Kρ

M+ containingevery ordered eigenspectrum corresponding to the rank ρ subset of a positivesemidefinite cone (§2.9.2.1). H

7.1.3.0.3 Proposition. (Hardy-Littlewood-Polya) Inequalities. [175, §X][53, §1.2] Any vectors σ and γ in RN−1 satisfy a tight inequality

π(σ)Tπ(γ) ≥ σTγ ≥ π(σ)TΞπ(γ) (1305)

where Ξ is the order-reversing permutation matrix defined in (1691),and permutator π(γ) is a nonlinear function that sorts vector γ intononincreasing order thereby providing the greatest upper bound and leastlower bound with respect to every possible sorting. ⋄

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7.1. FIRST PREVALENT PROBLEM: 551

7.1.3.0.4 Corollary. Monotone nonnegative sort.Any given vectors σ, γ∈RN−1 satisfy a tight Euclidean distance inequality

‖π(σ) − π(γ)‖ ≤ ‖σ − γ‖ (1306)

where nonlinear function π(γ) sorts vector γ into nonincreasing orderthereby providing the least lower bound with respect to every possiblesorting. ⋄

Given γ∈RN−1

infσ∈R

N−1+

‖σ−γ‖ = infσ∈R

N−1+

‖π(σ)−π(γ)‖ = infσ∈R

N−1+

‖σ−π(γ)‖ = infσ∈K

M+

‖σ−π(γ)‖

(1307)Yet for γ representing an arbitrary vector of eigenvalues, because

infσ∈

[

Rρ+

0

]

‖σ − γ‖2 ≥ infσ∈

[

Rρ+

0

]

‖σ − π(γ)‖2 = infσ∈

[

KρM+

0

]

‖σ − π(γ)‖2 (1308)

then projection of γ on the eigenspectra of a rank ρ subset can be tightenedsimply by presorting γ into nonincreasing order.

Proof. Simply because π(γ)1:ρ º π(γ1:ρ)

infσ∈

[

Rρ+

0

]

‖σ − γ‖2 = γTρ+1:N−1γρ+1:N−1 + inf

σ∈RN−1+

‖σ1:ρ − γ1:ρ‖2

= γTγ + infσ∈R

N−1+

σT1:ρσ1:ρ − 2σT

1:ργ1:ρ

≥ γTγ + infσ∈R

N−1+

σT1:ρσ1:ρ − 2σT

1:ρπ(γ)1:ρ

infσ∈

[

Rρ+

0

]

‖σ − γ‖2 ≥ infσ∈

[

Rρ+

0

]

‖σ − π(γ)‖2

(1309)

¨

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552 CHAPTER 7. PROXIMITY PROBLEMS

7.1.3.1 Orthant is best spectral cone for Problem 1

This means unique minimum-distance projection of γ on the nearestspectral member of the rank ρ subset is tantamount to presorting γ intononincreasing order. Only then does unique spectral projection on a subsetKρ

M+ of the monotone nonnegative cone become equivalent to unique spectralprojection on a subset Rρ

+ of the nonnegative orthant (which is simpler);in other words, unique minimum-distance projection of sorted γ on thenonnegative orthant in a ρ-dimensional subspace of RN is indistinguishablefrom its projection on the subset Kρ

M+ of the monotone nonnegative cone inthat same subspace.

7.1.4 Closest-EDM Problem 1, “nonconvex” case

Proof of solution (1292), for projection on a rank ρ subset of the positivesemidefinite cone SN−1

+ , can be algebraic in nature. [350, §2] Here we derivethat known result but instead using a more geometric argument via spectralprojection on a polyhedral cone (subsuming the proof in §7.1.1). In sodoing, we demonstrate how nonconvex Problem 1 is transformed to a convexoptimization:

7.1.4.0.1 Proof. Solution (1292), nonconvex case.As explained in §7.1.2, we may instead work with the more facile genericproblem (1299). With diagonalization of unknown

B , UΥUT∈ SN−1 (1310)

given desired affine dimension 0≤ ρ≤N−1 and diagonalizable

A , QΛQT∈ SN−1 (1311)

having eigenvalues in Λ arranged in nonincreasing order, by (48) the genericproblem is equivalent to

minimizeB∈SN−1

‖B − A‖2F

subject to rank B ≤ ρ

B º 0

minimizeR , Υ

‖Υ − RTΛR‖2F

subject to rank Υ ≤ ρΥ º 0R−1 = RT

(1312)

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7.1. FIRST PREVALENT PROBLEM: 553

where

R , QTU ∈ RN−1×N−1 (1313)

in U on the set of orthogonal matrices is a linear bijection. We proposesolving (1312) by instead solving the problem sequence:

minimizeΥ

‖Υ − RTΛR‖2F

subject to rank Υ ≤ ρΥ º 0

(a)

minimizeR

‖Υ⋆ − RTΛR‖2F

subject to R−1 = RT(b)

(1314)

Problem (1314a) is equivalent to: (1) orthogonal projection of RTΛRon an N− 1-dimensional subspace of isometrically isomorphic RN(N−1)/2

containing δ(Υ)∈ RN−1+ , (2) nonincreasingly ordering the result, (3) unique

minimum-distance projection of the ordered result on

[

Rρ+

0

]

. (§E.9.5)

Projection on that N−1-dimensional subspace amounts to zeroing RTΛR atall entries off the main diagonal; thus, the equivalent sequence leading witha spectral projection:

minimizeΥ

‖ δ(Υ) − π(

δ(RTΛR))

‖2

subject to δ(Υ) ∈[

Rρ+

0

]

(a)

minimizeR

‖Υ⋆ − RTΛR‖2F

subject to R−1 = RT(b)

(1315)

Because any permutation matrix is an orthogonal matrix, δ(RTΛR)∈ RN−1

can always be arranged in nonincreasing order without loss of generality;hence, permutation operator π . Unique minimum-distance projection

of vector π(

δ(RTΛR))

on the ρ-dimensional subset

[

Rρ+

0

]

of nonnegative

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554 CHAPTER 7. PROXIMITY PROBLEMS

orthant RN−1+ requires: (§E.9.2.0.1)

δ(Υ⋆)ρ+1:N−1 = 0

δ(Υ⋆) º 0

δ(Υ⋆)T(

δ(Υ⋆) − π(δ(RTΛR)))

= 0

δ(Υ⋆) − π(δ(RTΛR)) º 0

(1316)

which are necessary and sufficient conditions. Any value Υ⋆ satisfyingconditions (1316) is optimal for (1315a). So

δ(Υ⋆)i =

max

0 , π(

δ(RTΛR))

i

, i=1 . . . ρ

0 , i=ρ+1 . . . N−1(1317)

specifies an optimal solution. The lower bound on the objective with respectto R in (1315b) is tight: by (1283)

‖ |Υ⋆| − |Λ| ‖F ≤ ‖Υ⋆ − RTΛR‖F (1318)

where | | denotes absolute entry-value. For selection of Υ⋆ as in (1317), thislower bound is attained when (confer §C.4.2.2)

R⋆ = I (1319)

which is the known solution. ¨

7.1.4.1 Significance

Importance of this well-known [126] optimal solution (1292) for projectionon a rank ρ subset of a positive semidefinite cone should not be dismissed: Problem 1, as stated, is generally nonconvex. This analytical solution

at once encompasses projection on a rank ρ subset (207) of the positivesemidefinite cone (generally, a nonconvex subset of its boundary)from either the exterior or interior of that cone.7.10 By problemtransformation to the spectral domain, projection on a rank ρ subsetbecomes a convex optimization problem.

7.10Projection on the boundary from the interior of a convex Euclidean body is generallya nonconvex problem.

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7.1. FIRST PREVALENT PROBLEM: 555 This solution is closed form. This solution is equivalent to projection on a polyhedral cone inthe spectral domain (spectral projection, projection on a spectralcone §7.1.3.0.1); a necessary and sufficient condition (§A.3.1) formembership of a symmetric matrix to a rank ρ subset of a positivesemidefinite cone (§2.9.2.1). Because U⋆ = Q , a minimum-distance projection on a rank ρ subsetof the positive semidefinite cone is a positive semidefinite matrixorthogonal (in the Euclidean sense) to direction of projection.7.11 For the convex case problem (1295), this solution is always unique.Otherwise, distinct eigenvalues (multiplicity 1) in Λ guaranteesuniqueness of this solution by the reasoning in §A.5.0.1 .7.12

7.1.5 Problem 1 in spectral norm, convex case

When instead we pose the matrix 2-norm (spectral norm) in Problem 1(1290) for the convex case ρ = N−1, then the new problem

minimizeD

‖−V TN (D − H)VN‖2

subject to D ∈ EDMN(1320)

is convex although its solution is not necessarily unique;7.13 giving rise tononorthogonal projection (§E.1) on the positive semidefinite cone SN−1

+ .Indeed, its solution set includes the Frobenius solution (1292) for the convexcase whenever −V T

NHVN is a normal matrix. [180, §1] [173] [59, §8.1.1]Proximity problem (1320) is equivalent to

minimizeµ , D

µ

subject to −µI ¹ −V TN (D − H)VN ¹ µI

D ∈ EDMN

(1321)

7.11But Theorem E.9.2.0.1 for unique projection on a closed convex cone does not applyhere because the direction of projection is not necessarily a member of the dual PSD cone.This occurs, for example, whenever positive eigenvalues are truncated.7.12Uncertainty of uniqueness prevents the erroneous conclusion that a rank ρ subset(207) were a convex body by the Bunt-Motzkin theorem (§E.9.0.0.1).

7.13For each and every |t|≤ 2, for example,

[

2 00 t

]

has the same spectral-norm value.

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556 CHAPTER 7. PROXIMITY PROBLEMS

by (1667) where

µ⋆ = maxi

∣λ(

−V TN (D⋆ − H)VN

)

i

∣ , i = 1 . . . N−1

∈ R+ (1322)

the minimized largest absolute eigenvalue (due to matrix symmetry).For lack of a unique solution here, we prefer the Frobenius rather than

spectral norm.

7.2 Second prevalent problem:

Projection on EDM cone in√

dij

Let√

D , [√

dij ] ∈ K = SNh ∩ RN×N

+ (1323)

be an unknown matrix of absolute distance; id est,

D = [dij] ,√

D √

D ∈ EDMN (1324)

where denotes Hadamard product. The second prevalent proximityproblem is a Euclidean projection (in the natural coordinates

dij ) of matrixH on a nonconvex subset of the boundary of the nonconvex cone of Euclidean

absolute-distance matrices rel ∂√

EDMN : (§6.3, confer Figure 136b)

minimize√

D‖ √

D − H‖2F

subject to rank V TNDVN ≤ ρ

D ∈√

EDMN

Problem 2 (1325)

where√

EDMN = √

D | D∈ EDMN (1158)

This statement of the second proximity problem is considered difficult tosolve because of the constraint on desired affine dimension ρ (§5.7.2) andbecause the objective function

‖ √

D − H‖2F =

i,j

(√

dij − hij)2 (1326)

is expressed in the natural coordinates; projection on a doubly nonconvexset.

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7.2. SECOND PREVALENT PROBLEM: 557

Our solution to this second problem prevalent in the literature requiresmeasurement matrix H to be nonnegative;

H = [hij] ∈ RN×N+ (1327)

If the H matrix given has negative entries, then the technique of solutionpresented here becomes invalid. As explained in §7.0.1, projection of Hon K= SN

h ∩ RN×N+ (1277) prior to application of this proposed solution is

incorrect.

7.2.1 Convex case

When ρ = N − 1, the rank constraint vanishes and a convex problem thatis equivalent to (1284) emerges:7.14

minimize√

D‖ √

D − H‖2F

subject to √

D ∈√

EDMN⇔

minimizeD

i,j

dij − 2hij

dij + h2ij

subject to D ∈ EDMN (1328)

For any fixed i and j , the argument of summation is a convex functionof dij because (for nonnegative constant hij) the negative square root isconvex in nonnegative dij and because dij + h2

ij is affine (convex). Becausethe sum of any number of convex functions in D remains convex [59, §3.2.1]and because the feasible set is convex in D , we have a convex optimizationproblem:

minimizeD

1T(D − 2H √

D )1 + ‖H‖2F

subject to D ∈ EDMN(1329)

The objective function being a sum of strictly convex functions is,moreover, strictly convex in D on the nonnegative orthant. Existenceof a unique solution D⋆ for this second prevalent problem depends uponnonnegativity of H and a convex feasible set (§3.1.2).7.15

7.14 still thought to be a nonconvex problem as late as 1997 [351] even though discoveredconvex by de Leeuw in 1993. [100] [51, §13.6] Yet using methods from §3, it can be easilyascertained: ‖

√D − H‖F is not convex in D .

7.15The transformed problem in variable D no longer describes Euclidean projection on

an EDM cone. Otherwise we might erroneously conclude√

EDMN were a convex bodyby the Bunt-Motzkin theorem (§E.9.0.0.1).

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558 CHAPTER 7. PROXIMITY PROBLEMS

7.2.1.1 Equivalent semidefinite program, Problem 2, convex case

Convex problem (1328) is numerically solvable for its global minimum usingan interior-point method [388] [286] [273] [379] [11] [141]. We translate (1328)to an equivalent semidefinite program (SDP) for a pedagogical reason madeclear in §7.2.2.2 and because there exist readily available computer programsfor numerical solution [163] [381] [382] [356] [33] [380] [345] [333].

Substituting a new matrix variable Y , [yij]∈RN×N+

hij

dij ← yij (1330)

Boyd proposes: problem (1328) is equivalent to the semidefinite program

minimizeD , Y

i,j

dij − 2yij + h2ij

subject to

[

dij yij

yij h2ij

]

º 0 , i,j =1 . . . N

D ∈ EDMN

(1331)

To see that, recall dij ≥ 0 is implicit to D∈ EDMN (§5.8.1, (886)). Sowhen H∈ RN×N

+ is nonnegative as assumed,

[

dij yij

yij h2ij

]

º 0 ⇔ hij

dij ≥√

y2ij (1332)

Minimization of the objective function implies maximization of yij that isbounded above. Hence nonnegativity of yij is implicit to (1331) and, asdesired, yij →hij

dij as optimization proceeds. ¨

If the given matrix H is now assumed symmetric and nonnegative,

H = [hij] ∈ SN ∩ RN×N+ (1333)

then Y = H √

D must belong to K= SNh ∩ RN×N

+ (1277). Because Y ∈ SNh

(§B.4.2 no.20), then

‖ √

D −H‖2F =

i,j

dij − 2yij + h2ij = −N tr(V (D − 2Y )V ) + ‖H‖2

F (1334)

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7.2. SECOND PREVALENT PROBLEM: 559

So convex problem (1331) is equivalent to the semidefinite program

minimizeD , Y

− tr(V (D − 2Y )V )

subject to

[

dij yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

Y ∈ SNh

D ∈ EDMN

(1335)

where the constants h2ij and N have been dropped arbitrarily from the

objective.

7.2.1.2 Gram-form semidefinite program, Problem 2, convex case

There is great advantage to expressing problem statement (1335) inGram-form because Gram matrix G is a bidirectional bridge between pointlist X and distance matrix D ; e.g., Example 5.4.2.3.5, Example 6.4.0.0.1.This way, problem convexity can be maintained while simultaneouslyconstraining point list X , Gram matrix G , and distance matrix D at ourdiscretion.

Convex problem (1335) may be equivalently written via linear bijective(§5.6.1) EDM operator D(G) (879);

minimizeG∈SN

c , Y ∈ SNh

− tr(V (D(G) − 2Y )V )

subject to

[

〈Φij , G〉 yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

G º 0

(1336)

where distance-square D = [dij] ∈ SNh (863) is related to Gram matrix entries

G = [gij] ∈ SNc ∩ SN

+ by

dij = gii + gjj − 2gij

= 〈Φij , G〉(878)

where

Φij = (ei − ej)(ei − ej)T ∈ SN

+ (865)

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560 CHAPTER 7. PROXIMITY PROBLEMS

Confinement of G to the geometric center subspace provides numericalstability and no loss of generality (confer (1163)); implicit constraint G1 = 0is otherwise unnecessary.

To include constraints on the list X∈ Rn×N , we would first rewrite (1336)

minimizeG∈SN

c , Y ∈ SNh

, X∈Rn×N

− tr(V (D(G) − 2Y )V )

subject to

[

〈Φij , G〉 yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

[

I XXT G

]

º 0

X∈ C

(1337)

and then add the constraints, realized here in abstract membership to someconvex set C . This problem realization includes a convex relaxation of thenonconvex constraint G = XTX and, if desired, more constraints on G couldbe added. This technique is discussed in §5.4.2.3.5.

7.2.2 Minimization of affine dimension in Problem 2

When desired affine dimension ρ is diminished, the rank function becomesreinserted into problem (1331) that is then rendered difficult to solve becausefeasible set D , Y loses convexity in SN

h × RN×N . Indeed, the rank functionis quasiconcave (§3.9) on the positive semidefinite cone; (§2.9.2.6.2) id est,its sublevel sets are not convex.

7.2.2.1 Rank minimization heuristic

A remedy developed in [258] [132] [133] [131] introduces convex envelope ofthe quasiconcave rank function: (Figure 152)

7.2.2.1.1 Definition. Convex envelope. [195]Convex envelope cenv f of a function f : C→R is defined to be the largestconvex function g such that g ≤ f on convex domain C⊆Rn .7.16 7.16Provided f 6≡+∞ and there exists an affine function h≤ f on Rn, then the convexenvelope is equal to the convex conjugate (the Legendre-Fenchel transform) of the convexconjugate of f ; id est, the conjugate-conjugate function f∗∗. [196, §E.1]

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7.2. SECOND PREVALENT PROBLEM: 561

cenv rank X

rank X g

Figure 152: Abstraction of convex envelope of rank function. Rank is aquasiconcave function on positive semidefinite cone, but its convex envelopeis the largest convex function whose epigraph contains it. Vertical barlabelled g measures a trace/rank gap; id est, rank found always exceedsestimate; large decline in trace required here for only small decrease in rank.

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562 CHAPTER 7. PROXIMITY PROBLEMS [132] [131] Convex envelope of rank function: for σi a singular value,(1532)

cenv(rank A) on A∈Rm×n | ‖A‖2≤κ =1

κ1Tσ(A) =

1

κtr√

ATA (1338)

cenv(rank A) on A normal | ‖A‖2≤κ =1

κ‖λ(A)‖1 =

1

κtr√

ATA (1339)

cenv(rank A) on A∈Sn+ | ‖A‖2≤κ =

1

κ1Tλ(A) =

1

κtr(A) (1340)

A properly scaled trace thus represents the best convex lower bound on rankfor positive semidefinite matrices. The idea, then, is to substitute convexenvelope for rank of some variable A∈ SM

+ (§A.6.5)

rank A ← cenv(rank A) ∝ tr A =∑

i

σ(A)i =∑

i

λ(A)i (1341)

which is equivalent to the sum of all eigenvalues or singular values. [131] Convex envelope of the cardinality function is proportional to the1-norm:

cenv(card x) on x∈Rn | ‖x‖∞≤κ =1

κ‖x‖1 (1342)

cenv(card x) on x∈Rn+ | ‖x‖∞≤κ =

1

κ1Tx (1343)

7.2.2.2 Applying trace rank-heuristic to Problem 2

Substituting rank envelope for rank function in Problem 2, for D∈ EDMN

(confer (1018))

cenv rank(−V TNDVN ) = cenv rank(−V DV ) ∝ − tr(V DV ) (1344)

and for desired affine dimension ρ ≤ N− 1 and nonnegative H [sic] we geta convex optimization problem

minimizeD

‖ √

D − H‖2F

subject to − tr(V DV ) ≤ κ ρ

D ∈ EDMN

(1345)

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7.2. SECOND PREVALENT PROBLEM: 563

where κ ∈ R+ is a constant determined by cut-and-try. The equivalentsemidefinite program makes κ variable: for nonnegative and symmetric H

minimizeD , Y , κ

κ ρ + 2 tr(V Y V )

subject to

[

dij yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

− tr(V DV ) ≤ κ ρ

Y ∈ SNh

D ∈ EDMN

(1346)

which is the same as (1335), the problem with no explicit constraint on affinedimension. As the present problem is stated, the desired affine dimension ρyields to the variable scale factor κ ; ρ is effectively ignored.

Yet this result is an illuminant for problem (1335) and it equivalents(all the way back to (1328)): When the given measurement matrix His nonnegative and symmetric, finding the closest EDM D as in problem(1328), (1331), or (1335) implicitly entails minimization of affine dimension(confer §5.8.4, §5.14.4). Those non−rank-constrained problems are eachinherently equivalent to cenv(rank)-minimization problem (1346), in otherwords, and their optimal solutions are unique because of the strictly convexobjective function in (1328).

7.2.2.3 Rank-heuristic insight

Minimization of affine dimension by use of this trace rank-heuristic (1344)tends to find a list configuration of least energy; rather, it tends to optimizecompaction of the reconstruction by minimizing total distance. (891) It is bestused where some physical equilibrium implies such an energy minimization;e.g., [349, §5].

For this Problem 2, the trace rank-heuristic arose naturally in theobjective in terms of V . We observe: V (in contrast to V T

N ) spreads energyover all available distances (§B.4.2 no.20, contrast no.22) although the rankfunction itself is insensitive to choice of auxiliary matrix.

Trace rank-heuristic (1340) is useless when a main diagonal is constrainedto be constant. Such would be the case were optimization over anelliptope (§5.4.2.3), or when the diagonal represents a Boolean vector; e.g.,Example 4.2.3.1.1, Example 4.6.0.0.8.

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564 CHAPTER 7. PROXIMITY PROBLEMS

7.2.2.4 Rank minimization heuristic beyond convex envelope

Fazel, Hindi, & Boyd [133] [384] [134] propose a rank heuristic more potentthan trace (1341) for problems of rank minimization;

rank Y ← log det(Y + εI) (1347)

the concave surrogate function log det in place of quasiconcave rank Y(§2.9.2.6.2) when Y ∈ Sn

+ is variable and where ε is a small positive constant.They propose minimization of the surrogate by substituting a sequencecomprising infima of a linearized surrogate about the current estimate Yi ;id est, from the first-order Taylor series expansion about Yi on some openinterval of ‖Y ‖2 (§D.1.7)

log det(Y + εI) ≈ log det(Yi + εI) + tr(

(Yi + εI)−1(Y − Yi))

(1348)

we make the surrogate sequence of infima over bounded convex feasible set Carg inf

Y ∈Crank Y ← lim

i→∞Yi+1 (1349)

where, for i = 0 . . .

Yi+1 = arg infY ∈C

tr(

(Yi + εI)−1Y)

(1350)

a matrix analogue to the reweighting scheme disclosed in [203, §4.11.3].Choosing Y0 = I , the first step becomes equivalent to finding the infimum oftr Y ; the trace rank-heuristic (1341). The intuition underlying (1350) is thenew term in the argument of trace; specifically, (Yi + εI)−1 weights Y so thatrelatively small eigenvalues of Y found by the infimum are made even smaller.

To see that, substitute the nonincreasingly ordered diagonalizations

Yi + εI , Q(Λ + εI )QT (a)

Y , UΥUT (b)(1351)

into (1350). Then from (1664) we have,

infΥ∈U⋆TCU⋆

δ((Λ + εI )−1)Tδ(Υ) = inf

Υ∈UTCUinf

RT=R−1tr

(

(Λ + εI )−1RTΥR)

≤ infY ∈C

tr((Yi + εI)−1Y )(1352)

where R , QTU in U on the set of orthogonal matrices is a bijection. Therole of ε is, therefore, to limit maximum weight; the smallest entry on themain diagonal of Υ gets the largest weight. ¨

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7.2. SECOND PREVALENT PROBLEM: 565

7.2.2.5 Applying log det rank-heuristic to Problem 2

When the log det rank-heuristic is inserted into Problem 2, problem (1346)becomes the problem sequence in i

minimizeD , Y , κ

κ ρ + 2 tr(V Y V )

subject to

[

djl yjl

yjl h2jl

]

º 0 , l > j = 1 . . . N−1

− tr((−V DiV + εI )−1V DV ) ≤ κ ρ

Y ∈ SNh

D ∈ EDMN

(1353)

where Di+1 ,D⋆∈ EDMN and D0 ,11T− I .

7.2.2.6 Tightening this log det rank-heuristic

Like the trace method, this log det technique for constraining rank offersno provision for meeting a predetermined upper bound ρ . Yet sinceeigenvalues are simply determined, λ(Yi + εI)= δ(Λ + εI ) , we may certainlyforce selected weights to ε−1 by manipulating diagonalization (1351a).Empirically we find this sometimes leads to better results, although affinedimension of a solution cannot be guaranteed.

7.2.2.7 Cumulative summary of rank heuristics

We have studied a perturbation method of rank reduction in §4.3 as well asthe trace heuristic (convex envelope method §7.2.2.1.1) and log det heuristicin §7.2.2.4. There is another good contemporary method called LMIRank[282] based on alternating projection (§E.10).7.17

7.2.2.7.1 Example. Unidimensional scaling.We apply the convex iteration method from §4.4.1 to numerically solve aninstance of Problem 2; a method empirically superior to the foregoing convexenvelope and log det heuristics for rank regularization and enforcing affinedimension.

7.17 that does not solve the ball packing problem presented in §5.4.2.3.4.

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566 CHAPTER 7. PROXIMITY PROBLEMS

Unidimensional scaling, [102] a historically practical application ofmultidimensional scaling (§5.12), entails solution of an optimization problemhaving local minima whose multiplicity varies as the factorial of point-listcardinality; geometrically, it means reconstructing a list constrained to lie inone affine dimension. In terms of point list, the nonconvex problem is: givennonnegative symmetric matrix H = [hij] ∈ SN∩ RN×N

+ (1333) whose entrieshij are all known,

minimizexi∈R

N∑

i , j=1

(|xi − xj| − hij)2 (1284)

called a raw stress problem [51, p.34] which has an implicit constraint ondimensional embedding of points xi∈ R , i=1 . . . N. This problem hasproven NP-hard; e.g., [71].

As always, we first transform variables to distance-square D∈ SNh ; so

begin with convex problem (1335) on page 559

minimizeD , Y

− tr(V (D − 2Y )V )

subject to

[

dij yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

Y ∈ SNh

D ∈ EDMN

rank V TNDVN = 1

(1354)

that becomes equivalent to (1284) by making explicit the constraint on affinedimension via rank. The iteration is formed by moving the dimensionalconstraint to the objective:

minimizeD , Y

−〈V (D − 2Y )V , I 〉 − w〈V TNDVN , W 〉

subject to

[

dij yij

yij h2ij

]

º 0 , N ≥ j > i = 1 . . . N−1

Y ∈ SNh

D ∈ EDMN

(1355)

where w (≈ 10) is a positive scalar just large enough to make 〈V TNDVN , W 〉

vanish to within some numerical precision, and where direction matrix W is

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7.2. SECOND PREVALENT PROBLEM: 567

an optimal solution to semidefinite program (1663a)

minimizeW

−〈V TND⋆VN , W 〉

subject to 0 ¹ W ¹ I

tr W = N − 1

(1356)

one of which is known in closed form. Semidefinite programs (1355) and(1356) are iterated until convergence in the sense defined on page 301. Thisiteration is not a projection method. (§4.4.1.1) Convex problem (1355)is neither a relaxation of unidimensional scaling problem (1354); instead,problem (1355) is a convex equivalent to (1354) at convergence of theiteration.

Jan de Leeuw provided us with some test data

H =

0.000000 5.235301 5.499274 6.404294 6.486829 6.2632655.235301 0.000000 3.208028 5.840931 3.559010 5.3534895.499274 3.208028 0.000000 5.679550 4.020339 5.2398426.404294 5.840931 5.679550 0.000000 4.862884 4.5431206.486829 3.559010 4.020339 4.862884 0.000000 4.6187186.263265 5.353489 5.239842 4.543120 4.618718 0.000000

(1357)and a globally optimal solution

X⋆ = [−4.981494 −2.121026 −1.038738 4.555130 0.764096 2.822032 ]

= [ x⋆1 x⋆

2 x⋆3 x⋆

4 x⋆5 x⋆

6 ]

(1358)

found by searching 6! local minima of (1284) [102]. By iterating convexproblems (1355) and (1356) about twenty times (initial W = 0) we find theglobal infimum 98.12812 of stress problem (1284), and by (1108) we find acorresponding one-dimensional point list that is a rigid transformation in Rof X⋆.

Here we found the infimum to accuracy of the given data, but that ceasesto hold as problem size increases. Because of machine numerical precisionand an interior-point method of solution, we speculate, accuracy degradesquickly as problem size increases beyond this. 2

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568 CHAPTER 7. PROXIMITY PROBLEMS

7.3 Third prevalent problem:

Projection on EDM cone in dij

In summary, we find that the solution to problem [(1286.3) p.543]is difficult and depends on the dimension of the space as thegeometry of the cone of EDMs becomes more complex.

−Hayden, Wells, Liu, & Tarazaga (1991) [181, §3]

Reformulating Problem 2 (p.556) in terms of EDM D changes theproblem considerably:

minimizeD

‖D − H‖2F

subject to rank V TNDVN ≤ ρ

D ∈ EDMN

Problem 3 (1359)

This third prevalent proximity problem is a Euclidean projection of givenmatrix H on a generally nonconvex subset (ρ < N−1) of ∂EDMN theboundary of the convex cone of Euclidean distance matrices relativeto subspace SN

h (Figure 136d). Because coordinates of projection aredistance-square and H now presumably holds distance-square measurements,numerical solution to Problem 3 is generally different than that of Problem 2.

For the moment, we need make no assumptions regarding measurementmatrix H .

7.3.1 Convex case

minimizeD

‖D − H‖2F

subject to D ∈ EDMN(1360)

When the rank constraint disappears (for ρ = N−1), this third problembecomes obviously convex because the feasible set is then the entire EDMcone and because the objective function

‖D − H‖2F =

i,j

(dij − hij)2 (1361)

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7.3. THIRD PREVALENT PROBLEM: 569

is a strictly convex quadratic in D ;7.18

minimizeD

i,j

d2ij − 2hij dij + h2

ij

subject to D ∈ EDMN(1362)

Optimal solution D⋆ is therefore unique, as expected, for this simpleprojection on the EDM cone equivalent to (1285).

7.3.1.1 Equivalent semidefinite program, Problem 3, convex case

In the past, this convex problem was solved numerically by means ofalternating projection. (Example 7.3.1.1.1) [151] [144] [181, §1] We translate(1362) to an equivalent semidefinite program because we have a good solver:

Assume the given measurement matrix H to be nonnegative andsymmetric;7.19

H = [hij] ∈ SN ∩ RN×N+ (1333)

We then propose: Problem (1362) is equivalent to the semidefinite program,for

∂ , [d2ij] = D D (1363)

a matrix of distance-square squared,

minimize∂ , D

− tr(V (∂ − 2H D)V )

subject to

[

∂ij dij

dij 1

]

º 0 , N ≥ j > i = 1 . . . N−1

D ∈ EDMN

∂ ∈ SNh

(1364)

7.18For nonzero Y ∈ SNh and some open interval of t∈R (§3.8.3.0.2, §D.2.3)

d2

dt2‖(D + t Y ) − H‖2

F = 2 tr Y TY > 0 ¨

7.19If that H given has negative entries, then the technique of solution presented herebecomes invalid. Projection of H on K (1277) prior to application of this proposedtechnique, as explained in §7.0.1, is incorrect.

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570 CHAPTER 7. PROXIMITY PROBLEMS

where[

∂ij dij

dij 1

]

º 0 ⇔ ∂ij ≥ d2ij (1365)

Symmetry of input H facilitates trace in the objective (§B.4.2 no.20), whileits nonnegativity causes ∂ij →d2

ij as optimization proceeds.

7.3.1.1.1 Example. Alternating projection on nearest EDM.By solving (1364) we confirm the result from an example given by Glunt,Hayden, Hong, & Wells [151, §6] who found analytical solution to convexoptimization problem (1360) for particular cardinality N = 3 by using thealternating projection method of von Neumann (§E.10):

H =

0 1 11 0 91 9 0

, D⋆ =

0 199

199

199

0 769

199

769

0

(1366)

The original problem (1360) of projecting H on the EDM cone is transformedto an equivalent iterative sequence of projections on the two convex cones(1226) from §6.8.1.1. Using ordinary alternating projection, input H goes toD⋆ with an accuracy of four decimal places in about 17 iterations. Affinedimension corresponding to this optimal solution is r = 1.

Obviation of semidefinite programming’s computational expense is theprincipal advantage of this alternating projection technique. 2

7.3.1.2 Schur-form semidefinite program, Problem 3 convex case

Semidefinite program (1364) can be reformulated by moving the objectivefunction in

minimizeD

‖D − H‖2F

subject to D ∈ EDMN(1360)

to the constraints. This makes an equivalent epigraph form of the problem:for any measurement matrix H

minimizet∈R , D

t

subject to ‖D − H‖2F ≤ t

D ∈ EDMN

(1367)

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7.3. THIRD PREVALENT PROBLEM: 571

We can transform this problem to an equivalent Schur-form semidefiniteprogram; (§3.6.2)

minimizet∈R , D

t

subject to

[

tI vec(D − H)vec(D − H)T 1

]

º 0

D ∈ EDMN

(1368)

characterized by great sparsity and structure. The advantage of this SDP islack of conditions on input H ; e.g., negative entries would invalidate anysolution provided by (1364). (§7.0.1.2)

7.3.1.3 Gram-form semidefinite program, Problem 3 convex case

Further, this problem statement may be equivalently written in terms of aGram matrix via linear bijective (§5.6.1) EDM operator D(G) (879);

minimizeG∈SN

c , t∈R

t

subject to

[

tI vec(D(G) − H)

vec(D(G) − H)T 1

]

º 0

G º 0

(1369)

To include constraints on the list X∈ Rn×N , we would rewrite this:

minimizeG∈SN

c , t∈R , X∈Rn×N

t

subject to

[

tI vec(D(G) − H)

vec(D(G) − H)T 1

]

º 0

[

I XXT G

]

º 0

X∈ C

(1370)

where C is some abstract convex set. This technique is discussed in §5.4.2.3.5.

7.3.1.4 Dual interpretation, projection on EDM cone

From §E.9.1.1 we learn that projection on a convex set has a dual form. Inthe circumstance K is a convex cone and point x exists exterior to the cone

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572 CHAPTER 7. PROXIMITY PROBLEMS

or on its boundary, distance to the nearest point Px in K is found as theoptimal value of the objective

‖x − Px‖ = maximizea

aTx

subject to ‖a‖ ≤ 1

a ∈ K(1978)

where Kis the polar cone.

Applying this result to (1360), we get a convex optimization for any givensymmetric matrix H exterior to or on the EDM cone boundary:

minimizeD

‖D − H‖2F

subject to D ∈ EDMN≡

maximizeA

〈A, H 〉subject to ‖A‖F ≤ 1

A∈ EDMN(1371)

Then from (1980) projection of H on cone EDMN is

D⋆ = H − A⋆〈A⋆, H 〉 (1372)

Critchley proposed, instead, projection on the polar EDM cone in his 1980thesis [87, p.113]: In that circumstance, by projection on the algebraiccomplement (§E.9.2.2.1),

D⋆ = A⋆〈A⋆, H 〉 (1373)

which is equal to (1372) when A⋆ solves

maximizeA

〈A , H 〉subject to ‖A‖F = 1

A ∈ EDMN

(1374)

This projection of symmetric H on polar cone EDMNcan be made a convex

problem, of course, by relaxing the equality constraint (‖A‖F ≤ 1).

7.3.2 Minimization of affine dimension in Problem 3

When desired affine dimension ρ is diminished, Problem 3 (1359) is difficultto solve [181, §3] because the feasible set in RN(N−1)/2 loses convexity. By

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7.3. THIRD PREVALENT PROBLEM: 573

substituting rank envelope (1344) into Problem 3, then for any given H weget a convex problem

minimizeD

‖D − H‖2F

subject to − tr(V DV ) ≤ κ ρ

D ∈ EDMN

(1375)

where κ ∈ R+ is a constant determined by cut-and-try. Given κ , problem(1375) is a convex optimization having unique solution in any desiredaffine dimension ρ ; an approximation to Euclidean projection on thatnonconvex subset of the EDM cone containing EDMs with correspondingaffine dimension no greater than ρ .

The SDP equivalent to (1375) does not move κ into the variables as onpage 563: for nonnegative symmetric input H and distance-square squaredvariable ∂ as in (1363),

minimize∂ , D

− tr(V (∂ − 2H D)V )

subject to

[

∂ij dij

dij 1

]

º 0 , N ≥ j > i = 1 . . . N−1

− tr(V DV ) ≤ κ ρ

D ∈ EDMN

∂ ∈ SNh

(1376)

That means we will not see equivalence of this cenv(rank)-minimizationproblem to the non−rank-constrained problems (1362) and (1364) like wesaw for its counterpart (1346) in Problem 2.

Another approach to affine dimension minimization is to project insteadon the polar EDM cone; discussed in §6.8.1.5.

7.3.3 Constrained affine dimension, Problem 3

When one desires affine dimension diminished further below what can beachieved via cenv(rank)-minimization as in (1376), spectral projection can beconsidered a natural means in light of its successful application to projectionon a rank ρ subset of the positive semidefinite cone in §7.1.4.

Yet it is wrong here to zero eigenvalues of −V DV or −V GV or a variantto reduce affine dimension, because that particular method comes from

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574 CHAPTER 7. PROXIMITY PROBLEMS

projection on a positive semidefinite cone (1312); zeroing those eigenvalueshere in Problem 3 would place an elbow in the projection path (Figure 151)thereby producing a result that is necessarily suboptimal. Problem 3 isinstead a projection on the EDM cone whose associated spectral cone isconsiderably different. (§5.11.2.3) Proper choice of spectral cone is demandedby diagonalization of that variable argument to the objective:

7.3.3.1 Cayley-Menger form

We use Cayley-Menger composition of the Euclidean distance matrix to solvea problem that is the same as Problem 3 (1359): (§5.7.3.0.1)

minimizeD

[

0 1T

1 −D

]

−[

0 1T

1 −H

]∥

2

F

subject to rank

[

0 1T

1 −D

]

≤ ρ + 2

D ∈ EDMN

(1377)

a projection of H on a generally nonconvex subset (when ρ < N−1) of theEuclidean distance matrix cone boundary rel ∂EDMN ; id est, projectionfrom the EDM cone interior or exterior on a subset of its relative boundary(§6.6, (1155)).

Rank of an optimal solution is intrinsically bounded above and below;

2 ≤ rank

[

0 1T

1 −D⋆

]

≤ ρ + 2 ≤ N + 1 (1378)

Our proposed strategy for low-rank solution is projection on that subset

of a spectral cone λ

([

0 1T

1 −EDMN

])

(§5.11.2.3) corresponding to affine

dimension not in excess of that ρ desired; id est, spectral projection on

Rρ+1+

0R−

∩ ∂H ⊂ RN+1 (1379)

where∂H = λ ∈ RN+1 | 1Tλ = 0 (1088)

is a hyperplane through the origin. This pointed polyhedral cone (1379), towhich membership subsumes the rank constraint, is not full-dimensional.

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Given desired affine dimension 0≤ ρ≤N−1 and diagonalization (§A.5)of unknown EDM D

[

0 1T

1 −D

]

, UΥUT∈ SN+1h (1380)

and given symmetric H in diagonalization[

0 1T

1 −H

]

, QΛQT∈ SN+1 (1381)

having eigenvalues arranged in nonincreasing order, then by (1101) problem(1377) is equivalent to

minimizeΥ , R

∥δ(Υ) − π(

δ(RTΛR))∥

2

subject to δ(Υ) ∈

Rρ+1+

0R−

∩ ∂H

δ(QRΥRTQT) = 0

R−1 = RT

(1382)

where π is the permutation operator from §7.1.3 arranging its vectorargument in nonincreasing order,7.20 where

R , QTU ∈ RN+1×N+1 (1383)

in U on the set of orthogonal matrices is a bijection, and where ∂H insuresone negative eigenvalue. Hollowness constraint δ(QRΥRTQT) = 0 makesproblem (1382) difficult by making the two variables dependent.

Our plan is to instead divide problem (1382) into two and then iteratetheir solution:

minimizeΥ

∥δ(Υ) − π(

δ(RTΛR))∥

2

subject to δ(Υ) ∈

Rρ+1+

0R−

∩ ∂H(a)

minimizeR

‖R Υ⋆RT− Λ‖2F

subject to δ(QR Υ⋆RTQT) = 0

R−1 = RT

(b)

(1384)

7.20Recall, any permutation matrix is an orthogonal matrix.

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576 CHAPTER 7. PROXIMITY PROBLEMS

Proof. We justify disappearance of the hollowness constraint inconvex optimization problem (1384a): From the arguments in §7.1.3with regard to π the permutation operator, cone membership constraint

δ(Υ)∈

Rρ+1+

0R−

∩ ∂H from (1384a) is equivalent to

δ(Υ) ∈

Rρ+1+

0R−

∩ ∂H ∩ KM (1385)

where KM is the monotone cone (§2.13.9.4.3). Membership of δ(Υ) to thepolyhedral cone of majorization (Theorem A.1.2.0.1)

K∗λδ = ∂H ∩ K∗

M+ (1402)

where K∗M+ is the dual monotone nonnegative cone (§2.13.9.4.2), is a

condition (in absence of a hollowness constraint) that would insure existence

of a symmetric hollow matrix

[

0 1T

1 −D

]

. Curiously, intersection of

this feasible superset

Rρ+1+

0R−

∩ ∂H ∩ KM from (1385) with the cone of

majorization K∗λδ is a benign operation; id est,

∂H ∩ K∗M+ ∩ KM = ∂H ∩ KM (1386)

verifiable by observing conic dependencies (§2.10.3) among the aggregate ofhalfspace-description normals. The cone membership constraint in (1384a)therefore inherently insures existence of a symmetric hollow matrix. ¨

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7.3. THIRD PREVALENT PROBLEM: 577

Optimization (1384b) would be a Procrustes problem (§C.4) were itnot for the hollowness constraint; it is, instead, a minimization over theintersection of the nonconvex manifold of orthogonal matrices with anothernonconvex set in variable R specified by the hollowness constraint.

We solve problem (1384b) by a method introduced in §4.6.0.0.2: DefineR = [ r1 · · · rN+1 ]∈ RN+1×N+1 and make the assignment

G =

r1...

rN+1

1

[ rT1 · · · rT

N+1 1 ]

∈ S(N+1)2+1

=

R11 · · · R1,N+1 r1...

. . ....

RT1,N+1 RN+1,N+1 rN+1

rT1 · · · rT

N+1 1

,

r1rT1 · · · r1r

TN+1 r1

.... . .

...rN+1r

T1 rN+1r

TN+1 rN+1

rT1 · · · rT

N+1 1

(1387)

where Rij , rirTj ∈ RN+1×N+1 and Υ⋆

ii∈ R . Since R Υ⋆RT =N+1∑

i=1

Υ⋆ii Rii then

problem (1384b) is equivalently expressed:

minimizeRii∈S , Rij , ri

N+1∑

i=1

Υ⋆ii Rii − Λ

2

F

subject to tr Rii = 1 , i=1 . . . N+1tr Rij = 0 , i<j = 2 . . . N+1

G =

R11 · · · R1,N+1 r1...

. . ....

RT1,N+1 RN+1,N+1 rN+1

rT1 · · · rT

N+1 1

(º 0)

δ

(

QN+1∑

i=1

Υ⋆ii Rii QT

)

= 0

rank G = 1

(1388)

The rank constraint is regularized by method of convex iteration developedin §4.4. Problem (1388) is partitioned into two convex problems:

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578 CHAPTER 7. PROXIMITY PROBLEMS

minimizeRij , ri

N+1∑

i=1

Υ⋆ii Rii − Λ

2

F

+ 〈G , W 〉subject to tr Rii = 1 , i=1 . . . N+1

tr Rij = 0 , i<j = 2 . . . N+1

G =

R11 · · · R1,N+1 r1...

. . ....

RT1,N+1 RN+1,N+1 rN+1

rT1 · · · rT

N+1 1

º 0

δ

(

QN+1∑

i=1

Υ⋆ii Rii QT

)

= 0

(1389)

andminimize

W∈ S(N+1)2+1

〈G⋆, W 〉subject to 0 ¹ W ¹ I

tr W = (N + 1)2

(1390)

then iterated with convex problem (1384a) until a rank-1 G matrix is foundand the objective of (1384a) is minimized. An optimal solution to (1390)is known in closed form (p.646). The hollowness constraint in (1389) maycause numerical infeasibility; in that case, it can be moved to the objectivewithin an added weighted norm. Conditions for termination of the iterationwould then comprise a vanishing norm of hollowness.

7.4 Conclusion

The importance and application of solving rank- or cardinality-constrainedproblems are enormous, a conclusion generally accepted gratis by themathematics and engineering communities. Rank-constrained semidefiniteprograms arise in many vital feedback and control problems [168], optics [73](Figure 111), and communications [280] [250] (Figure 99). For example,one might be interested in the minimal order dynamic output feedback whichstabilizes a given linear time invariant plant (this problem is considered tobe among the most important open problems in control). [259] Rank andcardinality constraints also arise naturally in combinatorial optimization(§4.6.0.0.10), and find application to face recognition (Figure 4), cartography(Figure 132), video surveillance, latent semantic indexing [227], sparseor low-rank matrix completion for preference models and collaborative

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7.4. CONCLUSION 579

filtering, multidimensional scaling or principal component analysis (§5.12),medical imaging (Figure 105), molecular conformation (Figure 125), andsensor-network localization or wireless location (Figure 84).

There has been little progress in spectral projection since the discovery byEckart & Young in 1936 [126] leading to a formula for projection on a rank ρsubset of a positive semidefinite cone (§2.9.2.1). [142] The only closed-formspectral method presently available for solving proximity problems, having aconstraint on rank, is based on their discovery (Problem 1, §7.1, §5.13). One popular recourse is intentional misapplication of Eckart & Young’s

result by introducing spectral projection on a positive semidefinite coneinto Problem 3 via D(G) (879), for example. [71] Since Problem 3instead demands spectral projection on the EDM cone, any solutionacquired that way is necessarily suboptimal. A second recourse is problem redesign: A presupposition to allproximity problems in this chapter is that matrix H is given.We considered H having various properties such as nonnegativity,symmetry, hollowness, or lack thereof. It was assumed that if H didnot already belong to the EDM cone, then we wanted an EDM closestto H in some sense; id est, input-matrix H was assumed corruptedsomehow. For practical problems, it withstands reason that such aproximity problem could instead be reformulated so that some or allentries of H were unknown but bounded above and below by knownlimits; the norm objective is thereby eliminated as in the developmentbeginning on page 315. That particular redesign (the art, p.8), in termsof the Gram-matrix bridge between point-list X and EDM D , at onceencompasses proximity and completion problems. A third recourse is to apply the method of convex iteration just likewe did in §7.2.2.7.1. This technique is applicable to any semidefiniteproblem requiring a rank constraint; it places a regularization term inthe objective that enforces the rank constraint.

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580 CHAPTER 7. PROXIMITY PROBLEMS

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Appendix A

Linear algebra

A.1 Main-diagonal δ operator, λ , trace, vec

We introduce notation δ denoting the main-diagonal linear self-adjointoperator. When linear function δ operates on a square matrix A∈RN×N ,δ(A) returns a vector composed of all the entries from the main diagonal inthe natural order;

δ(A) ∈ RN (1391)

Operating on a vector y∈RN , δ naturally returns a diagonal matrix;

δ(y) ∈ SN (1392)

Operating recursively on a vector Λ∈RN or diagonal matrix Λ∈SN ,δ(δ(Λ)) returns Λ itself;

δ2(Λ) ≡ δ(δ(Λ)) , Λ (1393)

Defined in this manner, main-diagonal linear operator δ is self-adjoint[223, §3.10, §9.5-1];A.1 videlicet, (§2.2)

δ(A)Ty = 〈δ(A) , y〉 = 〈A , δ(y)〉 = tr(

ATδ(y))

(1394)

A.1Linear operator T : Rm×n→RM×N is self-adjoint when, for each and everyX1 , X2∈Rm×n

〈T (X1) , X2〉 = 〈X1 , T (X2)〉© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

581

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582 APPENDIX A. LINEAR ALGEBRA

A.1.1 Identities

This δ notation is efficient and unambiguous as illustrated in the followingexamples where: A B denotes Hadamard product [199] [155, §1.1.4] ofmatrices of like size, ⊗ Kronecker product [162] (§D.1.2.1), y a vector, Xa matrix, ei the ith member of the standard basis for Rn, SN

h the symmetrichollow subspace, σ(A) a vector of (nonincreasingly) ordered singular values,and λ(A) a vector of nonincreasingly ordered eigenvalues of matrix A :

1. δ(A) = δ(AT)

2. tr(A) = tr(AT) = δ(A)T1 = 〈I , A〉3. δ(cA) = c δ(A) c∈R

4. tr(cA) = c tr(A) = c1Tλ(A) c∈R

5. vec(cA) = c vec(A) c∈R

6. A cB = cA B c∈R

7. A ⊗ cB = cA ⊗ B c∈R

8. δ(A + B) = δ(A) + δ(B)

9. tr(A + B) = tr(A) + tr(B)

10. vec(A + B) = vec(A) + vec(B)

11. (A + B) C = A C + B CA (B + C) = A B + A C

12. (A + B) ⊗ C = A ⊗ C + B ⊗ CA ⊗ (B + C) = A ⊗ B + A ⊗ C

13. sgn(c)λ(|c|A) = c λ(A) c∈R

14. sgn(c)σ(|c|A) = c σ(A) c∈R

15. tr(c√

ATA ) = c tr√

ATA = c1Tσ(A) c∈R

16. π(δ(A)) = λ(I A) where π is the presorting function.

17. δ(AB) = (A BT)1 = (BT A)1

18. δ(AB)T = 1T(AT B) = 1T(B AT)

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A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC 583

19. δ(uvT) =

u1v1...

uN vN

= u v , u,v∈RN

20. tr(ATB) = tr(ABT) = tr(BAT) = tr(BTA)

= 1T(A B)1 = 1Tδ(ABT) = δ(ATB)T1 = δ(BAT)T1 = δ(BTA)T1

21. D = [dij] ∈ SNh , H = [hij] ∈ SN

h , V = I − 1N11T∈ SN (confer §B.4.2 no.20)

N tr(−V (D H)V ) = tr(DTH) = 1T(D H)1 = tr(11T(D H)) =∑

i,j

dij hij

22. tr(ΛA) = δ(Λ)Tδ(A) , δ2(Λ) , Λ ∈ SN

23. yTBδ(A) = tr(

Bδ(A)yT)

= tr(

δ(BTy)A)

= tr(

Aδ(BTy))

= δ(A)TBTy = tr(

y δ(A)TBT)

= tr(

ATδ(BTy))

= tr(

δ(BTy)AT)

24. δ2(ATA) =∑

i

eieTi A

TAeieTi

25. δ(

δ(A)1T)

= δ(

1 δ(A)T)

= δ(A)

26. δ(A1)1 = δ(A11T) = A1 , δ(y)1 = δ(y1T) = y

27. δ(I1) = δ(1) = I

28. δ(eieTj 1) = δ(ei) = eie

Ti

29. For ζ =[ζi]∈Rk and x=[xi]∈Rk,∑

i

ζi/xi = ζTδ(x)−11

30.vec(A B) = vec(A) vec(B) = δ(vec A) vec(B)

= vec(B) vec(A) = δ(vec B) vec(A)(42)(1750)

31. vec(AXB) = (BT⊗ A) vec X(

not H)

32. vec(BXA) = (AT⊗ B) vec X

33.tr(AXBXT) = vec(X)Tvec(AXB) = vec(X)T(BT⊗ A) vec X [162]

= δ(

vec(X) vec(X)T(BT⊗ A))T

1

34.tr(AXTBX) = vec(X)Tvec(BXA) = vec(X)T(AT⊗ B) vec X

= δ(

vec(X) vec(X)T(AT⊗ B))T

1

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584 APPENDIX A. LINEAR ALGEBRA

35. For any permutation matrix Ξ and dimensionally compatible vector yor matrix A

δ(Ξ y) = Ξ δ(y) ΞT (1395)

δ(Ξ A ΞT) = Ξ δ(A) (1396)

So given any permutation matrix Ξ and any dimensionally compatiblematrix B , for example,

δ2(B) = Ξ δ2(ΞTB Ξ)ΞT (1397)

36. A ⊗ 1 = 1 ⊗ A = A

37. A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C

38. (A ⊗ B)(C ⊗ D) = AC ⊗ BD

39. For A a vector, (A ⊗ B) = (A ⊗ I )B

40. For B a row vector, (A ⊗ B) = A(I ⊗ B)

41. (A ⊗ B)T = AT⊗ BT

42. (A ⊗ B)−1 = A−1⊗ B−1

43. tr(A ⊗ B) = tr A tr B

44. For A∈ Rm×m, B∈ Rn×n, det(A ⊗ B) = detn(A) detm(B)

45. There exist permutation matrices Ξ1 and Ξ2 such that [162, p.28]

A ⊗ B = Ξ1(B ⊗ A)Ξ2 (1398)

46. For eigenvalues λ(A)∈ Cn and eigenvectors v(A)∈ Cn×n such thatA = vδ(λ)v−1∈ Rn×n

λ(A ⊗ B) = λ(A) ⊗ λ(B) , v(A ⊗ B) = v(A) ⊗ v(B) (1399)

47. Given analytic function [80] f : Cn×n→ Cn×n, then f(I⊗A)= I⊗f(A)and f(A ⊗ I) = f(A) ⊗ I [162, p.28]

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A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC 585

A.1.2 Majorization

A.1.2.0.1 Theorem. (Schur) Majorization. [390, §7.4] [199, §4.3][200, §5.5] Let λ∈RN denote a given vector of eigenvalues and letδ∈RN denote a given vector of main diagonal entries, both arranged innonincreasing order. Then

∃A∈ SN Ä λ(A)=λ and δ(A)= δ ⇐ λ − δ ∈ K∗λδ (1400)

and converselyA∈ SN ⇒ λ(A) − δ(A) ∈ K∗

λδ (1401)

The difference belongs to the pointed polyhedral cone of majorization (nota full-dimensional cone, confer (291))

K∗λδ , K∗

M+ ∩ ζ1 | ζ ∈ R∗ (1402)

where K∗M+ is the dual monotone nonnegative cone (413), and where the

dual of the line is a hyperplane; ∂H= ζ1 | ζ ∈ R∗ = 1⊥. ⋄

Majorization cone K∗λδ is naturally consequent to the definition of

majorization; id est, vector y∈RN majorizes vector x if and only if

k∑

i=1

xi ≤k

i=1

yi ∀ 1 ≤ k ≤ N (1403)

and1Tx = 1Ty (1404)

Under these circumstances, rather, vector x is majorized by vector y .In the particular circumstance δ(A)=0 we get:

A.1.2.0.2 Corollary. Symmetric hollow majorization.Let λ∈RN denote a given vector of eigenvalues arranged in nonincreasingorder. Then

∃A∈ SNh Ä λ(A)=λ ⇐ λ ∈ K∗

λδ (1405)

and converselyA∈ SN

h ⇒ λ(A) ∈ K∗λδ (1406)

where K∗λδ is defined in (1402). ⋄

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586 APPENDIX A. LINEAR ALGEBRA

A.2 Semidefiniteness: domain of test

The most fundamental necessary, sufficient, and definitive test for positivesemidefiniteness of matrix A∈ Rn×n is: [200, §1]

xTAx ≥ 0 for each and every x ∈ Rn such that ‖x‖= 1 (1407)

Traditionally, authors demand evaluation over broader domain; namely,over all x ∈ Rn which is sufficient but unnecessary. Indeed, that standardtextbook requirement is far over-reaching because if xTAx is nonnegative forparticular x = xp , then it is nonnegative for any αxp where α∈R . Thus,only normalized x in Rn need be evaluated.

Many authors add the further requirement that the domain be complex;the broadest domain. By so doing, only Hermitian matrices (AH = A wheresuperscript H denotes conjugate transpose)A.2 are admitted to the set ofpositive semidefinite matrices (1410); an unnecessary prohibitive condition.

A.2.1 Symmetry versus semidefiniteness

We call (1407) the most fundamental test of positive semidefiniteness. Yetsome authors instead say, for real A and complex domain x∈Cn , thecomplex test xHAx≥ 0 is most fundamental. That complex broadening of thedomain of test causes nonsymmetric real matrices to be excluded from the setof positive semidefinite matrices. Yet admitting nonsymmetric real matricesor not is a matter of preferenceA.3 unless that complex test is adopted, as weshall now explain.

Any real square matrix A has a representation in terms of its symmetricand antisymmetric parts; id est,

A =(A +AT)

2+

(A −AT)

2(53)

Because, for all real A , the antisymmetric part vanishes under real test,

xT (A −AT)

2x = 0 (1408)

A.2Hermitian symmetry is the complex analogue; the real part of a Hermitian matrixis symmetric while its imaginary part is antisymmetric. A Hermitian matrix has realeigenvalues and real main diagonal.A.3Golub & Van Loan [155, §4.2.2], for example, admit nonsymmetric real matrices.

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A.2. SEMIDEFINITENESS: DOMAIN OF TEST 587

only the symmetric part of A , (A +AT)/2, has a role determining positivesemidefiniteness. Hence the oft-made presumption that only symmetricmatrices may be positive semidefinite is, of course, erroneous under (1407).Because eigenvalue-signs of a symmetric matrix translate unequivocally toits semidefiniteness, the eigenvalues that determine semidefiniteness arealways those of the symmetrized matrix. (§A.3) For that reason, andbecause symmetric (or Hermitian) matrices must have real eigenvalues,the convention adopted in the literature is that semidefinite matrices aresynonymous with symmetric semidefinite matrices. Certainly misleadingunder (1407), that presumption is typically bolstered with compellingexamples from the physical sciences where symmetric matrices occur withinthe mathematical exposition of natural phenomena.A.4 [137, §52]

Perhaps a better explanation of this pervasive presumption of symmetrycomes from Horn & Johnson [199, §7.1] whose perspectiveA.5 is the complexmatrix, thus necessitating the complex domain of test throughout theirtreatise. They explain, if A∈Cn×n

. . . and if xHAx is real for all x ∈ Cn, then A is Hermitian.Thus, the assumption that A is Hermitian is not necessary in thedefinition of positive definiteness. It is customary, however.

Their comment is best explained by noting, the real part of xHAx comesfrom the Hermitian part (A +AH)/2 of A ;

re(xHAx) = xHA +AH

2x (1409)

rather,xHAx ∈ R ⇔ AH = A (1410)

because the imaginary part of xHAx comes from the anti-Hermitian part(A −AH)/2 ;

im(xHAx) = xHA −AH

2x (1411)

that vanishes for nonzero x if and only if A = AH. So the Hermitiansymmetry assumption is unnecessary, according to Horn & Johnson, not

A.4Symmetric matrices are certainly pervasive in the our chosen subject as well.A.5A totally complex perspective is not necessarily more advantageous. The positive

semidefinite cone, for example, is not selfdual (§2.13.5) in the ambient space of Hermitianmatrices. [192, §II]

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588 APPENDIX A. LINEAR ALGEBRA

because nonHermitian matrices could be regarded positive semidefinite,rather because nonHermitian (includes nonsymmetric real) matrices are notcomparable on the real line under xHAx . Yet that complex edifice isdismantled in the test of real matrices (1407) because the domain of testis no longer necessarily complex; meaning, xTAx will certainly always bereal, regardless of symmetry, and so real A will always be comparable.

In summary, if we limit the domain of test to all x in Rn as in (1407),then nonsymmetric real matrices are admitted to the realm of semidefinitematrices because they become comparable on the real line. One importantexception occurs for rank-one matrices Ψ=uvT where u and v are realvectors: Ψ is positive semidefinite if and only if Ψ=uuT. (§A.3.1.0.7)

We might choose to expand the domain of test to all x in Cn so that onlysymmetric matrices would be comparable. The alternative to expandingdomain of test is to assume all matrices of interest to be symmetric; thatis commonly done, hence the synonymous relationship with semidefinitematrices.

A.2.1.0.1 Example. Nonsymmetric positive definite product.Horn & Johnson assert and Zhang agrees:

If A,B∈ Cn×n are positive definite, then we know that theproduct AB is positive definite if and only if AB is Hermitian.[199, §7.6, prob.10] [390, §6.2, §3.2]

Implicitly in their statement, A and B are assumed individually Hermitianand the domain of test is assumed complex.

We prove that assertion to be false for real matrices under (1407) thatadopts a real domain of test.

AT = A =

3 0 −1 00 5 1 0

−1 1 4 10 0 1 4

, λ(A) =

5.94.53.42.0

(1412)

BT = B =

4 4 −1 −14 5 0 0

−1 0 5 1−1 0 1 4

, λ(B) =

8.85.53.30.24

(1413)

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A.3. PROPER STATEMENTS 589

(AB)T 6= AB =

13 12 −8 −419 25 5 1−5 1 22 9−5 0 9 17

, λ(AB) =

36.29.10.0.72

(1414)

12(AB + (AB)T) =

13 15.5 −6.5 −4.515.5 25 3 0.5−6.5 3 22 9−4.5 0.5 9 17

, λ(

12(AB + (AB)T)

)

=

36.30.10.

0.014

(1415)

Whenever A∈ Sn+ and B∈ Sn

+ , then λ(AB)=λ(√AB

√A) will always

be a nonnegative vector by (1439) and Corollary A.3.1.0.5. Yet positivedefiniteness of product AB is certified instead by the nonnegative eigenvaluesλ(

12(AB + (AB)T)

)

in (1415) (§A.3.1.0.1) despite the fact that AB is notsymmetric.A.6 Horn & Johnson and Zhang resolve the anomaly by choosingto exclude nonsymmetric matrices and products; they do so by expandingthe domain of test to Cn. 2

A.3 Proper statements

of positive semidefiniteness

Unlike Horn & Johnson and Zhang, we never adopt a complex domain of testwith real matrices. So motivated is our consideration of proper statementsof positive semidefiniteness under real domain of test. This restriction,ironically, complicates the facts when compared to corresponding statementsfor the complex case (found elsewhere [199] [390]).

We state several fundamental facts regarding positive semidefiniteness ofreal matrix A and the product AB and sum A +B of real matrices underfundamental real test (1407); a few require proof as they depart from thestandard texts, while those remaining are well established or obvious.

A.6It is a little more difficult to find a counter-example in R2×2 or R3×3 ; which mayhave served to advance any confusion.

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590 APPENDIX A. LINEAR ALGEBRA

A.3.0.0.1 Theorem. Positive (semi)definite matrix.A∈ SM is positive semidefinite if and only if for each and every vector x∈RM

of unit norm, ‖x‖= 1 ,A.7 we have xTAx≥ 0 (1407);

A º 0 ⇔ tr(xxTA) = xTAx ≥ 0 ∀xxT (1416)

Matrix A∈ SM is positive definite if and only if for each and every ‖x‖= 1we have xTAx > 0 ;

A ≻ 0 ⇔ tr(xxTA) = xTAx > 0 ∀xxT, xxT 6= 0 (1417)

Proof. Statements (1416) and (1417) are each a particular instanceof dual generalized inequalities (§2.13.2) with respect to the positivesemidefinite cone; videlicet, [357]

A º 0 ⇔ 〈xxT, A〉 ≥ 0 ∀xxT(º 0)

A ≻ 0 ⇔ 〈xxT, A〉 > 0 ∀xxT(º 0) , xxT 6= 0(1418)

This says: positive semidefinite matrix A must belong to the normal sideof every hyperplane whose normal is an extreme direction of the positivesemidefinite cone. Relations (1416) and (1417) remain true when xxT isreplaced with “for each and every” positive semidefinite matrix X∈ SM

+

(§2.13.5) of unit norm, ‖X‖= 1, as in

A º 0 ⇔ tr(XA) ≥ 0 ∀X∈ SM+

A ≻ 0 ⇔ tr(XA) > 0 ∀X∈ SM+ , X 6= 0

(1419)

But that condition is more than what is necessary. By the discretizedmembership theorem in §2.13.4.2.1, the extreme directions xxT of the positivesemidefinite cone constitute a minimal set of generators necessary andsufficient for discretization of dual generalized inequalities (1419) certifyingmembership to that cone. ¨

A.7The traditional condition requiring all x∈RM for defining positive (semi)definitenessis actually more than what is necessary. The set of norm-1 vectors is necessary andsufficient to establish positive semidefiniteness; actually, any particular norm and anynonzero norm-constant will work.

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A.3. PROPER STATEMENTS 591

A.3.1 Semidefiniteness, eigenvalues, nonsymmetric

When A∈Rn×n, let λ(

12(A +AT)

)

∈ Rn denote eigenvalues of thesymmetrized matrixA.8 arranged in nonincreasing order. By positive semidefiniteness of A∈Rn×n we mean,A.9 [268, §1.3.1]

(confer §A.3.1.0.1)

xTAx ≥ 0 ∀x∈Rn ⇔ A +AT º 0 ⇔ λ(A +AT) º 0 (1420) (§2.9.0.1)A º 0 ⇒ AT = A (1421)

A º B ⇔ A−B º 0 ; A º 0 or B º 0 (1422)

xTAx≥ 0 ∀x ; AT = A (1423) Matrix symmetry is not intrinsic to positive semidefiniteness;

AT = A , λ(A) º 0 ⇒ xTAx ≥ 0 ∀x (1424)

λ(A) º 0 ⇐ AT = A , xTAx ≥ 0 ∀x (1425) If AT = A thenλ(A) º 0 ⇔ A º 0 (1426)

meaning, matrix A belongs to the positive semidefinite cone in thesubspace of symmetric matrices if and only if its eigenvalues belong tothe nonnegative orthant.

〈A , A〉 = 〈λ(A) , λ(A)〉 (45) For µ∈R , A∈Rn×n, and vector λ(A)∈Cn holding the orderedeigenvalues of A

λ(µI + A) = µ1 + λ(A) (1427)

Proof: A=MJM−1 and µI + A = M(µI + J )M−1 where Jis the Jordan form for A ; [328, §5.6, App.B] id est, δ(J ) = λ(A) ,so λ(µI + A) = δ(µI + J ) because µI + J is also a Jordan form. ¨

A.8The symmetrization of A is (A +AT)/2. λ(

12 (A +AT)

)

= λ(A +AT)/2.A.9Strang agrees [328, p.334] it is not λ(A) that requires observation. Yet he is mistaken

by proposing the Hermitian part alone xH(A+AH)x be tested, because the anti-Hermitianpart does not vanish under complex test unless A is Hermitian. (1411)

Page 592: Convex Optimization & Euclidean Distance Geometry

592 APPENDIX A. LINEAR ALGEBRA

By similar reasoning,

λ(I + µA) = 1 + λ(µA) (1428)

For vector σ(A) holding the singular values of any matrix A

σ(I + µATA) = π(|1 + µσ(ATA)|) (1429)

σ(µI + ATA) = π(|µ1 + σ(ATA)|) (1430)

where π is the nonlinear permutation-operator sorting its vectorargument into nonincreasing order. For A∈ SM and each and every ‖w‖= 1 [199, §7.7, prob.9]

wTAw ≤ µ ⇔ A ¹ µI ⇔ λ(A) ¹ µ1 (1431) [199, §2.5.4] (confer (44))

A is normal matrix ⇔ ‖A‖2F = λ(A)Tλ(A) (1432) For A∈Rm×n

ATA º 0 , AAT º 0 (1433)

because, for dimensionally compatible vector x , xTATAx = ‖Ax‖22 ,

xTAATx = ‖ATx‖22 . For A∈ Rn×n and c∈R

tr(cA) = c tr(A) = c1Tλ(A) (§A.1.1 no.4)

For m a nonnegative integer, (1831)

det(Am) =n

i=1

λ(A)mi (1434)

tr(Am) =n

i=1

λ(A)mi (1435)

Page 593: Convex Optimization & Euclidean Distance Geometry

A.3. PROPER STATEMENTS 593 For A diagonalizable (§A.5), A = SΛS−1, (confer [328, p.255])

rank A = rank δ(λ(A)) = rank Λ (1436)

meaning, rank is equal to the number of nonzero eigenvalues in vector

λ(A) , δ(Λ) (1437)

by the 0 eigenvalues theorem (§A.7.3.0.1). (Ky Fan) For A ,B∈ Sn [53, §1.2] (confer (1704))

tr(AB) ≤ λ(A)Tλ(B) (1690)

with equality (Theobald) when A and B are simultaneouslydiagonalizable [199] with the same ordering of eigenvalues. For A∈Rm×n and B∈Rn×m

tr(AB) = tr(BA) (1438)

and η eigenvalues of the product and commuted product are identical,including their multiplicity; [199, §1.3.20] id est,

λ(AB)1:η = λ(BA)1:η , η,minm , n (1439)

Any eigenvalues remaining are zero. By the 0 eigenvalues theorem(§A.7.3.0.1),

rank(AB) = rank(BA) , AB and BA diagonalizable (1440) For any compatible matrices A ,B [199, §0.4]

minrank A , rank B ≥ rank(AB) (1441) For A,B∈ Sn+

rank A + rankB ≥ rank(A + B) ≥ minrank A , rank B ≥ rank(AB)(1442) For linearly independent matrices A,B∈ Sn

+ (§2.1.2, R(A)∩R(B)=0,R(AT)∩R(BT)=0, §B.1.1),

rank A + rank B = rank(A + B) > minrank A , rank B ≥ rank(AB)(1443)

Page 594: Convex Optimization & Euclidean Distance Geometry

594 APPENDIX A. LINEAR ALGEBRA Because R(ATA)=R(AT) and R(AAT)=R(A) (p.624), for anyA∈Rm×n

rank(AAT) = rank(ATA) = rank A = rank AT (1444) For A∈Rm×n having no nullspace, and for any B∈Rn×k

rank(AB) = rank(B) (1445)

Proof. For any compatible matrix C , N (CAB)⊇ N (AB)⊇ N (B)is obvious. By assumption ∃A† Ä A†A = I . Let C = A† , thenN (AB)=N (B) and the stated result follows by conservation ofdimension (1549). ¨ For A∈ Sn and any nonsingular matrix Y

inertia(A) = inertia(YAY T) (1446)

a.k.a, Sylvester’s law of inertia. (1489) [110, §2.4.3] For A,B∈Rn×n square, [199, §0.3.5]

det(AB) = det(BA) (1447)

det(AB) = det A det B (1448)

Yet for A∈Rm×n and B∈Rn×m [75, p.72]

det(I + AB) = det(I + BA) (1449) For A,B∈ Sn, product AB is symmetric if and only if AB iscommutative;

(AB)T = AB ⇔ AB = BA (1450)

Proof. (⇒) Suppose AB=(AB)T. (AB)T=BTAT=BA .AB=(AB)T ⇒ AB=BA .(⇐) Suppose AB=BA . BA=BTAT=(AB)T. AB=BA ⇒AB=(AB)T. ¨

Commutativity alone is insufficient for symmetry of the product.[328, p.26]

Page 595: Convex Optimization & Euclidean Distance Geometry

A.3. PROPER STATEMENTS 595 Diagonalizable matrices A,B∈Rn×n commute if and only if they aresimultaneously diagonalizable. [199, §1.3.12] A product of diagonalmatrices is always commutative. For A,B∈ Rn×n and AB = BA

xTAx ≥ 0 , xTBx ≥ 0 ∀x ⇒ λ(A+AT)i λ(B+BT)i ≥ 0 ∀ i < xTABx ≥ 0 ∀x(1451)

the negative result arising because of the schism between the productof eigenvalues λ(A + AT)i λ(B + BT)i and the eigenvalues of thesymmetrized matrix product λ(AB + (AB)T)i . For example, X2 isgenerally not positive semidefinite unless matrix X is symmetric; then(1433) applies. Simply substituting symmetric matrices changes theoutcome: For A,B∈ Sn and AB = BA

A º 0 , B º 0 ⇒ λ(AB)i =λ(A)i λ(B)i≥ 0 ∀ i ⇔ AB º 0 (1452)

Positive semidefiniteness of A and B is sufficient but not a necessarycondition for positive semidefiniteness of the product AB .

Proof. Because all symmetric matrices are diagonalizable, (§A.5.1)[328, §5.6] we have A=SΛST and B=T∆TT, where Λ and ∆ arereal diagonal matrices while S and T are orthogonal matrices. Because(AB)T =AB , then T must equal S , [199, §1.3] and the eigenvalues ofA are ordered in the same way as those of B ; id est, λ(A)i =δ(Λ)i andλ(B)i =δ(∆)i correspond to the same eigenvector.(⇒) Assume λ(A)i λ(B)i≥ 0 for i=1 . . . n . AB=SΛ∆ST issymmetric and has nonnegative eigenvalues contained in diagonalmatrix Λ∆ by assumption; hence positive semidefinite by (1420). Nowassume A,Bº 0. That, of course, implies λ(A)i λ(B)i≥ 0 for all ibecause all the individual eigenvalues are nonnegative.(⇐) Suppose AB=SΛ∆STº 0. Then Λ∆º 0 by (1420),and so all products λ(A)i λ(B)i must be nonnegative; meaning,sgn(λ(A))= sgn(λ(B)). We may, therefore, conclude nothing aboutthe semidefiniteness of A and B . ¨

Page 596: Convex Optimization & Euclidean Distance Geometry

596 APPENDIX A. LINEAR ALGEBRA For A,B∈ Sn and A º 0 , B º 0 (Example A.2.1.0.1)

AB = BA ⇒ λ(AB)i =λ(A)i λ(B)i ≥ 0 ∀ i ⇒ AB º 0 (1453)

AB = BA ⇒ λ(AB)i ≥ 0 , λ(A)i λ(B)i ≥ 0 ∀ i ⇔ AB º 0 (1454) For A,B∈ Sn [200, §4.2.13]

A º 0 , B º 0 ⇒ A ⊗ B º 0 (1455)

A ≻ 0 , B ≻ 0 ⇒ A ⊗ B ≻ 0 (1456)

and the Kronecker product is symmetric. For A,B∈ Sn [390, §6.2]

A º 0 ⇒ tr A ≥ 0 (1457)

A º 0 , B º 0 ⇒ tr A tr B ≥ tr(AB)≥ 0 (1458)

Because A º 0 , B º 0 ⇒ λ(AB) = λ(√

AB√

A) º 0 by (1439) andCorollary A.3.1.0.5, then we have tr(AB)≥ 0.

A º 0 ⇔ tr(AB)≥ 0 ∀B º 0 (356) For A,B,C∈ Sn (Lowner)

A ¹ B , B ¹ C ⇒ A ¹ C (transitivity)A ¹ B ⇔ A + C ¹ B + C (additivity)A ¹ B , A º B ⇒ A = B (antisymmetry)A ¹ A (reflexivity)

(1459)

A ¹ B , B ≺ C ⇒ A ≺ C (strict transitivity)A ≺ B ⇔ A + C ≺ B + C (strict additivity)

(1460) For A,B∈ Rn×n

xTAx ≥ xTBx ∀x ⇒ tr A ≥ tr B (1461)

Proof. xTAx≥xTBx ∀x ⇔ λ((A−B) + (A−B)T)/2º 0 ⇒tr(A+AT− (B+BT))/2 = tr(A−B)≥ 0. There is no converse. ¨

Page 597: Convex Optimization & Euclidean Distance Geometry

A.3. PROPER STATEMENTS 597 For A,B∈ Sn [390, §6.2] (Theorem A.3.1.0.4)

A º B ⇒ tr A ≥ tr B (1462)

A º B ⇒ δ(A)º δ(B) (1463)

There is no converse, and restriction to the positive semidefinite conedoes not improve the situation. All-strict versions hold.

A º B º 0 ⇒ rank A ≥ rank B (1464)

A º B º 0 ⇒ det A ≥ det B ≥ 0 (1465)

A ≻ B º 0 ⇒ det A > det B ≥ 0 (1466) For A,B∈ int Sn+ [33, §4.2] [199, §7.7.4]

A º B ⇔ A−1 ¹ B−1, A ≻ 0 ⇔ A−1 ≻ 0 (1467) For A,B∈ Sn [390, §6.2]

A º B º 0 ⇒√

A º√

B (1468) For A,B∈ Sn and AB = BA [390, §6.2, prob.3]

A º B º 0 ⇒ Ak º Bk , k=1, 2, . . . (1469)

A.3.1.0.1 Theorem. Positive semidefinite ordering of eigenvalues.For A ,B∈ RM×M , place the eigenvalues of each symmetrized matrix intothe respective vectors λ

(

12(A +AT)

)

, λ(

12(B +BT)

)

∈ RM . Then, [328, §6]

xTAx ≥ 0 ∀x ⇔ λ(

A +AT)

º 0 (1470)

xTAx > 0 ∀x 6= 0 ⇔ λ(

A +AT)

≻ 0 (1471)

because xT(A −AT)x=0. (1408) Now arrange the entries of λ(

12(A +AT)

)

and λ(

12(B +BT)

)

in nonincreasing order so λ(

12(A +AT)

)

1holds the

largest eigenvalue of symmetrized A while λ(

12(B +BT)

)

1holds the largest

eigenvalue of symmetrized B , and so on. Then [199, §7.7, prob.1, prob.9]for κ ∈ R

xTAx ≥ xTBx ∀x ⇒ λ(

A +AT)

º λ(

B +BT)

xTAx ≥ xTI x κ ∀x ⇔ λ(

12(A +AT)

)

º κ1(1472)

Page 598: Convex Optimization & Euclidean Distance Geometry

598 APPENDIX A. LINEAR ALGEBRA

Now let A ,B ∈ SM have diagonalizations A=QΛQT and B =UΥUT withλ(A)= δ(Λ) and λ(B)= δ(Υ) arranged in nonincreasing order. Then

A º B ⇔ λ(A−B) º 0 (1473)

A º B ⇒ λ(A) º λ(B) (1474)

A º B : λ(A) º λ(B) (1475)

STAS º B ⇐ λ(A) º λ(B) (1476)

where S = QUT. [390, §7.5] ⋄

A.3.1.0.2 Theorem. (Weyl) Eigenvalues of sum. [199, §4.3.1]For A ,B∈ RM×M , place the eigenvalues of each symmetrized matrix intothe respective vectors λ

(

12(A +AT)

)

, λ(

12(B +BT)

)

∈ RM in nonincreasingorder so λ

(

12(A +AT)

)

1holds the largest eigenvalue of symmetrized A while

λ(

12(B +BT)

)

1holds the largest eigenvalue of symmetrized B , and so on.

Then, for any k∈1 . . . Mλ(

A +AT)

k+ λ

(

B +BT)

M≤ λ

(

(A +AT) + (B +BT))

k≤ λ

(

A +AT)

k+ λ

(

B +BT)

1

(1477)

Weyl’s theorem establishes: concavity of the smallest λM and convexityof the largest eigenvalue λ1 of a symmetric matrix, via (475), and positivesemidefiniteness of a sum of positive semidefinite matrices; for A ,B∈ SM

+

λ(A)k + λ(B)M ≤ λ(A + B)k ≤ λ(A)k + λ(B)1 (1478)

Because SM+ is a convex cone (§2.9.0.0.1), then by (166)

A ,B º 0 ⇒ ζA + ξB º 0 for all ζ , ξ ≥ 0 (1479)

A.3.1.0.3 Corollary. Eigenvalues of sum and difference. [199, §4.3]For A∈ SM and B∈ SM

+ , place the eigenvalues of each matrix into therespective vectors λ(A) , λ(B)∈ RM in nonincreasing order so λ(A)1 holdsthe largest eigenvalue of A while λ(B)1 holds the largest eigenvalue of B ,and so on. Then, for any k∈1 . . . M

λ(A − B)k ≤ λ(A)k ≤ λ(A +B)k (1480)

Page 599: Convex Optimization & Euclidean Distance Geometry

A.3. PROPER STATEMENTS 599

When B is rank-one positive semidefinite, the eigenvalues interlace; id est,for B = qqT

λ(A)k−1 ≤ λ(A − qqT)k ≤ λ(A)k ≤ λ(A + qqT)k ≤ λ(A)k+1 (1481)

A.3.1.0.4 Theorem. Positive (semi)definite principal submatrices.A.10 A∈SM is positive semidefinite if and only if all M principal submatricesof dimension M−1 are positive semidefinite and det A is nonnegative. A ∈ SM is positive definite if and only if any one principal submatrixof dimension M−1 is positive definite and detA is positive. ⋄

If any one principal submatrix of dimension M−1 is not positive definite,conversely, then A can neither be. Regardless of symmetry, if A∈RM×M ispositive (semi)definite, then the determinant of each and every principalsubmatrix is (nonnegative) positive. [268, §1.3.1]

A.3.1.0.5 Corollary. Positive (semi)definite symmetric products.[199, p.399] If A∈ SM is positive definite and any particular dimensionally

compatible matrix Z has no nullspace, then ZTAZ is positive definite. If matrix A∈ SM is positive (semi)definite then, for any matrix Z ofcompatible dimension, ZTAZ is positive semidefinite. A∈ SM is positive (semi)definite if and only if there exists a nonsingularZ such that ZTAZ is positive (semi)definite. If A∈ SM is positive semidefinite and singular it remains possible, forsome skinny Z∈ RM×N with N <M , that ZTAZ becomes positivedefinite.A.11 ⋄

A.10A recursive condition for positive (semi)definiteness, this theorem is a synthesis offacts from [199, §7.2] [328, §6.3] (confer [268, §1.3.1]). A principal submatrix is formedby discarding any subset of rows and columns having the same indices. There areM !/(1!(M−1)!) principal 1×1 submatrices, M !/(2!(M−2)!) principal 2×2 submatrices,and so on, totaling 2M − 1 principal submatrices including A itself. By loading y in yTAywith various patterns of ones and zeros, it follows that any principal submatrix must bepositive (semi)definite whenever A is.A.11Using the interpretation in §E.6.4.3, this means coefficients of orthogonal projection ofvectorized A on a subset of extreme directions from SM

+ determined by Z can be positive.

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600 APPENDIX A. LINEAR ALGEBRA

We can deduce from these, given nonsingular matrix Z and any particulardimensionally compatible Y : matrix A∈ SM is positive semidefinite if and

only if

[

ZT

Y T

]

A [Z Y ] is positive semidefinite. In other words, from the

Corollary it follows: for dimensionally compatible Z A º 0 ⇔ ZTAZ º 0 and ZT has a left inverse

Products such as Z†Z and ZZ† are symmetric and positive semidefinitealthough, given Aº 0, Z†AZ and ZAZ† are neither necessarily symmetricor positive semidefinite.

A.3.1.0.6 Theorem. Symmetric projector semidefinite. [20, §III][21, §6] [218, p.55] For symmetric idempotent matrices P and R

P ,R º 0

P º R ⇔ R(P ) ⊇ R(R) ⇔ N (P ) ⊆ N (R)(1482)

Projector P is never positive definite [330, §6.5, prob.20] unless it is theidentity matrix. ⋄

A.3.1.0.7 Theorem. Symmetric positive semidefinite.Given real matrix Ψ with rank Ψ = 1

Ψ º 0 ⇔ Ψ = uuT (1483)

where u is some real vector; id est, symmetry is necessary and sufficient forpositive semidefiniteness of a rank-1 matrix. ⋄

Proof. Any rank-one matrix must have the form Ψ = uvT. (§B.1)Suppose Ψ is symmetric; id est, v = u . For all y∈RM , yTuuTy ≥ 0.Conversely, suppose uvT is positive semidefinite. We know that can hold ifand only if uvT + vuT º 0 ⇔ for all normalized y∈RM , 2 yTu vTy ≥ 0 ;but that is possible only if v = u . ¨

The same does not hold true for matrices of higher rank, as Example A.2.1.0.1shows.

Page 601: Convex Optimization & Euclidean Distance Geometry

A.4. SCHUR COMPLEMENT 601

A.4 Schur complement

Consider Schur-form partitioned matrix G : Given AT = A and CT = C ,then [57]

G =

[

A BBT C

]

º 0

⇔ A º 0 , BT(I−AA†) = 0 , C−BTA†B º 0⇔ C º 0 , B(I−CC†) = 0 , A−BC†BT º 0

(1484)

where A† denotes the Moore-Penrose (pseudo)inverse (§E). In the firstinstance, I − AA† is a symmetric projection matrix orthogonally projectingon N (AT). (1873) It is apparently required

R(B) ⊥ N (AT) (1485)

which precludes A = 0 when B is any nonzero matrix. Note that A≻ 0 ⇒A† =A−1 ; thereby, the projection matrix vanishes. Likewise, in the secondinstance, I − CC† projects orthogonally on N (CT). It is required

R(BT) ⊥ N (CT) (1486)

which precludes C =0 for B nonzero. Again, C ≻ 0 ⇒ C† = C−1. So weget, for A or C nonsingular,

G =

[

A BBT C

]

º 0

⇔A ≻ 0 , C−BTA−1B º 0

or

C ≻ 0 , A−BC−1BT º 0

(1487)

When A is full-rank then, for all B of compatible dimension, R(B) is inR(A). Likewise, when C is full-rank, R(BT) is in R(C). Thus the flavor,for A and C nonsingular,

G =

[

A BBT C

]

≻ 0

⇔ A ≻ 0 , C−BTA−1B ≻ 0⇔ C ≻ 0 , A−BC−1BT ≻ 0

(1488)

where C − BTA−1B is called the Schur complement of A in G , while theSchur complement of C in G is A − BC−1BT. [148, §4.8]

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602 APPENDIX A. LINEAR ALGEBRA

Origin of the term Schur complement is from complementary inertia:[110, §2.4.4] Define

inertia(

G∈ SM)

, p , z , n (1489)

where p , z , n respectively represent number of positive, zero, and negativeeigenvalues of G ; id est,

M = p + z + n (1490)

Then, when A is invertible,

inertia(G) = inertia(A) + inertia(C − BTA−1B) (1491)

and when C is invertible,

inertia(G) = inertia(C) + inertia(A − BC−1BT) (1492)

When A=C =0, denoting by σ(B)∈ Rm+ the nonincreasingly ordered

singular values of matrix B∈ Rm×m, then we have the eigenvalues[53, §1.2, prob.17]

λ(G) = λ

([

0 BBT 0

])

=

[

σ(B)−Ξ σ(B)

]

(1493)

and

inertia(G) = inertia(BTB) + inertia(−BTB) (1494)

where Ξ is the order-reversing permutation matrix defined in (1691).

A.4.0.0.1 Example. Nonnegative polynomial. [33, p.163]Schur-form positive semidefiniteness is sufficient for quadratic polynomialconvexity in x , but it is necessary and sufficient for nonnegativity; videlicet,for all compatible x

[xT 1 ][

A bbT c

] [

x1

]

≥ 0 ⇔ xTAx + 2bTx + c ≥ 0 (1495)

if and only if it can be expressed as a sum of squares of polynomials.[33, p.157] Sublevel set x | xTAx + 2bTx + c ≤ 0 is convex if Aº 0, butthe quadratic polynomial is a convex function if and only if Aº 0. 2

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A.4. SCHUR COMPLEMENT 603

A.4.0.0.2 Example. Schur conditions.From (1457),

[

A BBT C

]

º 0 ⇒ tr(A + C) ≥ 0

m[

A 00T C−BTA−1B

]

º 0 ⇒ tr(C−BTA−1B) ≥ 0

m[

A−BC−1BT 00T C

]

º 0 ⇒ tr(A−BC−1BT) ≥ 0

(1496)

Since tr(C−BTA−1B)≥ 0 ⇔ tr C ≥ tr(BTA−1B) ≥ 0 for example, thenminimization of tr C is necessary and sufficient for minimization of

tr(C−BTA−1B) when both are under constraint

[

A BBT C

]

º 0. 2

A.4.0.0.3 Example. Sparse Schur conditions.Setting matrix A to the identity simplifies the Schur conditions. Oneconsequence relates the definiteness of three quantities:

[

I BBT C

]

º 0 ⇔ C − BTB º 0 ⇔[

I 00T C−BTB

]

º 0 (1497)

2

A.4.0.0.4 Exercise. Eigenvalues λ of sparse Schur-form.Prove: given C−BTB = 0, for B∈ Rm×n and C∈ Sn

λ

([

I BBT C

])

i

=

1 + λ(C)i , 1 ≤ i ≤ n

1 , n < i ≤ m

0 , otherwise

(1498)

H

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604 APPENDIX A. LINEAR ALGEBRA

A.4.0.0.5 Theorem. Rank of partitioned matrices.When symmetric matrix A is invertible and C is symmetric,

rank

[

A BBT C

]

= rank

[

A 00T C−BTA−1B

]

= rankA + rank(C−BTA−1B)

(1499)

equals rank of a block on the main diagonal plus rank of its Schur complement[390, §2.2, prob.7]. Similarly, when symmetric matrix C is invertible and Ais symmetric,

rank

[

A BBT C

]

= rank

[

A − BC−1BT 00T C

]

= rank(A − BC−1BT) + rank C

(1500)

Proof. The first assertion (1499) holds if and only if [199, §0.4.6(c)]

∃ nonsingular X,Y Ä X

[

A BBT C

]

Y =

[

A 00T C−BTA−1B

]

(1501)

Let [199, §7.7.6]

Y = XT =

[

I −A−1B0T I

]

(1502)

¨

A.4.0.0.6 Lemma. Rank of Schur-form block. [133] [131]Matrix B∈ Rm×n has rank B≤ ρ if and only if there exist matrices A∈ Sm

and C∈ Sn such that

rank

[

A 00T C

]

≤ 2ρ and G =

[

A BBT C

]

º 0 (1503)

Schur-form positive semidefiniteness alone implies rank A≥ rank B andrank C≥ rank B . But, even in absence of semidefiniteness, we must alwayshave rankG≥ rank A , rank B, rank C by fundamental linear algebra.

Page 605: Convex Optimization & Euclidean Distance Geometry

A.4. SCHUR COMPLEMENT 605

A.4.1 Determinant

G =

[

A BBT C

]

(1504)

We consider again a matrix G partitioned like (1484), but not necessarilypositive (semi)definite, where A and C are symmetric. When A is invertible,

det G = det A det(C − BTA−1B) (1505)

When C is invertible,

det G = det C det(A − BC−1BT) (1506) When B is full-rank and skinny, C = 0, and A º 0, then [59, §10.1.1]

det G 6= 0 ⇔ A + BBT ≻ 0 (1507)

When B is a (column) vector, then for all C∈ R and all A of dimensioncompatible with G

det G = det(A)C − BTATcofB (1508)

while for C 6= 0

det G = C det(A − 1

CBBT) (1509)

where Acof is the matrix of cofactors [328, §4] corresponding to A . When B is full-rank and fat, A = 0, and C º 0, then

det G 6= 0 ⇔ C + BTB ≻ 0 (1510)

When B is a row-vector, then for A 6= 0 and all C of dimensioncompatible with G

det G = A det(C − 1

ABTB) (1511)

while for all A∈R

det G = det(C)A − BCTcofB

T (1512)

where Ccof is the matrix of cofactors corresponding to C .

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606 APPENDIX A. LINEAR ALGEBRA

A.5 Eigenvalue decomposition

When a square matrix X∈ Rm×m is diagonalizable, [328, §5.6] then

X = SΛS−1 = [ s1 · · · sm ] Λ

wT1...

wTm

=m

i=1

λi siwTi (1513)

where si∈ N (X − λiI )⊆ Cm are l.i. (right-)eigenvectorsA.12 constitutingthe columns of S∈ Cm×m defined by

XS = SΛ rather Xsi , λisi , i = 1 . . . m (1514)

wi∈ N (XT− λiI )⊆ Cm are linearly independent left-eigenvectors of X(eigenvectors of XT) constituting the rows of S−1 defined by [199]

S−1X = ΛS−1 rather wTi X , λiw

Ti , i = 1 . . . m (1515)

and where λi∈ C are eigenvalues (1437)

δ(λ(X)) = Λ ∈ Cm×m (1516)

corresponding to both left and right eigenvectors; id est, λ(X) = λ(XT).There is no connection between diagonalizability and invertibility of X .

[328, §5.2] Diagonalizability is guaranteed by a full set of linearly independenteigenvectors, whereas invertibility is guaranteed by all nonzero eigenvalues.

distinct eigenvalues ⇒ l.i. eigenvectors ⇔ diagonalizablenot diagonalizable ⇒ repeated eigenvalue

(1517)

A.5.0.0.1 Theorem. Real eigenvector.Eigenvectors of a real matrix corresponding to real eigenvalues must be real.

Proof. Ax = λx . Given λ=λ∗, xHAx = λxHx = λ‖x‖2 = xTAx∗ ⇒x = x∗, where xH=x∗T. The converse is equally simple. ¨

A.12Eigenvectors must, of course, be nonzero. The prefix eigen is from the German; inthis context meaning, something akin to “characteristic”. [325, p.14]

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A.5. EIGENVALUE DECOMPOSITION 607

A.5.0.1 Uniqueness

From the fundamental theorem of algebra, [215] which guarantees existenceof zeros for a given polynomial, it follows: eigenvalues, including theirmultiplicity, for a given square matrix are unique; meaning, there is no otherset of eigenvalues for that matrix. (Conversely, many different matrices mayshare the same unique set of eigenvalues; e.g., for any X , λ(X) = λ(XT).)

Uniqueness of eigenvectors, in contrast, disallows multiplicity of the samedirection:

A.5.0.1.1 Definition. Unique eigenvectors.When eigenvectors are unique, we mean: unique to within a real nonzeroscaling, and their directions are distinct.

If S is a matrix of eigenvectors of X as in (1513), for example, then −Sis certainly another matrix of eigenvectors decomposing X with the sameeigenvalues.

For any square matrix, the eigenvector corresponding to a distincteigenvalue is unique; [325, p.220]

distinct eigenvalues ⇒ eigenvectors unique (1518)

Eigenvectors corresponding to a repeated eigenvalue are not unique for adiagonalizable matrix;

repeated eigenvalue ⇒ eigenvectors not unique (1519)

Proof follows from the observation: any linear combination of distincteigenvectors of diagonalizable X , corresponding to a particular eigenvalue,produces another eigenvector. For eigenvalue λ whose multiplicityA.13

dimN (X−λI ) exceeds 1, in other words, any choice of independentvectors from N (X−λI ) (of the same multiplicity) constitutes eigenvectorscorresponding to λ . ¨

A.13A matrix is diagonalizable iff algebraic multiplicity (number of occurrences of sameeigenvalue) equals geometric multiplicity dimN (X−λI ) = m − rank(X−λI ) [325, p.15](number of Jordan blocks w.r.t λ or number of corresponding l.i. eigenvectors).

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608 APPENDIX A. LINEAR ALGEBRA

Caveat diagonalizability insures linear independence which impliesexistence of distinct eigenvectors. We may conclude, for diagonalizablematrices,

distinct eigenvalues ⇔ eigenvectors unique (1520)

A.5.0.2 Invertible matrix

When diagonalizable matrix X∈ Rm×m is nonsingular (no zero eigenvalues),then it has an inverse obtained simply by inverting eigenvalues in (1513):

X−1 = SΛ−1S−1 (1521)

A.5.0.3 eigenmatrix

The (right-)eigenvectors si (1513) are naturally orthogonal wTi sj =0

to left-eigenvectors wi except, for i=1 . . . m , wTi si =1 ; called a

biorthogonality condition [362, §2.2.4] [199] because neither set of left or righteigenvectors is necessarily an orthogonal set. Consequently, each dyad from adiagonalization is an independent (§B.1.1) nonorthogonal projector because

siwTi siw

Ti = siw

Ti (1522)

(whereas the dyads of singular value decomposition are not inherentlyprojectors (confer (1529))).

Dyads of eigenvalue decomposition can be termed eigenmatrices because

X siwTi = λi siw

Ti (1523)

Sum of the eigenmatrices is the identity;

m∑

i=1

siwTi = I (1524)

A.5.1 Symmetric matrix diagonalization

The set of normal matrices is, precisely, that set of all real matrices havinga complete orthonormal set of eigenvectors; [390, §8.1] [330, prob.10.2.31]id est, any matrix X for which XXT = XTX ; [155, §7.1.3] [325, p.3]e.g., orthogonal and circulant matrices [164]. All normal matrices arediagonalizable.

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A.5. EIGENVALUE DECOMPOSITION 609

A symmetric matrix is a special normal matrix whose eigenvalues Λmust be realA.14 and whose eigenvectors S can be chosen to make a realorthonormal set; [330, §6.4] [328, p.315] id est, for X∈ Sm

X = SΛST = [ s1 · · · sm ] Λ

sT1...

sTm

=m

i=1

λi sisTi (1525)

where δ2(Λ) = Λ∈ Sm (§A.1) and S−1 = ST∈ Rm×m (orthogonal matrix,§B.5) because of symmetry: SΛS−1 = S−TΛST. By the 0 eigenvaluestheorem (§A.7.3.0.1),

Rsi |λi 6=0 = R(A) = R(AT)Rsi |λi =0 = N (AT) = N (A)

(1526)

A.5.1.1 Diagonal matrix diagonalization

Because arrangement of eigenvectors and their corresponding eigenvalues isarbitrary, we almost always arrange eigenvalues in nonincreasing order asis the convention for singular value decomposition. Then to diagonalize asymmetric matrix that is already a diagonal matrix, orthogonal matrix Sbecomes a permutation matrix.

A.5.1.2 Invertible symmetric matrix

When symmetric matrix X∈ Sm is nonsingular (invertible), then its inverse(obtained by inverting eigenvalues in (1525)) is also symmetric:

X−1 = SΛ−1ST∈ Sm (1527)

A.5.1.3 Positive semidefinite matrix square root

When X∈ Sm+ , its unique positive semidefinite matrix square root is defined

√X , S

√Λ ST ∈ Sm

+ (1528)

where the square root of nonnegative diagonal matrix√

Λ is taken entrywiseand positive. Then X =

√X√

X .

A.14Proof. Suppose λi is an eigenvalue corresponding to eigenvector si of real A=AT.Then sH

i Asi = sTi As∗i (by transposition) ⇒ s∗Ti λisi = sT

i λ∗is

∗i because (Asi)

∗= (λisi)∗ by

assumption. So we have λi‖si‖2 = λ∗i‖si‖2. There is no converse. ¨

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610 APPENDIX A. LINEAR ALGEBRA

A.6 Singular value decomposition, SVD

A.6.1 Compact SVD

[155, §2.5.4] For any A∈Rm×n

A = UΣQT = [u1 · · · uη ] Σ

qT1...

qTη

i=1

σi uiqTi

U ∈ Rm×η, Σ ∈ Rη×η, Q ∈ Rn×η

(1529)

where U and Q are always skinny-or-square each having orthonormalcolumns, and where

η , minm , n (1530)

Square matrix Σ is diagonal (§A.1.1)

δ2(Σ) = Σ ∈ Rη×η (1531)

holding the singular values σi∈ R of A which are always arranged innonincreasing order by convention and are related to eigenvalues λ byA.15

σ(A)i = σ(AT)i =

λ(ATA)i =√

λ(AAT)i = λ(√

ATA)

i= λ

(√AAT

)

i> 0 , 1 ≤ i ≤ ρ

0 , ρ < i ≤ η(1532)

of which the last η−ρ are 0 ,A.16 where

ρ , rank A = rank Σ (1533)

A point sometimes lost: Any real matrix may be decomposed in terms ofits real singular values σ(A)∈ Rη and real matrices U and Q as in (1529),where [155, §2.5.3]

Rui |σi 6=0 = R(A)Rui |σi =0 ⊆ N (AT)Rqi |σi 6=0 = R(AT)Rqi |σi =0 ⊆ N (A)

(1534)

A.15When matrix A is normal, σ(A) = |λ(A)|. [390, §8.1]A.16For η = n , σ(A) =

λ(ATA) = λ(

ATA)

.

For η = m , σ(A) =√

λ(AAT) = λ(

AAT)

.

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A.6. SINGULAR VALUE DECOMPOSITION, SVD 611

A.6.2 Subcompact SVD

Some authors allow only nonzero singular values. In that case the compactdecomposition can be made smaller; it can be redimensioned in terms ofrank ρ because, for any A∈Rm×n

ρ = rankA = rank Σ = max i∈1 . . . η | σi 6= 0 ≤ η (1535) There are η singular values. For any flavor SVD, rank is equivalent tothe number of nonzero singular values on the main diagonal of Σ .

Now

A = UΣQT = [u1 · · · uρ ] Σ

qT1...

qTρ

i=1

σi uiqTi

U ∈ Rm×ρ, Σ ∈ Rρ×ρ, Q ∈ Rn×ρ

(1536)

where the main diagonal of diagonal matrix Σ has no 0 entries, and

Rui = R(A)Rqi = R(AT)

(1537)

A.6.3 Full SVD

Another common and useful expression of the SVD makes U and Qsquare; making the decomposition larger than compact SVD. Completingthe nullspace bases in U and Q from (1534) provides what is called thefull singular value decomposition of A∈Rm×n [328, App.A]. Orthonormalmatrices U and Q become orthogonal matrices (§B.5):

Rui |σi 6=0 = R(A)Rui |σi =0 = N (AT)Rqi |σi 6=0 = R(AT)Rqi |σi =0 = N (A)

(1538)

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612 APPENDIX A. LINEAR ALGEBRA

For any matrix A having rank ρ (= rank Σ)

A = UΣQT = [u1 · · · um ] Σ

qT1...

qTn

i=1

σi uiqTi

=[

m×ρ basisR(A) m×m−ρ basisN (AT)]

σ1

σ2

. . .

(

n×ρ basisR(AT))T

(n×n−ρ basisN (A))T

U ∈ Rm×m, Σ ∈ Rm×n, Q ∈ Rn×n (1539)

where upper limit of summation η is defined in (1530). Matrix Σ is nolonger necessarily square, now padded with respect to (1531) by m−ηzero rows or n−η zero columns; the nonincreasingly ordered (possibly 0)singular values appear along its main diagonal as for compact SVD (1532).

An important geometrical interpretation of SVD is given in Figure 153

for m = n = 2 : The image of the unit sphere under any m× n matrixmultiplication is an ellipse. Considering the three factors of the SVDseparately, note that QT is a pure rotation of the circle. Figure 153 showshow the axes q1 and q2 are first rotated by QT to coincide with the coordinateaxes. Second, the circle is stretched by Σ in the directions of the coordinateaxes to form an ellipse. The third step rotates the ellipse by U into itsfinal position. Note how q1 and q2 are rotated to end up as u1 and u2 , theprincipal axes of the final ellipse. A direct calculation shows that Aqj = σj uj .Thus qj is first rotated to coincide with the j th coordinate axis, stretched bya factor σj , and then rotated to point in the direction of uj . All of thisis beautifully illustrated for 2×2 matrices by the Matlab code eigshow.m

(see [331]).

A direct consequence of the geometric interpretation is that the largestsingular value σ1 measures the “magnitude” of A (its 2-norm):

‖A‖2 = sup‖x‖2=1

‖Ax‖2 = σ1 (1540)

This means that ‖A‖2 is the length of the longest principal semiaxis of theellipse.

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A.6. SINGULAR VALUE DECOMPOSITION, SVD 613

u1

u2

q1

q1

q1

q2

q2

q2

U

QT

Σ

Figure 153: Geometrical interpretation of full SVD [267]: Image of circlex∈R2 | ‖x‖2 =1 under matrix multiplication Ax is, in general, an ellipse.For the example illustrated, U , [ u1 u2 ]∈ R2×2, Q, [ q1 q2 ]∈ R2×2.

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614 APPENDIX A. LINEAR ALGEBRA

Expressions for U , Q , and Σ follow readily from (1539),

AATU = UΣΣT and ATAQ = QΣTΣ (1541)

demonstrating that the columns of U are the eigenvectors of AAT and thecolumns of Q are the eigenvectors of ATA . −Muller, Magaia, & Herbst [267]

A.6.4 Pseudoinverse by SVD

Matrix pseudoinverse (§E) is nearly synonymous with singular valuedecomposition because of the elegant expression, given A = UΣQT∈ Rm×n

A† = QΣ†TUT ∈ Rn×m (1542)

that applies to all three flavors of SVD, where Σ† simply inverts nonzeroentries of matrix Σ .

Given symmetric matrix A∈ Sn and its diagonalization A = SΛST

(§A.5.1), its pseudoinverse simply inverts all nonzero eigenvalues:

A† = SΛ†ST∈ Sn (1543)

A.6.5 SVD of symmetric matrices

From (1532) and (1528) for A = AT

σ(A)i =

λ(A2)i = λ(√

A2)

i= |λ(A)i| > 0 , 1 ≤ i ≤ ρ

0 , ρ < i ≤ η(1544)

A.6.5.0.1 Definition. Step function. (confer §4.3.2.0.1)Define the signum-like quasilinear function ψ : Rn→ Rn that takes value 1corresponding to a 0-valued entry in its argument:

ψ(a) ,

[

limxi→ai

xi

|xi|=

1 , ai ≥ 0−1 , ai < 0

, i=1 . . . n

]

∈ Rn (1545)

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A.7. ZEROS 615

Eigenvalue signs of a symmetric matrix having diagonalizationA = SΛST (1525) can be absorbed either into real U or real Q from thefull SVD; [347, p.34] (confer §C.4.2.1)

A = SΛST = Sδ(ψ(δ(Λ))) |Λ|ST , U ΣQT ∈ Sn (1546)

or

A = SΛST = S|Λ| δ(ψ(δ(Λ)))ST , UΣ QT ∈ Sn (1547)

where matrix of singular values Σ = |Λ| denotes entrywise absolute value ofdiagonal eigenvalue matrix Λ .

A.7 Zeros

A.7.1 norm zero

For any given norm, by definition,

‖x‖ℓ= 0 ⇔ x = 0 (1548)

Consequently, a generally nonconvex constraint in x like ‖Ax − b‖ = κbecomes convex when κ = 0.

A.7.2 0 entry

If a positive semidefinite matrix A = [Aij] ∈ Rn×n has a 0 entry Aii on itsmain diagonal, then Aij + Aji = 0 ∀ j . [268, §1.3.1]

Any symmetric positive semidefinite matrix having a 0 entry on its maindiagonal must be 0 along the entire row and column to which that 0 entrybelongs. [155, §4.2.8] [199, §7.1, prob.2]

A.7.3 0 eigenvalues theorem

This theorem is simple, powerful, and widely applicable:

A.7.3.0.1 Theorem. Number of 0 eigenvalues.For any matrix A∈ Rm×n

rank(A) + dimN (A) = n (1549)

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616 APPENDIX A. LINEAR ALGEBRA

by conservation of dimension. [199, §0.4.4]For any square matrix A∈ Rm×m, the number of 0 eigenvalues is at least

equal to dimN (A)

dimN (A) ≤ number of 0 eigenvalues ≤ m (1550)

while the eigenvectors corresponding to those 0 eigenvalues belong to N (A).[328, §5.1]A.17

For diagonalizable matrix A (§A.5), the number of 0 eigenvalues isprecisely dimN (A) while the corresponding eigenvectors span N (A). Thereal and imaginary parts of the eigenvectors remaining span R(A).

(TRANSPOSE.)Likewise, for any matrix A∈ Rm×n

rank(AT) + dimN (AT) = m (1551)

For any square A∈ Rm×m, the number of 0 eigenvalues is at least equalto dimN (AT) = dimN (A) while the left-eigenvectors (eigenvectors of AT)corresponding to those 0 eigenvalues belong to N (AT).

For diagonalizable A , the number of 0 eigenvalues is preciselydimN (AT) while the corresponding left-eigenvectors span N (AT). The realand imaginary parts of the left-eigenvectors remaining span R(AT). ⋄

Proof. First we show, for a diagonalizable matrix, the number of 0eigenvalues is precisely the dimension of its nullspace while the eigenvectorscorresponding to those 0 eigenvalues span the nullspace:

Any diagonalizable matrix A∈ Rm×m must possess a complete set oflinearly independent eigenvectors. If A is full-rank (invertible), then allm=rank(A) eigenvalues are nonzero. [328, §5.1]

A.17We take as given the well-known fact that the number of 0 eigenvalues cannot be lessthan dimension of the nullspace. We offer an example of the converse:

A =

1 0 1 00 0 1 00 0 0 01 0 0 0

dimN (A) = 2, λ(A) = [ 0 0 0 1 ]T; three eigenvectors in the nullspace but only two areindependent. The right-hand side of (1550) is tight for nonzero matrices; e.g., (§B.1) dyaduvT∈ Rm×m has m 0-eigenvalues when u∈ v⊥.

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A.7. ZEROS 617

Suppose rank(A)< m . Then dimN (A) = m−rank(A). Thus there isa set of m−rank(A) linearly independent vectors spanning N (A). Eachof those can be an eigenvector associated with a 0 eigenvalue becauseA is diagonalizable ⇔ ∃ m linearly independent eigenvectors. [328, §5.2]Eigenvectors of a real matrix corresponding to 0 eigenvalues must be real.A.18

Thus A has at least m−rank(A) eigenvalues equal to 0.Now suppose A has more than m−rank(A) eigenvalues equal to 0.

Then there are more than m−rank(A) linearly independent eigenvectorsassociated with 0 eigenvalues, and each of those eigenvectors must be inN (A). Thus there are more than m−rank(A) linearly independent vectorsin N (A) ; a contradiction.

Diagonalizable A therefore has rank(A) nonzero eigenvalues and exactlym−rank(A) eigenvalues equal to 0 whose corresponding eigenvectorsspan N (A).

By similar argument, the left-eigenvectors corresponding to 0 eigenvaluesspan N (AT).

Next we show when A is diagonalizable, the real and imaginary parts ofits eigenvectors (corresponding to nonzero eigenvalues) span R(A) :

The (right-)eigenvectors of a diagonalizable matrix A∈ Rm×m are linearlyindependent if and only if the left-eigenvectors are. So, matrix A hasa representation in terms of its right- and left-eigenvectors; from thediagonalization (1513), assuming 0 eigenvalues are ordered last,

A =m

i=1

λi siwTi =

k ≤m∑

i=1λi 6=0

λi siwTi (1552)

From the linearly independent dyads theorem (§B.1.1.0.2), the dyads siwTi

must be independent because each set of eigenvectors are; hence rank A = k ,the number of nonzero eigenvalues. Complex eigenvectors and eigenvaluesare common for real matrices, and must come in complex conjugate pairs forthe summation to remain real. Assume that conjugate pairs of eigenvaluesappear in sequence. Given any particular conjugate pair from (1552), we getthe partial summation

λi siwTi + λ∗

i s∗i w∗Ti = 2 re(λi siw

Ti )

= 2(

re si re(λi wTi ) − im si im(λi w

Ti )

) (1553)

A.18Proof. Let ∗ denote complex conjugation. Suppose A=A∗ and Asi = 0. Thensi = s∗i ⇒ Asi =As∗i ⇒ As∗i = 0. Conversely, As∗i = 0 ⇒ Asi =As∗i ⇒ si = s∗i . ¨

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618 APPENDIX A. LINEAR ALGEBRA

whereA.19 λ∗i , λi+1 , s∗i , si+1 , and w∗

i , wi+1 . Then (1552) isequivalently written

A = 2∑

iλ∈C

λi 6=0

re s2i re(λ2i wT2i) − im s2i im(λ2i w

T2i) +

jλ∈R

λj 6=0

λj sjwTj (1554)

The summation (1554) shows: A is a linear combination of real and imaginaryparts of its (right-)eigenvectors corresponding to nonzero eigenvalues. Thek vectors re si∈ Rm, im si∈ Rm | λi 6=0, i∈1 . . . m must therefore spanthe range of diagonalizable matrix A .

The argument is similar regarding the span of the left-eigenvectors. ¨

A.7.4 0 trace and matrix product

For X,A∈RM×N+ (39)

tr(XTA) = 0 ⇔ X A = A X = 0 (1555)

For X,A∈SM+ [33, §2.6.1, exer.2.8] [356, §3.1]

tr(XA) = 0 ⇔ XA = AX = 0 (1556)

Proof. (⇐) Suppose XA = AX = 0. Then tr(XA)=0 is obvious.(⇒) Suppose tr(XA)=0. tr(XA)= tr(

√AX

√A) whose argument is

positive semidefinite by Corollary A.3.1.0.5. Trace of any square matrix isequivalent to the sum of its eigenvalues. Eigenvalues of a positive semidefinitematrix can total 0 if and only if each and every nonnegative eigenvalue is 0.The only positive semidefinite matrix, having all 0 eigenvalues, resides at theorigin; (confer (1580)) id est,

√AX

√A =

(√X√

A)T√

X√

A = 0 (1557)

implying√

X√

A = 0 which in turn implies√

X(√

X√

A)√

A = XA = 0.Arguing similarly yields AX = 0. ¨

Diagonalizable matrices A and X are simultaneously diagonalizable if andonly if they are commutative under multiplication; [199, §1.3.12] id est, iffthey share a complete set of eigenvectors.

A.19Complex conjugate of w is denoted w∗. Conjugate transpose is denoted wH = w∗T.

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A.7. ZEROS 619

A.7.4.0.1 Example. An equivalence in nonisomorphic spaces.Identity (1556) leads to an unusual equivalence relating convex geometry totraditional linear algebra: The convex sets, given Aº 0

X | 〈X , A〉 = 0 ∩ Xº 0 ≡ X | N (X) ⊇ R(A) ∩ Xº 0 (1558)

(one expressed in terms of a hyperplane, the other in terms of nullspace andrange) are equivalent only when symmetric matrix A is positive semidefinite.

We might apply this equivalence to the geometric center subspace, forexample,

SMc = Y ∈ SM | Y 1 = 0

= Y ∈ SM | N (Y ) ⊇ 1 = Y ∈ SM | R(Y ) ⊆ N (1T)(1960)

from which we derive (confer (970))

SMc ∩ SM

+ ≡ Xº 0 | 〈X , 11T〉 = 0 (1559)

2

A.7.5 Zero definite

The domain over which an arbitrary real matrix A is zero definite can exceedits left and right nullspaces. For any positive semidefinite matrix A∈RM×M

(for A +AT º 0)

x | xTAx = 0 = N (A +AT) (1560)

because ∃R Ä A+AT=RTR , ‖Rx‖=0 ⇔ Rx=0, and N (A+AT)=N (R).Then given any particular vector xp , xT

pAxp = 0 ⇔ xp∈ N (A +AT). Forany positive definite matrix A (for A +AT ≻ 0)

x | xTAx = 0 = 0 (1561)

Further, [390, §3.2, prob.5]

x | xTAx = 0 = RM ⇔ AT = −A (1562)

while

x | xHAx = 0 = CM ⇔ A = 0 (1563)

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620 APPENDIX A. LINEAR ALGEBRA

The positive semidefinite matrix

A =

[

1 20 1

]

(1564)

for example, has no nullspace. Yet

x | xTAx = 0 = x | 1Tx = 0 ⊂ R2 (1565)

which is the nullspace of the symmetrized matrix. Symmetric matrices arenot spared from the excess; videlicet,

B =

[

1 22 1

]

(1566)

has eigenvalues −1, 3 , no nullspace, but is zero definite onA.20

X , x∈R2 | x2 = (−2 ±√

3)x1 (1567)

A.7.5.0.1 Proposition. (Sturm/Zhang) Dyad-decompositions. [334, §5.2]Let positive semidefinite matrix X∈ SM

+ have rank ρ . Then given symmetricmatrix A∈ SM , 〈A , X 〉= 0 if and only if there exists a dyad-decomposition

X =

ρ∑

j=1

xjxTj (1568)

satisfying〈A , xjx

Tj 〉 = 0 for each and every j ∈ 1 . . . ρ (1569)

The dyad-decomposition of X proposed is generally not that obtainedfrom a standard diagonalization by eigenvalue decomposition, unless ρ =1or the given matrix A is simultaneously diagonalizable (§A.7.4) with X .That means, elemental dyads in decomposition (1568) constitute a generallynonorthogonal set. Sturm & Zhang give a simple procedure for constructingthe dyad-decomposition [Wıκımization]; matrix A may be regarded as aparameter.

A.20These two lines represent the limit in the union of two generally distinct hyperbolae;id est, for matrix B and set X as defined

limε→0+

x∈R2 | xTBx = ε = X

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A.7. ZEROS 621

A.7.5.0.2 Example. Dyad.The dyad uvT∈ RM×M (§B.1) is zero definite on all x for which eitherxTu=0 or xTv=0 ;

x | xTuvTx = 0 = x | xTu=0 ∪ x | vTx=0 (1570)

id est, on u⊥ ∪ v⊥. Symmetrizing the dyad does not change the outcome:

x | xT(uvT + vuT)x/2 = 0 = x | xTu=0 ∪ x | vTx=0 (1571)

2

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622 APPENDIX A. LINEAR ALGEBRA

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Appendix B

Simple matrices

Mathematicians also attempted to develop algebra of vectors butthere was no natural definition of the product of two vectorsthat held in arbitrary dimensions. The first vector algebra thatinvolved a noncommutative vector product (that is, v×w need notequal w×v) was proposed by Hermann Grassmann in his bookAusdehnungslehre (1844). Grassmann’s text also introduced theproduct of a column matrix and a row matrix, which resulted inwhat is now called a simple or a rank-one matrix. In the late19th century the American mathematical physicist Willard Gibbspublished his famous treatise on vector analysis. In that treatiseGibbs represented general matrices, which he called dyadics, assums of simple matrices, which Gibbs called dyads. Later thephysicist P. A. M. Dirac introduced the term “bra-ket” for whatwe now call the scalar product of a “bra” (row) vector times a“ket” (column) vector and the term “ket-bra” for the product of aket times a bra, resulting in what we now call a simple matrix, asabove. Our convention of identifying column matrices and vectorswas introduced by physicists in the 20th century.

−Marie A. Vitulli [365]

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

623

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624 APPENDIX B. SIMPLE MATRICES

B.1 Rank-one matrix (dyad)

Any matrix formed from the unsigned outer product of two vectors,

Ψ = uvT ∈ RM×N (1572)

where u∈RM and v ∈RN , is rank-one and called a dyad. Conversely, anyrank-one matrix must have the form Ψ . [199, prob.1.4.1] Product −uvT isa negative dyad. For matrix products ABT, in general, we have

R(ABT) ⊆ R(A) , N (ABT) ⊇ N (BT) (1573)

with equality when B =A [328, §3.3, §3.6]B.1 or respectively when B isinvertible and N (A)=0. Yet for all nonzero dyads we have

R(uvT) = R(u) , N (uvT) = N (vT) ≡ v⊥ (1574)

where dim v⊥=N−1.

It is obvious a dyad can be 0 only when u or v is 0 ;

Ψ = uvT = 0 ⇔ u = 0 or v = 0 (1575)

The matrix 2-norm for Ψ is equivalent to Frobenius’ norm;

‖Ψ‖2 = σ1 = ‖uvT‖F = ‖uvT‖2 = ‖u‖ ‖v‖ (1576)

When u and v are normalized, the pseudoinverse is the transposed dyad.Otherwise,

Ψ† = (uvT)† =vuT

‖u‖2 ‖v‖2(1577)

B.1Proof. R(AAT) ⊆ R(A) is obvious.

R(AAT) = AATy | y ∈ Rm⊇ AATy | ATy ∈ R(AT) = R(A) by (140) ¨

Page 625: Convex Optimization & Euclidean Distance Geometry

B.1. RANK-ONE MATRIX (DYAD) 625

R(v)

N (Ψ)=N (vT)

r r

N (uT)

R(Ψ) = R(u)

RN = R(v) ⊕ N (uvT) N (uT) ⊕ R(uvT) = RM

0 0

Figure 154: The four fundamental subspaces [330, §3.6] of any dyadΨ = uvT∈RM×N . Ψ(x), uvTx is a linear mapping from RN to RM . Themap from R(v) to R(u) is bijective. [328, §3.1]

When dyad uvT∈RN×N is square, uvT has at least N−1 0-eigenvaluesand corresponding eigenvectors spanning v⊥. The remaining eigenvector uspans the range of uvT with corresponding eigenvalue

λ = vTu = tr(uvT) ∈ R (1578)

Determinant is a product of the eigenvalues; so, it is always true that

det Ψ = det(uvT) = 0 (1579)

When λ = 1, the square dyad is a nonorthogonal projector projecting on itsrange (Ψ2 =Ψ , §E.6); a projector dyad. It is quite possible that u∈ v⊥ makingthe remaining eigenvalue instead 0 ;B.2 λ = 0 together with the first N−10-eigenvalues; id est, it is possible uvT were nonzero while all its eigenvaluesare 0. The matrix

[

1−1

]

[ 1 1 ]=

[

1 1−1 −1

]

(1580)

for example, has two 0-eigenvalues. In other words, eigenvector u maysimultaneously be a member of the nullspace and range of the dyad.The explanation is, simply, because u and v share the same dimension,dim u = M = dim v = N :

B.2A dyad is not always diagonalizable (§A.5) because its eigenvectors are not necessarilyindependent.

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626 APPENDIX B. SIMPLE MATRICES

Proof. Figure 154 shows the four fundamental subspaces for the dyad.Linear operator Ψ : RN→RM provides a map between vector spaces thatremain distinct when M =N ;

u ∈ R(uvT)

u ∈ N (uvT) ⇔ vTu = 0

R(uvT) ∩ N (uvT) = ∅(1581)

¨

B.1.0.1 rank-one modification

If A∈RN×N is any nonsingular matrix and 1 + vTA−1u 6= 0, then[217, App.6] [390, §2.3, prob.16] [148, §4.11.2] (Sherman-Morrison)

(A + uvT)−1 = A−1 − A−1 uvTA−1

1 + vTA−1u(1582)

B.1.0.2 dyad symmetry

In the specific circumstance that v = u , then uuT∈ RN×N is symmetric,rank-one, and positive semidefinite having exactly N−1 0-eigenvalues. Infact, (Theorem A.3.1.0.7)

uvT º 0 ⇔ v = u (1583)

and the remaining eigenvalue is almost always positive;

λ = uTu = tr(uuT) > 0 unless u=0 (1584)

The matrix[

Ψ uuT 1

]

(1585)

for example, is rank-1 positive semidefinite if and only if Ψ = uuT.

B.1.1 Dyad independence

Now we consider a sum of dyads like (1572) as encountered in diagonalizationand singular value decomposition:

R(

k∑

i=1

siwTi

)

=k

i=1

R(

siwTi

)

=k

i=1

R(si) ⇐ wi ∀ i are l.i. (1586)

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B.1. RANK-ONE MATRIX (DYAD) 627

range of summation is the vector sum of ranges.B.3 (Theorem B.1.1.1.1)Under the assumption the dyads are linearly independent (l.i.), then thevector sums are unique (p.758): for wi l.i. and si l.i.

R(

k∑

i=1

siwTi

)

= R(

s1wT1

)

⊕ . . . ⊕R(

skwTk

)

= R(s1) ⊕ . . . ⊕R(sk) (1587)

B.1.1.0.1 Definition. Linearly independent dyads. [207, p.29, thm.11][336, p.2] The set of k dyads

siwTi | i=1 . . . k

(1588)

where si∈ CM and wi∈ CN , is said to be linearly independent iff

rank

(

SWT ,

k∑

i=1

siwTi

)

= k (1589)

where S , [s1 · · · sk] ∈ CM×k and W , [w1 · · · wk] ∈ CN×k.

Dyad independence does not preclude existence of a nullspace N (SWT) ,as defined, nor does it imply SWT were full-rank. In absence of assumptionof independence, generally, rank SWT≤ k . Conversely, any rank-k matrixcan be written in the form SWT by singular value decomposition. (§A.6)

B.1.1.0.2 Theorem. Linearly independent (l.i.) dyads.Vectors si∈ CM , i=1 . . . k are l.i. and vectors wi∈ CN , i=1 . . . k arel.i. if and only if dyads siw

Ti ∈ CM×N , i=1 . . . k are l.i. ⋄

Proof. Linear independence of k dyads is identical to definition (1589).(⇒) Suppose si and wi are each linearly independent sets. InvokingSylvester’s rank inequality, [199, §0.4] [390, §2.4]

rank S+rank W − k ≤ rank(SWT) ≤ minrank S , rank W (≤ k) (1590)

Then k≤ rank(SWT)≤k which implies the dyads are independent.(⇐) Conversely, suppose rank(SWT)=k . Then

k≤minrank S , rank W ≤ k (1591)

implying the vector sets are each independent. ¨

B.3Move of range R to inside the summation depends on linear independence of wi.

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628 APPENDIX B. SIMPLE MATRICES

B.1.1.1 Biorthogonality condition, Range and Nullspace of Sum

Dyads characterized by biorthogonality condition WTS = I are independent;id est, for S∈CM×k and W ∈ CN×k, if WTS = I then rank(SWT)=k bythe linearly independent dyads theorem because (confer §E.1.1)

WTS = I ⇒ rank S =rankW =k≤M =N (1592)

To see that, we need only show: N (S)=0 ⇔ ∃ B Ä BS =I .B.4

(⇐) Assume BS =I . Then N (BS)=0=x | BSx = 0 ⊇ N (S). (1573)(⇒) If N (S)=0 then S must be full-rank skinny-or-square.

∴ ∃ A,B,C Ä

[

BC

]

[S A ] = I (id est, [S A ] is invertible) ⇒ BS = I .

Left inverse B is given as WT here. Because of reciprocity with S , itimmediately follows: N (W )=0 ⇔ ∃ S Ä STW = I . ¨

Dyads produced by diagonalization, for example, are independent becauseof their inherent biorthogonality. (§A.5.0.3) The converse is generally false;id est, linearly independent dyads are not necessarily biorthogonal.

B.1.1.1.1 Theorem. Nullspace and range of dyad sum.Given a sum of dyads represented by SWT where S∈CM×k and W ∈ CN×k

N (SWT) = N (WT) ⇐ ∃ B Ä BS = I

R(SWT) = R(S) ⇐ ∃ Z Ä WTZ = I(1593)

Proof. (⇒) N (SWT)⊇N (WT) and R(SWT)⊆R(S) are obvious.(⇐) Assume the existence of a left inverse B∈Rk×N and a right inverseZ∈ RN×k .B.5

N (SWT) = x | SWTx = 0 ⊆ x | BSWTx = 0 = N (WT) (1594)

R(SWT) = SWTx | x∈RN ⊇ SWTZy | Zy∈RN = R(S) (1595)

¨

B.4Left inverse is not unique, in general.B.5By counter-example, the theorem’s converse cannot be true; e.g., S = W = [1 0 ] .

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B.2. DOUBLET 629

R([ u v ])

N (Π)= v⊥ ∩ u⊥

r r

v⊥ ∩ u⊥

R(Π)=R([u v])

RN = R([ u v ]) ⊕ N([

vT

uT

])

N([

vT

uT

])

⊕ R([ u v ]) = RN

0 0

Figure 155: Four fundamental subspaces [330, §3.6] of a doubletΠ = uvT + vuT∈ SN . Π(x) = (uvT + vuT)x is a linear bijective mappingfrom R([ u v ]) to R([ u v ]).

B.2 Doublet

Consider a sum of two linearly independent square dyads, one a transpositionof the other:

Π = uvT + vuT =[u v ]

[

vT

uT

]

= SWT ∈ SN (1596)

where u , v∈RN . Like the dyad, a doublet can be 0 only when u or v is 0;

Π = uvT + vuT = 0 ⇔ u = 0 or v = 0 (1597)

By assumption of independence, a nonzero doublet has two nonzeroeigenvalues

λ1 , uTv + ‖uvT‖ , λ2 , uTv − ‖uvT‖ (1598)

where λ1 > 0 >λ2 , with corresponding eigenvectors

x1 ,u

‖u‖ +v

‖v‖ , x2 ,u

‖u‖ − v

‖v‖ (1599)

spanning the doublet range. Eigenvalue λ1 cannot be 0 unless u and v haveopposing directions, but that is antithetical since then the dyads would nolonger be independent. Eigenvalue λ2 is 0 if and only if u and v share thesame direction, again antithetical. Generally we have λ1 > 0 and λ2 < 0, soΠ is indefinite.

By the nullspace and range of dyad sum theorem, doublet Π hasN−2 zero-eigenvalues remaining and corresponding eigenvectors spanning

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630 APPENDIX B. SIMPLE MATRICES

N (uT)

N (E)=R(u)

r r

R(v)

R(E) = N (vT)

RN = N (uT) ⊕ N (E) R(v) ⊕ R(E) = RN

0 0

Figure 156: vTu = 1/ζ . The four fundamental subspaces [330, §3.6] of

elementary matrix E as a linear mapping E(x)=

(

I − uvT

vTu

)

x .

N([

vT

uT

])

. We therefore have

R(Π) = R([ u v ]) , N (Π) = v⊥ ∩ u⊥ (1600)

of respective dimension 2 and N−2.

B.3 Elementary matrix

A matrix of the formE = I − ζ uvT ∈ RN×N (1601)

where ζ ∈ R is finite and u,v ∈ RN , is called an elementary matrix or arank-one modification of the identity. [201] Any elementary matrix in RN×N

has N−1 eigenvalues equal to 1 corresponding to real eigenvectors thatspan v⊥. The remaining eigenvalue

λ = 1− ζ vTu (1602)

corresponds to eigenvector u .B.6 From [217, App.7.A.26] the determinant:

det E = 1 − tr(

ζ uvT)

= λ (1603)

If λ 6= 0 then E is invertible; [148] (confer §B.1.0.1)

E−1 = I +ζ

λuvT (1604)

B.6Elementary matrix E is not always diagonalizable because eigenvector u need not beindependent of the others; id est, u∈ v⊥ is possible.

Page 631: Convex Optimization & Euclidean Distance Geometry

B.3. ELEMENTARY MATRIX 631

Eigenvectors corresponding to 0 eigenvalues belong to N (E) , andthe number of 0 eigenvalues must be at least dimN (E) which, here,can be at most one. (§A.7.3.0.1) The nullspace exists, therefore,when λ=0 ; id est, when vTu=1/ζ ; rather, whenever u belongs tohyperplane z∈RN | vTz=1/ζ. Then (when λ=0) elementary matrix Eis a nonorthogonal projector projecting on its range (E2 =E , §E.1) andN (E)=R(u) ; eigenvector u spans the nullspace when it exists. Byconservation of dimension, dimR(E)=N−dimN (E). It is apparentfrom (1601) that v⊥⊆R(E) , but dim v⊥=N−1. Hence R(E)≡ v⊥ whenthe nullspace exists, and the remaining eigenvectors span it.

In summary, when a nontrivial nullspace of E exists,

R(E) = N (vT), N (E) = R(u), vTu = 1/ζ (1605)

illustrated in Figure 156, which is opposite to the assignment of subspacesfor a dyad (Figure 154). Otherwise, R(E)= RN .

When E = ET, the spectral norm is

‖E‖2 = max1 , |λ| (1606)

B.3.1 Householder matrix

An elementary matrix is called a Householder matrix when it has the definingform, for nonzero vector u [155, §5.1.2] [148, §4.10.1] [328, §7.3] [199, §2.2]

H = I − 2uuT

uTu∈ SN (1607)

which is a symmetric orthogonal (reflection) matrix (H−1 =HT=H(§B.5.2)). Vector u is normal to an N−1-dimensional subspace u⊥ throughwhich this particular H effects pointwise reflection; e.g., Hu⊥ = u⊥ whileHu =−u .

Matrix H has N−1 orthonormal eigenvectors spanning that reflectingsubspace u⊥ with corresponding eigenvalues equal to 1. The remainingeigenvector u has corresponding eigenvalue −1 ; so

det H = −1 (1608)

Due to symmetry of H , the matrix 2-norm (the spectral norm) is equal to thelargest eigenvalue-magnitude. A Householder matrix is thus characterized,

HT = H , H−1 = HT , ‖H‖2 = 1 , H ² 0 (1609)

Page 632: Convex Optimization & Euclidean Distance Geometry

632 APPENDIX B. SIMPLE MATRICES

For example, the permutation matrix

Ξ =

1 0 00 0 10 1 0

(1610)

is a Householder matrix having u=[ 0 1 −1 ]T/√

2 . Not all permutationmatrices are Householder matrices, although all permutation matricesare orthogonal matrices (§B.5.1, ΞTΞ = I ) [328, §3.4] because they aremade by permuting rows and columns of the identity matrix. Neitherare all symmetric permutation matrices Householder matrices; e.g.,

Ξ =

0 0 0 10 0 1 00 1 0 01 0 0 0

(1691) is not a Householder matrix.

B.4 Auxiliary V -matrices

B.4.1 Auxiliary projector matrix V

It is convenient to define a matrix V that arises naturally as a consequence oftranslating the geometric center αc (§5.5.1.0.1) of some list X to the origin.In place of X − αc1

T we may write XV as in (950) where

V = I − 1

N11T ∈ SN (889)

is an elementary matrix called the geometric centering matrix.Any elementary matrix in RN×N has N−1 eigenvalues equal to 1. For the

particular elementary matrix V , the N th eigenvalue equals 0. The numberof 0 eigenvalues must equal dimN (V ) = 1, by the 0 eigenvalues theorem(§A.7.3.0.1), because V =V T is diagonalizable. Because

V 1 = 0 (1611)

the nullspace N (V )=R(1) is spanned by the eigenvector 1. The remainingeigenvectors span R(V ) ≡ 1⊥ = N (1T) that has dimension N−1.

Because

V 2 = V (1612)

Page 633: Convex Optimization & Euclidean Distance Geometry

B.4. AUXILIARY V -MATRICES 633

and V T = V , elementary matrix V is also a projection matrix (§E.3)projecting orthogonally on its range N (1T) which is a hyperplane containingthe origin in RN

V = I − 1(1T1)−11T (1613)

The 0, 1 eigenvalues also indicate diagonalizable V is a projectionmatrix. [390, §4.1, thm.4.1] Symmetry of V denotes orthogonal projection;from (1878),

V T = V , V † = V , ‖V ‖2 = 1 , V º 0 (1614)

Matrix V is also circulant [164].

B.4.1.0.1 Example. Relationship of auxiliary to Householder matrix.Let H∈ SN be a Householder matrix (1607) defined by

u =

1...1

1 +√

N

∈ RN (1615)

Then we have [151, §2]

V = H

[

I 00T 0

]

H (1616)

Let D∈ SNh and define

−HDH , −[

A bbT c

]

(1617)

where b is a vector. Then because H is nonsingular (§A.3.1.0.5) [180, §3]

−V DV = −H

[

A 00T 0

]

H º 0 ⇔ −A º 0 (1618)

and affine dimension is r = rankA when D is a Euclidean distance matrix.

2

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634 APPENDIX B. SIMPLE MATRICES

B.4.2 Schoenberg auxiliary matrix VN

1. VN =1√2

[

−1T

I

]

∈ RN×N−1

2. V TN 1 = 0

3. I − e11T =

[

0√

2VN]

4.[

0√

2VN]

VN = VN

5.[

0√

2VN]

V = V

6. V[

0√

2VN]

=[

0√

2VN]

7.[

0√

2VN] [

0√

2VN]

=[

0√

2VN]

8.[

0√

2VN]†

=

[

0 0T

0 I

]

V

9.[

0√

2VN]†

V =[

0√

2VN]†

10.[

0√

2VN] [

0√

2VN]†

= V

11.[

0√

2VN]† [

0√

2VN]

=

[

0 0T

0 I

]

12.[

0√

2VN]

[

0 0T

0 I

]

=[

0√

2VN]

13.

[

0 0T

0 I

]

[

0√

2VN]

=

[

0 0T

0 I

]

14. [ VN1√21 ]−1 =

[

V †N

√2

N1T

]

15. V †N =

√2[

− 1N1 I− 1

N11T

]

∈ RN−1×N ,(

I− 1N11T ∈ SN−1

)

16. V †N1 = 0

17. V †NVN = I

Page 635: Convex Optimization & Euclidean Distance Geometry

B.4. AUXILIARY V -MATRICES 635

18. V T = V = VNV †N = I − 1

N11T ∈ SN

19. −V †N (11T− I )VN = I ,

(

11T− I ∈ EDMN)

20. D = [dij] ∈ SNh (891)

tr(−V DV ) = tr(−V D) = tr(−V †NDVN ) = 1

N1TD 1 = 1

Ntr(11TD) = 1

N

i,j

dij

Any elementary matrix E∈ SN of the particular form

E = k1 I − k2 11T (1619)

where k1 , k2∈R ,B.7 will make tr(−ED) proportional to∑

dij .

21. D = [dij] ∈ SN

tr(−V DV ) = 1N

i,ji6=j

dij − N−1N

i

dii = 1TD1 1N− tr D

22. D = [dij] ∈ SNh

tr(−V TNDVN ) =

j

d1j

23. For Y ∈ SN

V (Y − δ(Y 1))V = Y − δ(Y 1)

B.4.3 Orthonormal auxiliary matrix VW

The skinny matrix

VW ,

−1√N

−1√N

· · · −1√N

1 + −1N+

√N

−1N+

√N

· · · −1N+

√N

−1N+

√N

. . . . . . −1N+

√N

.... . . . . .

...

−1N+

√N

−1N+

√N

· · · 1 + −1N+

√N

∈ RN×N−1 (1620)

B.7If k1 is 1−ρ while k2 equals −ρ∈R , then all eigenvalues of E for −1/(N−1) < ρ < 1are guaranteed positive and therefore E is guaranteed positive definite. [297]

Page 636: Convex Optimization & Euclidean Distance Geometry

636 APPENDIX B. SIMPLE MATRICES

has R(VW)=N (1T) and orthonormal columns. [6] We defined threeauxiliary V -matrices: V , VN (873), and VW sharing some attributes listedin Table B.4.4. For example, V can be expressed

V = VWV TW = VNV †

N (1621)

but V TWVW = I means V is an orthogonal projector (1875) and

V †W = V T

W , ‖VW‖2 = 1 , V TW1 = 0 (1622)

B.4.4 Auxiliary V -matrix Table

dim V rank V R(V ) N (V T) V TV V V T V V †

V N×N N−1 N (1T) R(1) V V V

VN N×(N−1) N−1 N (1T) R(1) 12(I + 11T) 1

2

[

N−1 −1T

−1 I

]

V

VW N×(N−1) N−1 N (1T) R(1) I V V

B.4.5 More auxiliary matrices

Mathar shows [255, §2] that any elementary matrix (§B.3) of the form

VS = I − b1T ∈ RN×N (1623)

such that bT1 = 1 (confer [158, §2]), is an auxiliary V -matrix having

R(V TS ) = N (bT) , R(VS) = N (1T)

N (VS) = R(b) , N (V TS ) = R(1)

(1624)

Given X ∈ Rn×N , the choice b = 1N1 (VS = V ) minimizes ‖X(I − b1T)‖F .

[160, §3.2.1]

Page 637: Convex Optimization & Euclidean Distance Geometry

B.5. ORTHOGONAL MATRIX 637

B.5 Orthogonal matrix

B.5.1 Vector rotation

The property Q−1 = QT completely defines an orthogonal matrix Q∈Rn×n

employed to effect vector rotation; [328, §2.6, §3.4] [330, §6.5] [199, §2.1]for any x ∈ Rn

‖Qx‖2 = ‖x‖2 (1625)

In other words, the 2-norm is orthogonally invariant. A unitary matrix is acomplex generalization of the orthogonal matrix. The conjugate transposedefines it: U−1 = UH. An orthogonal matrix is simply a real unitary matrix.

Orthogonal matrix Q is a normal matrix further characterized:

Q−1 = QT , ‖Q‖2 = 1 (1626)

Applying characterization (1626) to QT we see it too is an orthogonal matrix.Hence the rows and columns of Q respectively form an orthonormal set.Normalcy guarantees diagonalization (§A.5.1) so, for Q , SΛSH

SΛ−1SH = S∗ΛST , ‖δ(Λ)‖∞ = 1 (1627)

characterizes an orthogonal matrix in terms of eigenvalues and eigenvectors.All permutation matrices Ξ , for example, are orthogonal matrices.

Any product of permutation matrices remains a permutator. Any productof a permutation matrix with an orthogonal matrix remains orthogonal.In fact, any product of orthogonal matrices AQ remains orthogonal bydefinition. Given any other dimensionally compatible orthogonal matrix U ,the mapping g(A)= UTAQ is a linear bijection on the domain of orthogonalmatrices (a nonconvex manifold of dimension 1

2n(n−1) [50]). [235, §2.1] [236]

The largest magnitude entry of an orthogonal matrix is 1; for each andevery j∈ 1 . . . n

‖Q(j , :)‖∞ ≤ 1‖Q(: , j)‖∞ ≤ 1

(1628)

Each and every eigenvalue of a (real) orthogonal matrix has magnitude 1

λ(Q) ∈ Cn , |λ(Q)| = 1 (1629)

while only the identity matrix can be simultaneously positive definite andorthogonal.

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638 APPENDIX B. SIMPLE MATRICES

B.5.2 Reflection

A matrix for pointwise reflection is defined by imposing symmetry uponthe orthogonal matrix; id est, a reflection matrix is completely defined byQ−1 = QT = Q . The reflection matrix is a symmetric orthogonal matrix,characterized:

QT = Q , Q−1 = QT , ‖Q‖2 = 1 (1630)

The Householder matrix (§B.3.1) is an example of symmetric orthogonal(reflection) matrix.

Reflection matrices have eigenvalues equal to ±1 and so det Q=±1. Itis natural to expect a relationship between reflection and projection matricesbecause all projection matrices have eigenvalues belonging to 0, 1. Infact, any reflection matrix Q is related to some orthogonal projector P by[201, §1, prob.44]

Q = I − 2P (1631)

Yet P is, generally, neither orthogonal or invertible. (§E.3.2)

λ(Q) ∈ Rn , |λ(Q)| = 1 (1632)

Reflection is with respect to R(P )⊥. Matrix 2P−I represents antireflection.

Every orthogonal matrix can be expressed as the product of a rotation anda reflection. The collection of all orthogonal matrices of particular dimensiondoes not form a convex set.

B.5.3 Rotation of range and rowspace

Given orthogonal matrix Q , column vectors of a matrix X are simultaneouslyrotated about the origin via product QX . In three dimensions (X∈ R3×N),the precise meaning of rotation is best illustrated in Figure 157 where thegimbal aids visualization of what is achievable; mathematically, (§5.5.2.0.1)

Q =

cos θ 0 − sin θ0 1 0

sin θ 0 cos θ

1 0 00 cos ψ − sin ψ0 sin ψ cos ψ

cos φ − sin φ 0sin φ cos φ 0

0 0 1

(1633)

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B.5. ORTHOGONAL MATRIX 639

Figure 157: Gimbal : a mechanism imparting three degrees of dimensionalfreedom to a Euclidean body suspended at the device’s center. Each ring isfree to rotate about one axis. (Courtesy of The MathWorks Inc.)

B.5.3.0.1 Example. One axis of revolution.

Partition an n+1-dimensional Euclidean space Rn+1 ,

[

Rn

R

]

and define

an n-dimensional subspace

R , λ∈Rn+1 | 1Tλ = 0 (1634)

(a hyperplane through the origin). We want an orthogonal matrix thatrotates a list in the columns of matrix X∈ Rn+1×N through the dihedralangle between Rn and R (§2.4.3)

Á(Rn, R) = arccos

( 〈en+1 , 1〉‖en+1‖ ‖1‖

)

= arccos

(

1√n+1

)

radians (1635)

The vertex-description of the nonnegative orthant in Rn+1 is

[ e1 e2 · · · en+1 ] a | a º 0 = a º 0 = Rn+1+ ⊂ Rn+1 (1636)

Consider rotation of these vertices via orthogonal matrix

Q , [1 1√n+1

ΞVW ]Ξ ∈ Rn+1×n+1 (1637)

where permutation matrix Ξ∈Sn+1 is defined in (1691), and VW ∈Rn+1×n

is the orthonormal auxiliary matrix defined in §B.4.3. This particularorthogonal matrix is selected because it rotates any point in subspace Rn

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640 APPENDIX B. SIMPLE MATRICES

about one axis of revolution onto R ; e.g., rotation Qen+1 aligns the laststandard basis vector with subspace normal R⊥= 1. The rotated standardbasis vectors remaining are orthonormal spanning R . 2

Another interpretation of product QX is rotation/reflection of R(X).Rotation of X as in QXQT is the simultaneous rotation/reflection of rangeand rowspace.B.8

Proof. Any matrix can be expressed as a singular value decompositionX = UΣWT (1529) where δ2(Σ) = Σ , R(U)⊇R(X) , and R(W )⊇R(XT).

¨

B.5.4 Matrix rotation

Orthogonal matrices are also employed to rotate/reflect other matrices likevectors: [155, §12.4.1] Given orthogonal matrix Q , the product QTA will

rotate A∈Rn×n in the Euclidean sense in Rn2

because Frobenius’ norm isorthogonally invariant (§2.2.1);

‖QTA‖F =√

tr(ATQQTA) = ‖A‖F (1638)

(likewise for AQ). Were A symmetric, such a rotation would depart from Sn.One remedy is to instead form the product QTAQ because

‖QTAQ‖F =√

tr(QTATQQTAQ) = ‖A‖F (1639)

Matrix A is orthogonally equivalent to B if B=STAS for someorthogonal matrix S . Every square matrix, for example, is orthogonallyequivalent to a matrix having equal entries along the main diagonal.[199, §2.2, prob.3]

B.8The product QTAQ can be regarded as a coordinate transformation; e.g., givenlinear map y =Ax : Rn→Rn and orthogonal Q , the transformation Qy =AQx is arotation/reflection of the range and rowspace (139) of matrix A where Qy∈R(A) andQx∈R(AT) (140).

Page 641: Convex Optimization & Euclidean Distance Geometry

Appendix C

Some analytical optimal results

People have been working on Optimization since the ancientGreeks learned that a string encloses the most area when it isformed into the shape of a circle.

−Roman Polyak

We speculate that optimization problems possessing analytical solutionhave convex transformation or constructive global optimality conditions,perhaps yet unknown; e.g., §7.1.4, (1663), §C.3.0.1.

C.1 Properties of infimainf ∅ , ∞sup ∅ , −∞ (1640) Given f(x) : X →R defined on arbitrary set X [196, §0.1.2]

infx∈X

f(x) = − supx∈X

−f(x)

supx∈X

f(x) = − infx∈X

−f(x)(1641)

arg infx∈X

f(x) = arg supx∈X

−f(x)

arg supx∈X

f(x) = arg infx∈X

−f(x)(1642)© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

641

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642 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS Given scalar κ and f(x) : X →R and g(x) : X →R defined onarbitrary set X [196, §0.1.2]

infx∈X

(κ + f(x)) = κ + infx∈X

f(x)

arg infx∈X

(κ + f(x)) = arg infx∈X

f(x)(1643)

infx∈X

κ f(x) = κ infx∈X

f(x)

arg infx∈X

κ f(x) = arg infx∈X

f(x)

, κ > 0 (1644)

infx∈X

(f(x) + g(x)) ≥ infx∈X

f(x) + infx∈X

g(x) (1645) Given f(x) : X ∪ Y→R and arbitrary sets X and Y [196, §0.1.2]

X ⊂ Y ⇒ infx∈X

f(x) ≥ infx∈Y

f(x) (1646)

infx∈X∪Y

f(x) = min infx∈X

f(x) , infx∈Y

f(x) (1647)

infx∈X∩Y

f(x) ≥ max infx∈X

f(x) , infx∈Y

f(x) (1648)

C.2 Trace, singular and eigen values For A∈Rm×n and σ(A)1 connoting spectral norm,

σ(A)1 =√

λ(ATA)1 = ‖A‖2 = sup‖x‖=1

‖Ax‖2 = minimizet∈R

t

subject to

[

tI AAT tI

]

º 0

(615) For A∈Rm×n and σ(A) denoting its singular values, the nuclear norm(Ky Fan norm) of matrix A (confer (44), (1532), [200, p.200]) is

i

σ(A)i = tr√

ATA = sup‖X‖2≤1

tr(XTA) = maximizeX∈R

m×ntr(XTA)

subject to

[

I XXT I

]

º 0

= 12

minimizeX∈S

m, Y ∈S

ntr X + tr Y

subject to

[

X AAT Y

]

º 0

(1649)

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C.2. TRACE, SINGULAR AND EIGEN VALUES 643

This nuclear norm is convexC.1 and dual to spectral norm. [200, p.214][59, §A.1.6] Given singular value decomposition A = SΣQT∈ Rm×n

(A.6), then X⋆ = SQT∈ Rm×n is an optimal solution to maximization(confer §2.3.2.0.5) while X⋆ = SΣST∈ Sm and Y ⋆ = QΣQT∈ Sn is anoptimal solution to minimization [132]. Srebro [320] asserts

i

σ(A)i = 12minimize

U,V‖U‖2

F + ‖V ‖2F

subject to A = UV T

= minimizeU,V

‖U‖F‖V ‖F

subject to A = UV T

(1650)

C.2.0.0.1 Exercise. Optimal matrix factorization.Prove (1650).C.2 H For X∈ Sm, Y ∈ Sn, A∈ C⊆Rm×n for set C convex, and σ(A)denoting the singular values of A [132, §3]

minimizeA

i

σ(A)i

subject to A ∈ C≡

12minimize

A , X , Ytr X + tr Y

subject to

[

X AAT Y

]

º 0

A ∈ C

(1651)

C.1 discernible as envelope of the rank function (1338) or as supremum of functions linearin A (Figure 73).C.2Hint: Write A = SΣQT∈ Rm×n and

[

X AAT Y

]

=

[

UV

]

[UT V T ] º 0

Show U⋆ = S√

Σ∈Rm×minm,n and V ⋆ = Q√

Σ∈Rn×minm,n, hence ‖U⋆‖2F = ‖V ⋆‖2

F .

Page 644: Convex Optimization & Euclidean Distance Geometry

644 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS For A∈SN+ and β∈R

β tr A = maximizeX∈ SN

tr(XA)

subject to X ¹ βI(1652)

But the following statement is numerically stable, preventing anunbounded solution in direction of a 0 eigenvalue:

maximizeX∈ SN

sgn(β) tr(XA)

subject to X ¹ |β|IX º −|β|I

(1653)

where β tr A = tr(X⋆A). If β≥ 0, then Xº−|β|I ← Xº 0. For symmetric A∈ SN , its smallest and largest eigenvalue in λ(A)∈ RN

is respectively [11, §4.1] [42, §I.6.15] [199, §4.2] [235, §2.1] [236]

miniλ(A)i = inf

‖x‖=1xTAx = minimize

X∈ SN+

tr(XA) = maximizet∈R

t

subject to tr X = 1 subject to A º t I(1654)

maxi

λ(A)i = sup‖x‖=1

xTAx = maximizeX∈ SN

+

tr(XA) = minimizet∈R

t

subject to tr X = 1 subject to A ¹ t I(1655)

The largest eigenvalue λ1 is always convex in A∈ SN because, givenparticular x , xTAx is linear in matrix A ; supremum of a family oflinear functions is convex, as illustrated in Figure 73. So for A ,B∈ SN ,λ1(A + B)≤ λ1(A) + λ1(B). (1477) Similarly, the smallest eigenvalueλN of any symmetric matrix is a concave function of its entries;λN(A + B)≥ λN(A) + λN(B). (1477) For v1 a normalized eigenvectorof A corresponding to the largest eigenvalue, and vN a normalizedeigenvector corresponding to the smallest eigenvalue,

vN = arg inf‖x‖=1

xTAx (1656)

v1 = arg sup‖x‖=1

xTAx (1657)

Page 645: Convex Optimization & Euclidean Distance Geometry

C.2. TRACE, SINGULAR AND EIGEN VALUES 645 For A∈SN having eigenvalues λ(A)∈ RN , consider the unconstrainednonconvex optimization that is a projection on the rank-1 subset(§2.9.2.1, §3.7.0.0.1) of the boundary of positive semidefinite cone SN

+ :

Defining λ1 , maxiλ(A)i and corresponding eigenvector v1

minimizex

‖xxT− A‖2F = minimize

xtr(xxT(xTx) − 2AxxT + ATA)

=

‖λ(A)‖2 , λ1 ≤ 0

‖λ(A)‖2 − λ21 , λ1 > 0

(1658)

arg minimizex

‖xxT− A‖2F =

0 , λ1 ≤ 0

v1

√λ1 , λ1 > 0

(1659)

Proof. This is simply the Eckart & Young solution from §7.1.2:

x⋆x⋆T =

0 , λ1 ≤ 0

λ1 v1vT1 , λ1 > 0

(1660)

Given nonincreasingly ordered diagonalization A = QΛQT whereΛ = δ(λ(A)) (§A.5), then (1658) has minimum value

minimizex

‖xxT−A‖2F =

‖QΛQT‖2F = ‖δ(Λ)‖2 , λ1 ≤ 0

Q

λ10

. . .0

− Λ

QT

2

F

=

λ1

0...0

− δ(Λ)

2

, λ1 > 0

(1661)

¨

C.2.0.0.2 Exercise. Rank-1 approximation.Given symmetric matrix A∈SN , prove:

v1 = arg minimizex

‖xxT− A‖2F

subject to ‖x‖ = 1(1662)

where v1 is a normalized eigenvector of A corresponding to its largesteigenvalue. What is the objective’s optimal value? H

Page 646: Convex Optimization & Euclidean Distance Geometry

646 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS (Ky Fan, 1949) For B∈ SN whose eigenvalues λ(B)∈ RN are arrangedin nonincreasing order, and for 1≤k≤N [11, §4.1] [211] [199, §4.3.18][356, §2] [235, §2.1] [236]

N∑

i=N−k+1

λ(B)i = infU∈RN×k

U TU=I

tr(UUTB) = minimizeX∈ SN

+

tr(XB)

subject to X ¹ I

tr X = k

(a)

= maximizeµ∈R , Z∈SN

+

(k − N)µ + tr(B − Z)

subject to µI + Z º B

(b)

k∑

i=1

λ(B)i = supU∈RN×k

U TU=I

tr(UUTB) = maximizeX∈ SN

+

tr(XB)

subject to X ¹ I

tr X = k

(c)

= minimizeµ∈R , Z∈SN

+

kµ + tr Z

subject to µI + Z º B

(d)

(1663)

Given ordered diagonalization B = QΛQT, (§A.5.1) then an optimalU for the infimum is U⋆ = Q(: , N− k+1:N)∈RN×k whereasU⋆ = Q(: , 1:k)∈ RN×k for the supremum is more reliably computed.In both cases, X⋆ = U⋆U⋆T. Optimization (a) searches the convex hullof outer product UUT of all N×k orthonormal matrices. (§2.3.2.0.1) For B∈ SN whose eigenvalues λ(B)∈ RN are arranged in nonincreasingorder, and for diagonal matrix Υ∈ Sk whose diagonal entries arearranged in nonincreasing order where 1≤k≤N , we utilize themain-diagonal δ operator’s self-adjointness property (1394): [12, §4.2]

k∑

i=1

Υii λ(B)N−i+1 = infU∈RN×k

U TU=I

tr(Υ UTBU) = infU∈RN×k

U TU=I

δ(Υ)Tδ(UTBU)

= minimizeVi∈SN

tr

(

Bk

i=1

(Υii−Υi+1,i+1)Vi

)

subject to tr Vi = i , i=1 . . . kI º Vi º 0 , i=1 . . . k

(1664)

Page 647: Convex Optimization & Euclidean Distance Geometry

C.2. TRACE, SINGULAR AND EIGEN VALUES 647

where Υk+1,k+1 , 0. We speculate,

k∑

i=1

Υii λ(B)i = supU∈RN×k

U TU=I

tr(Υ UTBU) = supU∈RN×k

U TU=I

δ(Υ)Tδ(UTBU) (1665)

Alizadeh shows: [11, §4.2]

k∑

i=1

Υii λ(B)i = minimizeµ∈Rk , Zi∈SN

k∑

i=1

iµi + tr Zi

subject to µiI + Zi − (Υii−Υi+1,i+1)B º 0 , i=1 . . . k

Zi º 0 , i=1 . . . k

= maximizeVi∈SN

tr

(

Bk

i=1

(Υii−Υi+1,i+1)Vi

)

subject to tr Vi = i , i=1 . . . kI º Vi º 0 , i=1 . . . k (1666)

where Υk+1,k+1 , 0. The largest eigenvalue magnitude µ of A∈ SN

maxi

|λ(A)i| = minimizeµ∈R

µ

subject to −µI ¹ A ¹ µI(1667)

is minimized over convex set C by semidefinite program: (confer §7.1.5)

minimizeA

‖A‖2

subject to A ∈ C≡

minimizeµ , A

µ

subject to −µI ¹ A ¹ µI

A ∈ C(1668)

id est,

µ⋆ , maxi

|λ(A⋆)i| , i = 1 . . . N ∈ R+ (1669)

Page 648: Convex Optimization & Euclidean Distance Geometry

648 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS For B∈ SN whose eigenvalues λ(B)∈ RN are arranged in nonincreasingorder, let Πλ(B) be a permutation of eigenvalues λ(B) suchthat their absolute value becomes arranged in nonincreasingorder: |Πλ(B)|1 ≥ |Πλ(B)|2 ≥ · · · ≥ |Πλ(B)|N . Then, for 1≤k≤N[11, §4.3]C.3

k∑

i=1

|Πλ(B)|i = minimizeµ∈R , Z∈SN

+

kµ + tr Z

subject to µI + Z + B º 0µI + Z − B º 0

= maximizeV ,W∈ SN

+

〈B , V − W 〉

subject to I º V , Wtr(V + W )=k

(1670)

For diagonal matrix Υ∈ Sk whose diagonal entries are arranged innonincreasing order where 1≤k≤N

k∑

i=1

Υii|Πλ(B)|i = minimizeµ∈Rk , Zi∈SN

k∑

i=1

iµi + tr Zi

subject to µiI + Zi + (Υii−Υi+1,i+1)B º 0 , i=1 . . . k

µiI + Zi − (Υii−Υi+1,i+1)B º 0 , i=1 . . . k

Zi º 0 , i=1 . . . k

= maximizeVi ,Wi∈SN

tr

(

Bk

i=1

(Υii−Υi+1,i+1)(Vi − Wi)

)

subject to tr(Vi + Wi) = i , i=1 . . . kI º Vi º 0 , i=1 . . . kI º Wi º 0 , i=1 . . . k (1671)

where Υk+1,k+1 , 0.

C.2.0.0.3 Exercise. Weighted sum of largest eigenvalues.Prove (1665). H

C.3We eliminate a redundant positive semidefinite variable from Alizadeh’s minimization.There exist typographical errors in [289, (6.49) (6.55)] for this minimization.

Page 649: Convex Optimization & Euclidean Distance Geometry

C.3. ORTHOGONAL PROCRUSTES PROBLEM 649 For A ,B∈ SN whose eigenvalues λ(A) , λ(B)∈ RN are respectivelyarranged in nonincreasing order, and for nonincreasingly ordereddiagonalizations A = WAΥWT

A and B = WBΛWTB [197] [235, §2.1]

[236]

λ(A)Tλ(B) = supU∈RN×N

U TU=I

tr(ATUTBU) ≥ tr(ATB) (1690)

(confer (1695)) where optimal U is

U⋆ = WBWTA ∈ RN×N (1687)

We can push that upper bound higher using a result in §C.4.2.1:

|λ(A)|T|λ(B)| = supU∈CN×N

U HU=I

re tr(ATU HBU) (1672)

For step function ψ as defined in (1545), optimal U becomes

U⋆ = WB

δ(ψ(δ(Λ)))H√

δ(ψ(δ(Υ))) WTA ∈ CN×N (1673)

C.3 Orthogonal Procrustes problem

Given matrices A ,B∈ Rn×N , their product having full singular valuedecomposition (§A.6.3)

ABT , UΣQT ∈ Rn×n (1674)

then an optimal solution R⋆ to the orthogonal Procrustes problem

minimizeR

‖A − RTB‖F

subject to RT = R−1(1675)

maximizes tr(ATRTB) over the nonconvex manifold of orthogonal matrices:[199, §7.4.8]

R⋆ = QUT ∈ Rn×n (1676)

A necessary and sufficient condition for optimality

ABTR⋆ º 0 (1677)

holds whenever R⋆ is an orthogonal matrix. [160, §4]

Page 650: Convex Optimization & Euclidean Distance Geometry

650 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

Optimal solution R⋆ can reveal rotation/reflection (§5.5.2, §B.5) of onelist in the columns of matrix A with respect to another list in B . Solution isunique if rank BVN = n . [110, §2.4.1] In the case that A is a vector andpermutation of B , solution R⋆ is not necessarily a permutation matrix(§4.6.0.0.3) although the optimal objective will be zero. More generally, theoptimal value for objective of minimization is

tr(

ATA + BTB − 2ABTR⋆)

= tr(ATA) + tr(BTB) − 2 tr(UΣUT)

= ‖A‖2F + ‖B‖2

F − 2δ(Σ)T1(1678)

while the optimal value for corresponding trace maximization is

supRT=R−1

tr(ATRTB) = tr(ATR⋆TB) = δ(Σ)T1 ≥ tr(ATB) (1679)

The same optimal solution R⋆ solves

maximizeR

‖A + RTB‖F

subject to RT = R−1(1680)

C.3.0.1 Procrustes relaxation

By replacing its feasible set with the convex hull of orthogonal matrices(Example 2.3.2.0.5), we relax Procrustes problem (1675) to a convex problem

minimizeR

‖A − RTB‖2F = tr(ATA + BTB) − 2 maximize

Rtr(ATRTB)

subject to RT = R−1 subject to

[

I RRT I

]

º 0(1681)

whose adjusted objective must always equal Procrustes’.C.4

C.3.1 Effect of translation

Consider the impact on problem (1675) of dc offset in known listsA ,B∈ Rn×N . Rotation of B there is with respect to the origin, so betterresults may be obtained if offset is first accounted. Because the geometriccenters of the lists AV and BV are the origin, instead we solve

minimizeR

‖AV − RTBV ‖F

subject to RT = R−1(1682)

C.4 (because orthogonal matrices are the extreme points of this hull) and whose optimalnumerical solution (SDPT3 [345]) [163] is reliably observed to be orthogonal for n≤N .

Page 651: Convex Optimization & Euclidean Distance Geometry

C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 651

where V ∈ SN is the geometric centering matrix (§B.4.1). Now we define thefull singular value decomposition

AV BT , UΣQT ∈ Rn×n (1683)

and an optimal rotation matrix

R⋆ = QUT ∈ Rn×n (1676)

The desired result is an optimally rotated offset list

R⋆TBV + A(I − V ) ≈ A (1684)

which most closely matches the list in A . Equality is attained when the listsare precisely related by a rotation/reflection and an offset. When R⋆TB=Aor B1=A1=0, this result (1684) reduces to R⋆TB ≈A .

C.3.1.1 Translation of extended list

Suppose an optimal rotation matrix R⋆∈ Rn×n were derived as beforefrom matrix B∈ Rn×N , but B is part of a larger list in the columns of[ C B ]∈ Rn×M+N where C∈ Rn×M . In that event, we wish to applythe rotation/reflection and translation to the larger list. The expressionsupplanting the approximation in (1684) makes 1T of compatible dimension;

R⋆T[ C−B11T 1N

BV ] + A11T 1N

(1685)

id est, C−B11T 1N∈ Rn×M and A11T 1

N∈ Rn×M+N .

C.4 Two-sided orthogonal Procrustes

C.4.0.1 Minimization

Given symmetric A ,B∈ SN , each having diagonalization (§A.5.1)

A , QAΛAQTA , B , QBΛBQT

B (1686)

where eigenvalues are arranged in their respective diagonal matrix Λ innonincreasing order, then an optimal solution [127]

R⋆ = QBQTA ∈ RN×N (1687)

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652 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

to the two-sided orthogonal Procrustes problem

minimizeR

‖A − RTBR‖F

subject to RT = R−1=

minimizeR

tr(

ATA − 2ATRTBR + BTB)

subject to RT = R−1 (1688)

maximizes tr(ATRTBR) over the nonconvex manifold of orthogonal matrices.Optimal product R⋆TBR⋆ has the eigenvectors of A but the eigenvalues of B .[160, §7.5.1] The optimal value for the objective of minimization is, by (48)

‖QAΛAQTA −R⋆TQBΛBQT

BR⋆‖F = ‖QA(ΛA−ΛB)QTA‖F = ‖ΛA−ΛB‖F (1689)

while the corresponding trace maximization has optimal value

supRT=R−1

tr(ATRTBR) = tr(ATR⋆TBR⋆) = tr(ΛAΛB) ≥ tr(ATB) (1690)

C.4.0.2 Maximization

Any permutation matrix is an orthogonal matrix. Defining a row- andcolumn-swapping permutation matrix (a reflection matrix, §B.5.2)

Ξ = ΞT =

0 1·

·1

1 0

(1691)

then an optimal solution R⋆ to the maximization problem [sic]

maximizeR

‖A − RTBR‖F

subject to RT = R−1(1692)

minimizes tr(ATRTBR) : [197] [235, §2.1] [236]

R⋆ = QBΞQTA ∈ RN×N (1693)

The optimal value for the objective of maximization is

‖QAΛAQTA − R⋆TQBΛBQT

BR⋆‖F = ‖QAΛAQTA − QAΞTΛBΞQT

A‖F

= ‖ΛA − ΞΛBΞ‖F

(1694)

while the corresponding trace minimization has optimal value

infRT=R−1

tr(ATRTBR) = tr(ATR⋆TBR⋆) = tr(ΛAΞΛBΞ) (1695)

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C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 653

C.4.1 Procrustes’ relation to linear programming

Although these two-sided Procrustes problems are nonconvex, a connectionwith linear programming [92] was discovered by Anstreicher & Wolkowicz[12, §3] [235, §2.1] [236]: Given A ,B∈ SN , this semidefinite program in Sand T

minimizeR

tr(ATRTBR) = maximizeS , T∈SN

tr(S + T )

subject to RT = R−1 subject to AT⊗ B − I ⊗ S − T ⊗ I º 0

(1696)

(where ⊗ signifies Kronecker product (§D.1.2.1)) has optimal objectivevalue (1695). These two problems in (1696) are strong duals (§2.13.1.0.3).Given ordered diagonalizations (1686), make the observation:

infR

tr(ATRTBR) = infR

tr(ΛARTΛBR) (1697)

because R,QTBRQA on the set of orthogonal matrices (which includes the

permutation matrices) is a bijection. This means, basically, diagonal matricesof eigenvalues ΛA and ΛB may be substituted for A and B , so only the maindiagonals of S and T come into play;

maximizeS,T∈SN

1Tδ(S + T )

subject to δ(ΛA ⊗ (ΞΛBΞ) − I ⊗ S − T ⊗ I) º 0(1698)

a linear program in δ(S) and δ(T ) having the same optimal objective valueas the semidefinite program (1696).

We relate their results to Procrustes problem (1688) by manipulatingsigns (1641) and permuting eigenvalues:

maximizeR

tr(ATRTBR) = minimizeS , T∈SN

1Tδ(S + T )

subject to RT = R−1 subject to δ(I ⊗ S + T ⊗ I − ΛA ⊗ ΛB) º 0

= minimizeS , T∈SN

tr(S + T )

subject to I ⊗ S + T ⊗ I − AT⊗ B º 0

(1699)

This formulation has optimal objective value identical to that in (1690).

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654 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

C.4.2 Two-sided orthogonal Procrustes via SVD

By making left- and right-side orthogonal matrices independent, we can pushthe upper bound on trace (1690) a little further: Given real matrices A ,Beach having full singular value decomposition (§A.6.3)

A , UAΣAQTA ∈ Rm×n , B , UBΣBQT

B ∈ Rm×n (1700)

then a well-known optimal solution R⋆, S⋆ to the problem

minimizeR , S

‖A − SBR‖F

subject to RH = R−1

SH = S−1

(1701)

maximizes re tr(ATSBR) : [310] [285] [50] [191] optimal orthogonal matrices

S⋆ = UAUHB ∈ Rm×m , R⋆ = QBQH

A ∈ Rn×n (1702)

[sic] are not necessarily unique [199, §7.4.13] because the feasible set is notconvex. The optimal value for the objective of minimization is, by (48)

‖UAΣAQHA − S⋆UBΣBQH

B R⋆‖F = ‖UA(ΣA − ΣB)QHA ‖F = ‖ΣA − ΣB‖F (1703)

while the corresponding trace maximization has optimal value [42, §III.6.12]

supRH=R−1

SH=S−1

| tr(ATSBR) | = supRH=R−1

SH=S−1

re tr(ATSBR) = re tr(ATS⋆BR⋆) = tr(ΣTA ΣB) ≥ tr(ATB)

(1704)

for which it is necessary

ATS⋆BR⋆ º 0 , BR⋆ATS⋆ º 0 (1705)

The lower bound on inner product of singular values in (1704) is due tovon Neumann. Equality is attained if UH

A UB = I and QHB QA = I .

C.4.2.1 Symmetric matrices

Now optimizing over the complex manifold of unitary matrices (§B.5.1),the upper bound on trace (1690) is thereby raised: Suppose we are givendiagonalizations for (real) symmetric A ,B (§A.5)

A = WAΥWTA ∈ Sn , δ(Υ) ∈ KM (1706)

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C.4. TWO-SIDED ORTHOGONAL PROCRUSTES 655

B = WBΛWTB ∈ Sn , δ(Λ) ∈ KM (1707)

having their respective eigenvalues in diagonal matrices Υ, Λ ∈ Sn arrangedin nonincreasing order (membership to the monotone cone KM (414)). Thenby splitting eigenvalue signs, we invent a symmetric SVD-like decomposition

A , UAΣAQHA ∈ Sn , B , UBΣBQH

B ∈ Sn (1708)

where UA , UB , QA , QB ∈ Cn×n are unitary matrices defined by (confer §A.6.5)

UA , WA

δ(ψ(δ(Υ))) , QA , WA

δ(ψ(δ(Υ)))H, ΣA = |Υ| (1709)

UB , WB

δ(ψ(δ(Λ))) , QB , WB

δ(ψ(δ(Λ)))H, ΣB = |Λ| (1710)

where step function ψ is defined in (1545). In this circumstance,

S⋆ = UAUHB = R⋆T ∈ Cn×n (1711)

optimal matrices (1702) now unitary are related by transposition. Theoptimal value of objective (1703) is

‖UAΣAQHA − S⋆UBΣBQH

B R⋆‖F = ‖ |Υ| − |Λ| ‖F (1712)

while the corresponding optimal value of trace maximization (1704) is

supRH=R−1

SH=S−1

re tr(ATSBR) = tr(|Υ| |Λ|) (1713)

C.4.2.2 Diagonal matrices

Now suppose A and B are diagonal matrices

A = Υ = δ2(Υ) ∈ Sn , δ(Υ) ∈ KM (1714)

B = Λ = δ2(Λ) ∈ Sn , δ(Λ) ∈ KM (1715)

both having their respective main diagonal entries arranged in nonincreasingorder:

minimizeR , S

‖Υ − SΛR‖F

subject to RH = R−1

SH = S−1

(1716)

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656 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS

Then we have a symmetric decomposition from unitary matrices as in (1708)where

UA ,√

δ(ψ(δ(Υ))) , QA ,√

δ(ψ(δ(Υ)))H, ΣA = |Υ| (1717)

UB ,√

δ(ψ(δ(Λ))) , QB ,√

δ(ψ(δ(Λ)))H, ΣB = |Λ| (1718)

Procrustes solution (1702) again sees the transposition relationship

S⋆ = UAUHB = R⋆T ∈ Cn×n (1711)

but both optimal unitary matrices are now themselves diagonal. So,

S⋆ΛR⋆ = δ(ψ(δ(Υ)))Λδ(ψ(δ(Λ))) = δ(ψ(δ(Υ)))|Λ| (1719)

C.5 Nonconvex quadratics

[343, §2] [317, §2] Given A∈ Sn, b∈Rn, and ρ>0

minimizex

12xTAx + bTx

subject to ‖x‖ ≤ ρ(1720)

is a nonconvex problem for symmetric A unless Aº 0. But necessaryand sufficient global optimality conditions are known for any symmetric A :vector x⋆ solves (1720) iff ∃ Lagrange multiplier λ⋆≥ 0 Ä

i) (A + λ⋆I )x⋆ = −b

ii) λ⋆(‖x⋆‖ − ρ) = 0 , ‖x⋆‖ ≤ ρ

iii) A + λ⋆I º 0

λ⋆ is unique.C.5 x⋆ is unique if A+λ⋆I≻0. Equality constrained problem

i) (A + λ⋆I )x⋆ = −bii) ‖x⋆‖ = ρiii) A + λ⋆I º 0

⇔minimize

x

12xTAx + bTx

subject to ‖x‖ = ρ(1721)

is nonconvex for any symmetric A matrix. x⋆ solves (1721) iff ∃ λ⋆∈ Rsatisfying the associated conditions. λ⋆ and x⋆ are unique as before. In 1998,Hiriart-Urruty [194] disclosed global optimality conditions for maximizing aconstrained convex quadratic; a nonconvex problem.

C.5The first two are necessary KKT conditions [59, §5.5.3] while condition iii governspassage to nonconvex global optimality.

Page 657: Convex Optimization & Euclidean Distance Geometry

Appendix D

Matrix calculus

From too much study, and from extreme passion, cometh madnesse.

−Isaac Newton [150, §5]

D.1 Directional derivative, Taylor series

D.1.1 Gradients

Gradient of a differentiable real function f(x) : RK→R with respect to itsvector argument is defined in terms of partial derivatives

∇f(x) ,

∂f(x)∂x1

∂f(x)∂x2...

∂f(x)∂xK

∈ RK (1722)

while the second-order gradient of the twice differentiable real function withrespect to its vector argument is traditionally called the Hessian ;

∇2f(x) ,

∂2f(x)

∂x21

∂2f(x)∂x1∂x2

· · · ∂2f(x)∂x1∂xK

∂2f(x)∂x2∂x1

∂2f(x)

∂x22

· · · ∂2f(x)∂x2∂xK

......

. . ....

∂2f(x)∂xK∂x1

∂2f(x)∂xK∂x2

· · · ∂2f(x)

∂x2K

∈ SK (1723)

© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

657

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658 APPENDIX D. MATRIX CALCULUS

The gradient of vector-valued function v(x) : R→RN on real domain isa row-vector

∇v(x) ,[

∂v1(x)∂x

∂v2(x)∂x

· · · ∂vN (x)∂x

]

∈ RN (1724)

while the second-order gradient is

∇2v(x) ,[

∂2v1(x)∂x2

∂2v2(x)∂x2 · · · ∂2vN (x)

∂x2

]

∈ RN (1725)

Gradient of vector-valued function h(x) : RK→RN on vector domain is

∇h(x) ,

∂h1(x)∂x1

∂h2(x)∂x1

· · · ∂hN (x)

∂x1

∂h1(x)∂x2

∂h2(x)∂x2

· · · ∂hN (x)

∂x2...

......

∂h1(x)∂xK

∂h2(x)∂xK

· · · ∂hN (x)

∂xK

= [∇h1(x) ∇h2(x) · · · ∇hN(x) ] ∈ RK×N

(1726)

while the second-order gradient has a three-dimensional representationdubbed cubix ;D.1

∇2h(x) ,

∇∂h1(x)∂x1

∇∂h2(x)∂x1

· · · ∇∂hN (x)

∂x1

∇∂h1(x)∂x2

∇∂h2(x)∂x2

· · · ∇∂hN (x)

∂x2...

......

∇∂h1(x)∂xK

∇∂h2(x)∂xK

· · · ∇∂hN (x)

∂xK

= [∇2h1(x) ∇2h2(x) · · · ∇2hN(x) ] ∈ RK×N×K

(1727)

where the gradient of each real entry is with respect to vector x as in (1722).

D.1The word matrix comes from the Latin for womb ; related to the prefix matri- derivedfrom mater meaning mother.

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 659

The gradient of real function g(X) : RK×L→R on matrix domain is

∇g(X) ,

∂g(X)∂X11

∂g(X)∂X12

· · · ∂g(X)∂X1L

∂g(X)∂X21

∂g(X)∂X22

· · · ∂g(X)∂X2L

......

...∂g(X)∂XK1

∂g(X)∂XK2

· · · ∂g(X)∂XKL

∈ RK×L

=

[

∇X(:,1) g(X)

∇X(:,2) g(X). . .

∇X(:,L) g(X)]

∈ RK×1×L

(1728)

where the gradient ∇X(:,i) is with respect to the ith column of X . Thestrange appearance of (1728) in RK×1×L is meant to suggest a third dimensionperpendicular to the page (not a diagonal matrix). The second-order gradienthas representation

∇2g(X) ,

∇∂g(X)∂X11

∇∂g(X)∂X12

· · · ∇∂g(X)∂X1L

∇∂g(X)∂X21

∇∂g(X)∂X22

· · · ∇∂g(X)∂X2L

......

...

∇∂g(X)∂XK1

∇∂g(X)∂XK2

· · · ∇∂g(X)∂XKL

∈ RK×L×K×L

=

[

∇∇X(:,1) g(X)

∇∇X(:,2) g(X). . .

∇∇X(:,L) g(X)]

∈ RK×1×L×K×L

(1729)

where the gradient ∇ is with respect to matrix X .

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660 APPENDIX D. MATRIX CALCULUS

Gradient of vector-valued function g(X) : RK×L→RN on matrix domainis a cubix

∇g(X) ,

[

∇X(:,1) g1(X) ∇X(:,1) g2(X) · · · ∇X(:,1) gN(X)

∇X(:,2) g1(X) ∇X(:,2) g2(X) · · · ∇X(:,2) gN(X). . . . . . . . .

∇X(:,L) g1(X) ∇X(:,L) g2(X) · · · ∇X(:,L) gN(X)]

= [∇g1(X) ∇g2(X) · · · ∇gN(X) ] ∈ RK×N×L (1730)

while the second-order gradient has a five-dimensional representation;

∇2g(X) ,

[

∇∇X(:,1) g1(X) ∇∇X(:,1) g2(X) · · · ∇∇X(:,1) gN(X)

∇∇X(:,2) g1(X) ∇∇X(:,2) g2(X) · · · ∇∇X(:,2) gN(X). . . . . . . . .

∇∇X(:,L) g1(X) ∇∇X(:,L) g2(X) · · · ∇∇X(:,L) gN(X)]

= [∇2g1(X) ∇2g2(X) · · · ∇2gN(X) ] ∈ RK×N×L×K×L (1731)

The gradient of matrix-valued function g(X) : RK×L→RM×N on matrixdomain has a four-dimensional representation called quartix (fourth-ordertensor)

∇g(X) ,

∇g11(X) ∇g12(X) · · · ∇g1N(X)

∇g21(X) ∇g22(X) · · · ∇g2N(X)...

......

∇gM1(X) ∇gM2(X) · · · ∇gMN(X)

∈ RM×N×K×L (1732)

while the second-order gradient has six-dimensional representation

∇2g(X) ,

∇2g11(X) ∇2g12(X) · · · ∇2g1N(X)

∇2g21(X) ∇2g22(X) · · · ∇2g2N(X)...

......

∇2gM1(X) ∇2gM2(X) · · · ∇2gMN(X)

∈ RM×N×K×L×K×L

(1733)and so on.

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 661

D.1.2 Product rules for matrix-functions

Given dimensionally compatible matrix-valued functions of matrix variablef(X) and g(X)

∇X

(

f(X)Tg(X))

= ∇X(f) g + ∇X(g) f (1734)

while [51, §8.3] [311]

∇X tr(

f(X)Tg(X))

= ∇X

(

tr(

f(X)Tg(Z))

+ tr(

g(X) f(Z)T)

)∣

Z←X(1735)

These expressions implicitly apply as well to scalar-, vector-, or matrix-valuedfunctions of scalar, vector, or matrix arguments.

D.1.2.0.1 Example. Cubix.Suppose f(X) : R2×2→R2 = XTa and g(X) : R2×2→R2 = Xb . We wishto find

∇X

(

f(X)Tg(X))

= ∇X aTX2b (1736)

using the product rule. Formula (1734) calls for

∇X aTX2b = ∇X(XTa) Xb + ∇X(Xb) XTa (1737)

Consider the first of the two terms:

∇X(f) g = ∇X(XTa) Xb

=[

∇(XTa)1 ∇(XTa)2

]

Xb(1738)

The gradient of XTa forms a cubix in R2×2×2 ; a.k.a, third-order tensor.

∂(XTa)1∂X11 IIIIII ∂(XTa)2

∂X11 IIIIII∂(XTa)1

∂X12

∂(XTa)2∂X12

∂(XTa)1∂X21 IIIIII ∂(XTa)2

∂X21 IIIIII∂(XTa)1

∂X22

∂(XTa)2∂X22

∇X(XTa) Xb =

(Xb)1

(Xb)2

∈ R2×1×2

(1739)

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662 APPENDIX D. MATRIX CALCULUS

Because gradient of the product (1736) requires total change with respectto change in each entry of matrix X , the Xb vector must make an innerproduct with each vector in the second dimension of the cubix (indicated bydotted line segments);

∇X(XTa) Xb =

a1 00 a1

a2 00 a2

[

b1X11 + b2X12

b1X21 + b2X22

]

∈ R2×1×2

=

[

a1(b1X11 + b2X12) a1(b1X21 + b2X22)a2(b1X11 + b2X12) a2(b1X21 + b2X22)

]

∈ R2×2

= abTXT

(1740)

where the cubix appears as a complete 2×2×2 matrix. In like manner forthe second term ∇X(g) f

∇X(Xb) XTa =

b1 0b2 0

0 b1

0 b2

[

X11a1 + X21a2

X12a1 + X22a2

]

∈ R2×1×2

= XTabT ∈ R2×2

(1741)

The solution∇X aTX2b = abTXT + XTabT (1742)

can be found from Table D.2.1 or verified using (1735). 2

D.1.2.1 Kronecker product

A partial remedy for venturing into hyperdimensional matrix representations,such as the cubix or quartix, is to first vectorize matrices as in (37). Thisdevice gives rise to the Kronecker product of matrices ⊗ ; a.k.a, directproduct or tensor product. Although it sees reversal in the literature,[324, §2.1] we adopt the definition: for A∈Rm×n and B∈Rp×q

B ⊗ A ,

B11A B12A · · · B1qAB21A B22A · · · B2qA

......

...Bp1A Bp2A · · · BpqA

∈ Rpm×qn (1743)

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 663

for which A ⊗ 1 = 1 ⊗ A = A (real unity acts like identity).One advantage to vectorization is existence of a traditional

two-dimensional matrix representation (second-order tensor) for thesecond-order gradient of a real function with respect to a vectorized matrix.For example, from §A.1.1 no.33 (§D.2.1) for square A ,B ∈Rn×n [162, §5.2][12, §3]

∇2vec X tr(AXBXT) = ∇2

vec X vec(X)T(BT⊗A) vec X = B⊗AT+ BT⊗A ∈ Rn2×n2

(1744)

To disadvantage is a large new but known set of algebraic rules (§A.1.1)and the fact that its mere use does not generally guarantee two-dimensionalmatrix representation of gradients.

Another application of the Kronecker product is to reverse order ofappearance in a matrix product: Suppose we wish to weight the columnsof a matrix S∈ RM×N , for example, by respective entries wi from the maindiagonal in

W ,

w1 0. . .

0T wN

∈ SN (1745)

A conventional means for accomplishing column weighting is to multiply Sby diagonal matrix W on the right-hand side:

SW = S

w1 0. . .

0T wN

=[

S(: , 1)w1 · · · S(: , N)wN

]

∈ RM×N (1746)

To reverse product order such that diagonal matrix W instead appears tothe left of S : for I∈ SM (Sze Wan)

SW = (δ(W )T ⊗ I)

S(: , 1) 0 0

0 S(: , 2). . .

. . . . . . 00 0 S(: , N)

∈ RM×N (1747)

To instead weight the rows of S via diagonal matrix W ∈ SM , for I∈ SN

WS =

S(1 , :) 0 0

0 S(2 , :). . .

. . . . . . 00 0 S(M , :)

(δ(W ) ⊗ I) ∈ RM×N (1748)

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664 APPENDIX D. MATRIX CALCULUS

For any matrices of like size, S , Y ∈ RM×N

SY =[

δ(Y (: , 1)) · · · δ(Y (: , N))]

S(: , 1) 0 0

0 S(: , 2). . .

. . . . . . 00 0 S(: , N)

∈ RM×N

(1749)

which converts a Hadamard product into a standard matrix product. In thespecial case that S = s and Y = y are vectors in RM

s y = δ(s)y (1750)

sT⊗ y = ysT

s ⊗ yT = syT (1751)

D.1.3 Chain rules for composite matrix-functions

Given dimensionally compatible matrix-valued functions of matrix variablef(X) and g(X) [215, §15.7]

∇X g(

f(X)T)

= ∇XfT∇f g (1752)

∇2X g

(

f(X)T)

= ∇X

(

∇XfT∇f g)

= ∇2Xf ∇f g + ∇XfT∇2

f g ∇Xf (1753)

D.1.3.1 Two arguments

∇X g(

f(X)T, h(X)T)

= ∇XfT∇f g + ∇XhT∇h g (1754)

D.1.3.1.1 Example. Chain rule for two arguments. [41, §1.1]

g(

f(x)T, h(x)T)

= (f(x) + h(x))TA(f(x) + h(x)) (1755)

f(x) =

[

x1

εx2

]

, h(x) =

[

εx1

x2

]

(1756)

∇x g(

f(x)T, h(x)T)

=

[

1 00 ε

]

(A +AT)(f + h) +

[

ε 00 1

]

(A +AT)(f + h)

(1757)

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 665

∇x g(

f(x)T, h(x)T)

=

[

1 + ε 00 1 + ε

]

(A +AT)

([

x1

εx2

]

+

[

εx1

x2

])

(1758)

limε→0

∇x g(

f(x)T, h(x)T)

= (A +AT)x (1759)

from Table D.2.1. 2

These foregoing formulae remain correct when gradient produceshyperdimensional representation:

D.1.4 First directional derivative

Assume that a differentiable function g(X) : RK×L→RM×N has continuousfirst- and second-order gradients ∇g and ∇2g over dom g which is an openset. We seek simple expressions for the first and second directional derivatives

in direction Y∈RK×L : respectively,→Y

dg ∈ RM×N and→Y

dg2 ∈ RM×N .

Assuming that the limit exists, we may state the partial derivative of themnth entry of g with respect to the klth entry of X ;

∂gmn(X)

∂Xkl

= lim∆t→0

gmn(X + ∆t ekeTl ) − gmn(X)

∆t∈ R (1760)

where ek is the kth standard basis vector in RK while el is the lth standardbasis vector in RL. The total number of partial derivatives equals KLMNwhile the gradient is defined in their terms; the mnth entry of the gradient is

∇gmn(X) =

∂gmn(X)∂X11

∂gmn(X)∂X12

· · · ∂gmn(X)∂X1L

∂gmn(X)∂X21

∂gmn(X)∂X22

· · · ∂gmn(X)∂X2L

......

...∂gmn(X)

∂XK1

∂gmn(X)∂XK2

· · · ∂gmn(X)∂XKL

∈ RK×L (1761)

while the gradient is a quartix

∇g(X) =

∇g11(X) ∇g12(X) · · · ∇g1N(X)

∇g21(X) ∇g22(X) · · · ∇g2N(X)...

......

∇gM1(X) ∇gM2(X) · · · ∇gMN(X)

∈ RM×N×K×L (1762)

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666 APPENDIX D. MATRIX CALCULUS

By simply rotating our perspective of the four-dimensional representation ofgradient matrix, we find one of three useful transpositions of this quartix(connoted T1):

∇g(X)T1 =

∂g(X)∂X11

∂g(X)∂X12

· · · ∂g(X)∂X1L

∂g(X)∂X21

∂g(X)∂X22

· · · ∂g(X)∂X2L...

......

∂g(X)∂XK1

∂g(X)∂XK2

· · · ∂g(X)∂XKL

∈ RK×L×M×N (1763)

When the limit for ∆t∈R exists, it is easy to show by substitution ofvariables in (1760)

∂gmn(X)

∂Xkl

Ykl = lim∆t→0

gmn(X + ∆t Ykl ekeTl ) − gmn(X)

∆t∈ R (1764)

which may be interpreted as the change in gmn at X when the change in Xkl

is equal to Ykl , the klth entry of any Y ∈ RK×L. Because the total changein gmn(X) due to Y is the sum of change with respect to each and everyXkl , the mnth entry of the directional derivative is the corresponding totaldifferential [215, §15.8]

dgmn(X)|dX→Y =∑

k,l

∂gmn(X)

∂Xkl

Ykl = tr(

∇gmn(X)TY)

(1765)

=∑

k,l

lim∆t→0

gmn(X + ∆t Ykl ekeTl ) − gmn(X)

∆t(1766)

= lim∆t→0

gmn(X + ∆t Y ) − gmn(X)

∆t(1767)

=d

dt

t=0

gmn(X+ t Y ) (1768)

where t∈R . Assuming finite Y , equation (1767) is called the Gateauxdifferential [40, App.A.5] [196, §D.2.1] [355, §5.28] whose existence is impliedby existence of the Frechet differential (the sum in (1765)). [246, §7.2] Eachmay be understood as the change in gmn at X when the change in X is equal

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 667

in magnitude and direction to Y .D.2 Hence the directional derivative,

→Y

dg(X) ,

dg11(X) dg12(X) · · · dg1N(X)

dg21(X) dg22(X) · · · dg2N(X)...

......

dgM1(X) dgM2(X) · · · dgMN(X)

dX→Y

∈ RM×N

=

tr(

∇g11(X)TY)

tr(

∇g12(X)TY)

· · · tr(

∇g1N(X)TY)

tr(

∇g21(X)TY)

tr(

∇g22(X)TY)

· · · tr(

∇g2N(X)TY)

......

...tr

(

∇gM1(X)TY)

tr(

∇gM2(X)TY)

· · · tr(

∇gMN(X)TY)

=

k,l

∂g11(X)∂Xkl

Ykl

k,l

∂g12(X)∂Xkl

Ykl · · · ∑

k,l

∂g1N (X)

∂XklYkl

k,l

∂g21(X)∂Xkl

Ykl

k,l

∂g22(X)∂Xkl

Ykl · · · ∑

k,l

∂g2N (X)

∂XklYkl

......

...∑

k,l

∂gM1(X)∂Xkl

Ykl

k,l

∂gM2(X)∂Xkl

Ykl · · · ∑

k,l

∂gMN (X)

∂XklYkl

(1769)

from which it follows→Y

dg(X) =∑

k,l

∂g(X)

∂Xkl

Ykl (1770)

Yet for all X∈ dom g , any Y∈RK×L, and some open interval of t∈R

g(X+ t Y ) = g(X) + t→Y

dg(X) + o(t2) (1771)

which is the first-order Taylor series expansion about X . [215, §18.4][149, §2.3.4] Differentiation with respect to t and subsequent t-zeroingisolates the second term of expansion. Thus differentiating and zeroingg(X+ t Y ) in t is an operation equivalent to individually differentiating andzeroing every entry gmn(X+ t Y ) as in (1768). So the directional derivativeof g(X) : RK×L→RM×N in any direction Y ∈ RK×L evaluated at X∈ dom gbecomes

→Y

dg(X) =d

dt

t=0

g(X+ t Y ) ∈ RM×N (1772)

D.2Although Y is a matrix, we may regard it as a vector in RKL.

Page 668: Convex Optimization & Euclidean Distance Geometry

668 APPENDIX D. MATRIX CALCULUS

(f(α), α)

∂H

υ T

f(x)

f(α + t y)

υ ,

∇xf(α)

12

→∇xf(α)

df (α)

­­­­­­­­­­

Figure 158: Strictly convex quadratic bowl in R2×R ; f(x)= xTx : R2→ Rversus x on some open disc in R2. Plane slice ∂H is perpendicularto function domain. Slice intersection with domain connotes bidirectionalvector y . Slope of tangent line T at point (α , f(α)) is value of ∇xf(α)Tydirectional derivative (1797) at α in slice direction y . Negative gradient−∇xf(x)∈ R2 is direction of steepest descent. [372] [215, §15.6] [149] Whenvector υ∈R3 entry υ3 is half directional derivative in gradient direction at α

and when

[

υ1

υ2

]

= ∇xf(α) , then −υ points directly toward bowl bottom.

[273, §2.1, §5.4.5] [33, §6.3.1] which is simplest. In case of a real functiong(X) : RK×L→R

→Y

dg(X) = tr(

∇g(X)TY)

(1794)

In case g(X) : RK→R

→Y

dg(X) = ∇g(X)TY (1797)

Unlike gradient, directional derivative does not expand dimension;directional derivative (1772) retains the dimensions of g . The derivativewith respect to t makes the directional derivative resemble ordinary calculus

(§D.2); e.g., when g(X) is linear,→Y

dg(X) = g(Y ). [246, §7.2]

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 669

D.1.4.1 Interpretation of directional derivative

In the case of any differentiable real function g(X) : RK×L→R , thedirectional derivative of g(X) at X in any direction Y yields the slope ofg along the line X+ t Y | t ∈ R through its domain evaluated at t = 0.For higher-dimensional functions, by (1769), this slope interpretation can beapplied to each entry of the directional derivative.

Figure 158, for example, shows a plane slice of a real convex bowl-shapedfunction f(x) along a line α + t y | t ∈ R through its domain. The slicereveals a one-dimensional real function of t ; f(α + t y). The directionalderivative at x = α in direction y is the slope of f(α + t y) with respectto t at t = 0. In the case of a real function having vector argumenth(X) : RK→R , its directional derivative in the normalized direction of itsgradient is the gradient magnitude. (1797) For a real function of real variable,the directional derivative evaluated at any point in the function domain isjust the slope of that function there scaled by the real direction. (confer §3.7)

Directional derivative generalizes our one-dimensional notion of derivativeto a multidimensional domain. When direction Y coincides with a memberof the standard Cartesian basis eke

Tl (60), then a single partial derivative

∂g(X)/∂Xkl is obtained from directional derivative (1770); such is each entryof gradient ∇g(X) in equalities (1794) and (1797), for example.

D.1.4.1.1 Theorem. Directional derivative optimality condition.[246, §7.4] Suppose f(X) : RK×L→R is minimized on convex set C⊆Rp×k

by X⋆, and the directional derivative of f exists there. Then for all X∈ C

→X−X⋆

df(X) ≥ 0 (1773)

D.1.4.1.2 Example. Simple bowl.Bowl function (Figure 158)

f(x) : RK → R , (x − a)T(x − a) − b (1774)

has function offset −b ∈ R , axis of revolution at x = a , and positive definiteHessian (1723) everywhere in its domain (an open hyperdisc in RK ); id est,strictly convex quadratic f(x) has unique global minimum equal to −b at

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670 APPENDIX D. MATRIX CALCULUS

x = a . A vector −υ based anywhere in dom f × R pointing toward theunique bowl-bottom is specified:

υ ∝[

x − af(x) + b

]

∈ RK× R (1775)

Such a vector is

υ =

∇xf(x)

12

→∇xf(x)

df(x)

(1776)

since the gradient is∇xf(x) = 2(x − a) (1777)

and the directional derivative in the direction of the gradient is (1797)→∇xf(x)

df(x) = ∇xf(x)T∇xf(x) = 4(x − a)T(x − a) = 4(f(x) + b) (1778)2

D.1.5 Second directional derivative

By similar argument, it so happens: the second directional derivative isequally simple. Given g(X) : RK×L→RM×N on open domain,

∇∂gmn(X)

∂Xkl

=∂∇gmn(X)

∂Xkl

=

∂2gmn(X)∂Xkl∂X11

∂2gmn(X)∂Xkl∂X12

· · · ∂2gmn(X)∂Xkl∂X1L

∂2gmn(X)∂Xkl∂X21

∂2gmn(X)∂Xkl∂X22

· · · ∂2gmn(X)∂Xkl∂X2L

......

...∂2gmn(X)∂Xkl∂XK1

∂2gmn(X)∂Xkl∂XK2

· · · ∂2gmn(X)∂Xkl∂XKL

∈ RK×L (1779)

∇2gmn(X) =

∇∂gmn(X)∂X11

∇∂gmn(X)∂X12

· · · ∇∂gmn(X)∂X1L

∇∂gmn(X)∂X21

∇∂gmn(X)∂X22

· · · ∇∂gmn(X)∂X2L

......

...

∇∂gmn(X)∂XK1

∇∂gmn(X)∂XK2

· · · ∇∂gmn(X)∂XKL

∈ RK×L×K×L

=

∂∇gmn(X)∂X11

∂∇gmn(X)∂X12

· · · ∂∇gmn(X)∂X1L

∂∇gmn(X)∂X21

∂∇gmn(X)∂X22

· · · ∂∇gmn(X)∂X2L...

......

∂∇gmn(X)∂XK1

∂∇gmn(X)∂XK2

· · · ∂∇gmn(X)∂XKL

(1780)

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 671

Rotating our perspective, we get several views of the second-order gradient:

∇2g(X) =

∇2g11(X) ∇2g12(X) · · · ∇2g1N(X)

∇2g21(X) ∇2g22(X) · · · ∇2g2N(X)...

......

∇2gM1(X) ∇2gM2(X) · · · ∇2gMN(X)

∈ RM×N×K×L×K×L (1781)

∇2g(X)T1 =

∇∂g(X)∂X11

∇∂g(X)∂X12

· · · ∇∂g(X)∂X1L

∇∂g(X)∂X21

∇∂g(X)∂X22

· · · ∇∂g(X)∂X2L...

......

∇∂g(X)∂XK1

∇∂g(X)∂XK2

· · · ∇∂g(X)∂XKL

∈ RK×L×M×N×K×L (1782)

∇2g(X)T2 =

∂∇g(X)∂X11

∂∇g(X)∂X12

· · · ∂∇g(X)∂X1L

∂∇g(X)∂X21

∂∇g(X)∂X22

· · · ∂∇g(X)∂X2L...

......

∂∇g(X)∂XK1

∂∇g(X)∂XK2

· · · ∂∇g(X)∂XKL

∈ RK×L×K×L×M×N (1783)

Assuming the limits exist, we may state the partial derivative of the mnth

entry of g with respect to the klth and ij th entries of X ;

∂2gmn(X)∂Xkl ∂Xij

= lim∆τ,∆t→0

gmn(X+∆t ekeTl +∆τ eie

Tj )−gmn(X+∆t ekeT

l )−(gmn(X+∆τ eieTj )−gmn(X))

∆τ ∆t

(1784)Differentiating (1764) and then scaling by Yij

∂2gmn(X)∂Xkl ∂Xij

YklYij = lim∆t→0

∂gmn(X+∆t Ykl ekeTl )−∂gmn(X)

∂Xij ∆tYij

= lim∆τ,∆t→0

gmn(X+∆t Ykl ekeTl +∆τ Yij eie

Tj )−gmn(X+∆t Ykl ekeT

l )−(gmn(X+∆τ Yij eieTj )−gmn(X))

∆τ ∆t

(1785)

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672 APPENDIX D. MATRIX CALCULUS

which can be proved by substitution of variables in (1784). The mnth

second-order total differential due to any Y∈RK×L is

d2gmn(X)|dX→Y =∑

i,j

k,l

∂2gmn(X)

∂Xkl ∂Xij

YklYij = tr(

∇X tr(

∇gmn(X)TY)T

Y)

(1786)

=∑

i,j

lim∆t→0

∂gmn(X + ∆t Y ) − ∂gmn(X)

∂Xij ∆tYij (1787)

= lim∆t→0

gmn(X + 2∆t Y ) − 2gmn(X + ∆t Y ) + gmn(X)

∆t2(1788)

=d2

dt2

t=0

gmn(X+ t Y ) (1789)

Hence the second directional derivative,

→Y

dg2(X) ,

d2g11(X) d2g12(X) · · · d2g1N(X)

d2g21(X) d2g22(X) · · · d2g2N(X)...

......

d2gM1(X) d2gM2(X) · · · d2gMN(X)

dX→Y

∈ RM×N

=

tr(

∇tr(

∇g11(X)TY)T

Y)

tr(

∇tr(

∇g12(X)TY)T

Y)

· · · tr(

∇tr(

∇g1N(X)TY)T

Y)

tr(

∇tr(

∇g21(X)TY)T

Y)

tr(

∇tr(

∇g22(X)TY)T

Y)

· · · tr(

∇tr(

∇g2N(X)TY)T

Y)

......

...

tr(

∇tr(

∇gM1(X)TY)T

Y)

tr(

∇tr(

∇gM2(X)TY)T

Y)

· · · tr(

∇tr(

∇gMN(X)TY)T

Y)

=

i,j

k,l

∂2g11(X)∂Xkl ∂Xij

YklYij

i,j

k,l

∂2g12(X)∂Xkl ∂Xij

YklYij · · · ∑

i,j

k,l

∂2g1N (X)

∂Xkl ∂XijYklYij

i,j

k,l

∂2g21(X)∂Xkl ∂Xij

YklYij

i,j

k,l

∂2g22(X)∂Xkl ∂Xij

YklYij · · · ∑

i,j

k,l

∂2g2N (X)

∂Xkl ∂XijYklYij

......

...∑

i,j

k,l

∂2gM1(X)∂Xkl ∂Xij

YklYij

i,j

k,l

∂2gM2(X)∂Xkl ∂Xij

YklYij · · · ∑

i,j

k,l

∂2gMN (X)

∂Xkl ∂XijYklYij

(1790)

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 673

from which it follows

→Y

dg2(X) =∑

i,j

k,l

∂2g(X)

∂Xkl ∂Xij

YklYij =∑

i,j

∂Xij

→Y

dg(X) Yij (1791)

Yet for all X∈ dom g , any Y∈RK×L, and some open interval of t∈R

g(X+ t Y ) = g(X) + t→Y

dg(X) +1

2!t2

→Y

dg2(X) + o(t3) (1792)

which is the second-order Taylor series expansion about X . [215, §18.4][149, §2.3.4] Differentiating twice with respect to t and subsequent t-zeroingisolates the third term of the expansion. Thus differentiating and zeroingg(X+ t Y ) in t is an operation equivalent to individually differentiating andzeroing every entry gmn(X+ t Y ) as in (1789). So the second directionalderivative of g(X) : RK×L→RM×N becomes [273, §2.1, §5.4.5] [33, §6.3.1]

→Y

dg2(X) =d2

dt2

t=0

g(X+ t Y ) ∈ RM×N (1793)

which is again simplest. (confer (1772)) Directional derivative retains thedimensions of g .

D.1.6 directional derivative expressions

In the case of a real function g(X) : RK×L→R , all its directional derivativesare in R :

→Y

dg(X) = tr(

∇g(X)TY)

(1794)

→Y

dg2(X) = tr(

∇X tr(

∇g(X)TY)T

Y)

= tr

(

∇X

→Y

dg(X)TY

)

(1795)

→Y

dg3(X) = tr

(

∇X tr(

∇X tr(

∇g(X)TY)T

Y)T

Y

)

= tr

(

∇X

→Y

dg2(X)TY

)

(1796)

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674 APPENDIX D. MATRIX CALCULUS

In the case g(X) : RK→R has vector argument, they further simplify:→Y

dg(X) = ∇g(X)TY (1797)

→Y

dg2(X) = Y T∇2g(X)Y (1798)→Y

dg3(X) = ∇X

(

Y T∇2g(X)Y)T

Y (1799)

and so on.

D.1.7 Taylor series

Series expansions of the differentiable matrix-valued function g(X) , ofmatrix argument, were given earlier in (1771) and (1792). Assuming g(X)has continuous first-, second-, and third-order gradients over the open setdom g , then for X∈ dom g and any Y ∈ RK×L the complete Taylor seriesis expressed on some open interval of µ∈R

g(X+µY ) = g(X) + µ→Y

dg(X) +1

2!µ2

→Y

dg2(X) +1

3!µ3

→Y

dg3(X) + o(µ4) (1800)

or on some open interval of ‖Y ‖2

g(Y ) = g(X) +→Y −X

dg(X) +1

2!

→Y −X

dg2(X) +1

3!

→Y −X

dg3(X) + o(‖Y ‖4) (1801)

which are third-order expansions about X . The mean value theorem fromcalculus is what insures finite order of the series. [215] [41, §1.1] [40, App.A.5][196, §0.4] These somewhat unbelievable formulae imply that a function canbe determined over the whole of its domain by knowing its value and all itsdirectional derivatives at a single point X .

D.1.7.0.1 Example. Inverse-matrix function.Say g(Y )= Y −1. From the table on page 680,

→Y

dg(X) =d

dt

t=0

g(X+ t Y ) = −X−1Y X−1 (1802)

→Y

dg2(X) =d2

dt2

t=0

g(X+ t Y ) = 2X−1Y X−1Y X−1 (1803)

→Y

dg3(X) =d3

dt3

t=0

g(X+ t Y ) = −6X−1Y X−1Y X−1Y X−1 (1804)

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 675

Let’s find the Taylor series expansion of g about X = I : Since g(I )= I , for‖Y ‖2 < 1 (µ = 1 in (1800))

g(I + Y ) = (I + Y )−1 = I − Y + Y 2− Y 3 + . . . (1805)

If Y is small, (I + Y )−1 ≈ I − Y . D.3 Now we find Taylor series expansionabout X :

g(X + Y ) = (X + Y )−1 = X−1 − X−1Y X−1 + 2X−1Y X−1Y X−1 − . . .(1806)

If Y is small, (X + Y )−1 ≈X−1 − X−1Y X−1. 2

D.1.7.0.2 Exercise. log det . (confer [59, p.644])Find the first three terms of a Taylor series expansion for log detY . Specifyan open interval over which the expansion holds in vicinity of X . H

D.1.8 Correspondence of gradient to derivative

From the foregoing expressions for directional derivative, we derive arelationship between the gradient with respect to matrix X and the derivativewith respect to real variable t :

D.1.8.1 first-order

Removing evaluation at t = 0 from (1772),D.4 we find an expression for thedirectional derivative of g(X) in direction Y evaluated anywhere along aline X+ t Y | t ∈ R intersecting dom g

→Y

dg(X+ t Y ) =d

dtg(X+ t Y ) (1807)

In the general case g(X) : RK×L→RM×N , from (1765) and (1768) we find

tr(

∇X gmn(X+ t Y )TY)

=d

dtgmn(X+ t Y ) (1808)

D.3Had we instead set g(Y )=(I + Y )−1, then the equivalent expansion would have beenabout X = 0.D.4Justified by replacing X with X+ t Y in (1765)-(1767); beginning,

dgmn(X+ t Y )|dX→Y =∑

k , l

∂gmn(X+ t Y )

∂XklYkl

Page 676: Convex Optimization & Euclidean Distance Geometry

676 APPENDIX D. MATRIX CALCULUS

which is valid at t = 0, of course, when X ∈ dom g . In the important caseof a real function g(X) : RK×L→R , from (1794) we have simply

tr(

∇X g(X+ t Y )TY)

=d

dtg(X+ t Y ) (1809)

When, additionally, g(X) : RK→R has vector argument,

∇X g(X+ t Y )TY =d

dtg(X+ t Y ) (1810)

D.1.8.1.1 Example. Gradient.g(X) = wTXTXw , X∈ RK×L, w∈RL. Using the tables in §D.2,

tr(

∇X g(X+ t Y )TY)

= tr(

2wwT(XT+ t Y T)Y)

(1811)

= 2wT(XTY + t Y TY )w (1812)

Applying the equivalence (1809),

d

dtg(X+ t Y ) =

d

dtwT(X+ t Y )T(X+ t Y )w (1813)

= wT(

XTY + Y TX + 2t Y TY)

w (1814)

= 2wT(XTY + t Y TY )w (1815)

which is the same as (1812); hence, equivalence is demonstrated.It is easy to extract ∇g(X) from (1815) knowing only (1809):

tr(

∇X g(X+ t Y )TY)

= 2wT(XTY + t Y TY )w= 2 tr

(

wwT(XT+ t Y T)Y)

tr(

∇X g(X)TY)

= 2 tr(

wwTXTY)

⇔∇X g(X) = 2XwwT

(1816)

2

D.1.8.2 second-order

Likewise removing the evaluation at t = 0 from (1793),

→Y

dg2(X+ t Y ) =d2

dt2g(X+ t Y ) (1817)

we can find a similar relationship between the second-order gradient and thesecond derivative: In the general case g(X) : RK×L→RM×N from (1786) and(1789),

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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 677

tr(

∇X tr(

∇X gmn(X+ t Y )TY)T

Y)

=d2

dt2gmn(X+ t Y ) (1818)

In the case of a real function g(X) : RK×L→R we have, of course,

tr(

∇X tr(

∇X g(X+ t Y )TY)T

Y)

=d2

dt2g(X+ t Y ) (1819)

From (1798), the simpler case, where the real function g(X) : RK→R hasvector argument,

Y T∇2Xg(X+ t Y )Y =

d2

dt2g(X+ t Y ) (1820)

D.1.8.2.1 Example. Second-order gradient.Given real function g(X) = log det X having domain int SK

+ , we want tofind ∇2g(X)∈ RK×K×K×K . From the tables in §D.2,

h(X) , ∇g(X) = X−1∈ int SK+ (1821)

so ∇2g(X)=∇h(X). By (1808) and (1771), for Y ∈ SK

tr(

∇hmn(X)TY)

=d

dt

t=0

hmn(X+ t Y ) (1822)

=

(

d

dt

t=0

h(X+ t Y )

)

mn

(1823)

=

(

d

dt

t=0

(X+ t Y )−1

)

mn

(1824)

= −(

X−1 Y X−1)

mn(1825)

Setting Y to a member of ekeTl ∈ RK×K | k, l=1 . . . K , and employing a

property (39) of the trace function we find

∇2g(X)mnkl = tr(

∇hmn(X)TekeTl

)

= ∇hmn(X)kl = −(

X−1ekeTl X−1

)

mn

(1826)

∇2g(X)kl = ∇h(X)kl = −(

X−1ekeTl X−1

)

∈ RK×K (1827)2

From all these first- and second-order expressions, we may generate newones by evaluating both sides at arbitrary t (in some open interval) but onlyafter the differentiation.

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678 APPENDIX D. MATRIX CALCULUS

D.2 Tables of gradients and derivatives Results may be numerically proven by Romberg extrapolation. [105]When proving results for symmetric matrices algebraically, it is criticalto take gradients ignoring symmetry and to then substitute symmetricentries afterward. [162] [63] a , b∈Rn, x, y∈Rk, A ,B∈ Rm×n, X,Y ∈ RK×L, t , µ∈R ,i, j, k, ℓ,K, L ,m, n ,M,N are integers, unless otherwise noted. xµ means δ(δ(x)µ) for µ∈R ; id est, entrywise vector exponentiation.

δ is the main-diagonal linear operator (1391). x0 , 1, X0 , I if square. ddx

,

ddx1...d

dxk

,

→y

dg(x) ,→y

dg2(x) (directional derivatives §D.1), log x , ex,

|x| , sgn x , x/y (Hadamard quotient),√

x (entrywise square root),etcetera, are maps f : Rk→ Rk that maintain dimension; e.g., (§A.1.1)

d

dxx−1 , ∇x 1Tδ(x)−11 (1828) For A a scalar or square matrix, we have the Taylor series [75, §3.6]

eA ,

∞∑

k=0

1

k!Ak (1829)

Further, [328, §5.4]

eA ≻ 0 ∀A ∈ Sm (1830) For all square A and integer k

detkA = det Ak (1831)

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D.2. TABLES OF GRADIENTS AND DERIVATIVES 679

D.2.1 algebraic

∇x x = ∇x xT = I ∈ Rk×k ∇XX = ∇XXT , I ∈ RK×L×K×L (identity)

∇x(Ax − b) = AT

∇x

(

xTA − bT)

= A

∇x(Ax − b)T(Ax − b) = 2AT(Ax − b)∇2

x (Ax − b)T(Ax − b) = 2ATA

∇x‖Ax − b‖ = AT(Ax − b)/‖Ax − b‖

∇xzT|Ax − b| = ATδ(z) sgn(Ax − b) , zi 6= 0 ⇒ (Ax − b)i 6= 0

∇x1Tf(|Ax − b|) = ATδ

(

df(y)dy

y=|Ax−b|

)

sgn(Ax − b)

∇x

(

xTAx + 2xTBy + yTCy)

=(

A +AT)

x + 2By∇x(x + y)TA(x + y) = (A +AT)(x + y)∇2

x

(

xTAx + 2xTBy + yTCy)

= A +AT

∇X aTXb = ∇X bTXTa = abT

∇X aTX2b = XTabT+ abTXT

∇X aTX−1b = −X−TabTX−T

∇X(X−1)kl =∂X−1

∂Xkl= −X−1ekeT

l X−1,confer(1763)(1827)

∇x aTxTxb = 2xaTb ∇X aTXTXb = X(abT+ baT)

∇x aTxxTb = (abT+ baT)x ∇X aTXXTb = (abT+ baT)X

∇x aTxTxa = 2xaTa ∇X aTXTXa = 2XaaT

∇x aTxxTa = 2aaTx ∇X aTXXTa = 2aaTX

∇x aTyxTb = baTy ∇X aTYXTb = baTY

∇x aTyTxb = ybTa ∇X aTY TXb = Y abT

∇x aTxyTb = abTy ∇X aTXY Tb = abTY

∇x aTxTyb = yaTb ∇X aTXTY b = Y baT

Page 680: Convex Optimization & Euclidean Distance Geometry

680 APPENDIX D. MATRIX CALCULUS

algebraic continued

ddt

(X+ t Y ) = Y

ddt

BT(X+ t Y )−1A = −BT(X+ t Y )−1 Y (X+ t Y )−1Addt

BT(X+ t Y )−TA = −BT(X+ t Y )−T Y T(X+ t Y )−TAddt

BT(X+ t Y )µA = ... , −1 ≤ µ ≤ 1, X , Y ∈ SM+

d2

dt2BT(X+ t Y )−1A = 2BT(X+ t Y )−1 Y (X+ t Y )−1 Y (X+ t Y )−1A

d3

dt3BT(X+ t Y )−1A = −6BT(X+ t Y )−1 Y (X+ t Y )−1 Y (X+ t Y )−1Y (X+ t Y )−1A

ddt

(

(X+ t Y )TA(X+ t Y ))

= Y TAX + XTAY + 2 t Y TAYd2

dt2

(

(X+ t Y )TA(X+ t Y ))

= 2Y TAY

ddt

(

(X+ t Y )TA(X+ t Y ))−1

=−(

(X+ t Y )TA(X+ t Y ))−1

(Y TAX + XTAY + 2 t Y TAY )(

(X+ t Y )TA(X+ t Y ))−1

ddt((X+ t Y )A(X+ t Y )) = YAX + XAY + 2 t YAY

d2

dt2((X+ t Y )A(X+ t Y )) = 2YAY

D.2.1.0.1 Exercise. Expand these tables.Provide unfinished table entries indicated by . . . throughout §D.2. H

D.2.1.0.2 Exercise. log . (§D.1.7)Find the first four terms of the Taylor series expansion for log x about x = 1.Prove that log x ≤ x−1; alternatively, plot the supporting hyperplane to

the hypograph of log x at

[

xlog x

]

=

[

10

]

. H

D.2.2 trace Kronecker

∇vec X tr(AXBXT) = ∇vec X vec(X)T(BT⊗ A) vec X = (B ⊗AT + BT⊗A) vec X

∇2vec X tr(AXBXT) = ∇2

vec X vec(X)T(BT⊗ A) vec X = B ⊗AT + BT⊗A

Page 681: Convex Optimization & Euclidean Distance Geometry

D.2. TABLES OF GRADIENTS AND DERIVATIVES 681

D.2.3 trace

∇x µ x = µI ∇X tr µX = ∇X µ trX = µI

∇x 1Tδ(x)−11 = ddxx−1 = −x−2 ∇X tr X−1 = −X−2T

∇x 1Tδ(x)−1y = −δ(x)−2y ∇X tr(X−1 Y ) = ∇X tr(Y X−1) = −X−TY TX−T

ddxxµ = µxµ−1 ∇X tr Xµ = µXµ−1 , X∈ SM

∇X tr Xj = jX(j−1)T

∇x(b − aTx)−1 = (b − aTx)−2a ∇X tr(

(B − AX)−1)

=(

(B − AX)−2A)T

∇x(b − aTx)µ = −µ(b − aTx)µ−1a

∇x xTy = ∇x yTx = y ∇X tr(XTY ) = ∇X tr(Y XT) = ∇X tr(Y TX) = ∇X tr(XY T) = Y

∇X tr(AXBXT) = ∇X tr(XBXTA) = ATXBT + AXB∇X tr(AXBX) = ∇X tr(XBXA) = ATXTBT+ BTXTAT

∇X tr(AXAXAX) = ∇X tr(XAXAXA) = 3(AXAXA)T

∇X tr(Y Xk) = ∇X tr(Xk Y ) =k−1∑

i=0

(

XiY Xk−1−i)T

∇X tr(Y TXXTY ) = ∇X tr(XTY Y TX) = 2 Y Y TX∇X tr(Y TXTXY ) = ∇X tr(XY Y TXT) = 2XY Y T

∇X tr(

(X + Y )T(X + Y ))

= 2(X + Y ) = ∇X‖X + Y ‖2F

∇X tr((X + Y )(X + Y )) = 2(X + Y )T

∇X tr(ATXB) = ∇X tr(XTABT) = ABT

∇X tr(ATX−1B) = ∇X tr(X−TABT) = −X−TABTX−T

∇X aTXb = ∇X tr(baTX) = ∇X tr(XbaT) = abT

∇X bTXTa = ∇X tr(XTabT) = ∇X tr(abTXT) = abT

∇X aTX−1b = ∇X tr(X−TabT) = −X−TabTX−T

∇X aTXµb = ...

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682 APPENDIX D. MATRIX CALCULUS

trace continued

ddt

tr g(X+ t Y ) = tr ddt

g(X+ t Y ) [200, p.491]

ddt

tr(X+ t Y ) = tr Y

ddt

trj(X+ t Y ) = j trj−1(X+ t Y ) tr Y

ddt

tr(X+ t Y )j = j tr((X+ t Y )j−1 Y ) (∀ j)

ddt

tr((X+ t Y )Y ) = tr Y 2

ddt

tr(

(X+ t Y )k Y)

= ddt

tr(Y (X+ t Y )k) = k tr(

(X+ t Y )k−1 Y 2)

, k∈0, 1, 2

ddt

tr(

(X+ t Y )k Y)

= ddt

tr(Y (X+ t Y )k) = trk−1∑

i=0

(X+ t Y )i Y (X+ t Y )k−1−i Y

ddt

tr((X+ t Y )−1 Y ) = − tr((X+ t Y )−1 Y (X+ t Y )−1 Y )ddt

tr(

BT(X+ t Y )−1A)

= − tr(

BT(X+ t Y )−1 Y (X+ t Y )−1A)

ddt

tr(

BT(X+ t Y )−TA)

= − tr(

BT(X+ t Y )−T Y T(X+ t Y )−TA)

ddt

tr(

BT(X+ t Y )−kA)

= ... , k>0ddt

tr(

BT(X+ t Y )µA)

= ... , −1 ≤ µ ≤ 1, X , Y ∈ SM+

d2

dt2tr

(

BT(X+ t Y )−1A)

= 2 tr(

BT(X+ t Y )−1 Y (X+ t Y )−1 Y (X+ t Y )−1A)

ddt

tr(

(X+ t Y )TA(X+ t Y ))

= tr(

Y TAX + XTAY + 2 t Y TAY)

d2

dt2tr

(

(X+ t Y )TA(X+ t Y ))

= 2 tr(

Y TAY)

ddt

tr(

(

(X+ t Y )TA(X+ t Y ))−1

)

=− tr(

(

(X+ t Y )TA(X+ t Y ))−1

(Y TAX + XTAY + 2 t Y TAY )(

(X+ t Y )TA(X+ t Y ))−1

)

ddt

tr((X+ t Y )A(X+ t Y )) = tr(YAX + XAY + 2 t YAY )d2

dt2tr((X+ t Y )A(X+ t Y )) = 2 tr(YAY )

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D.2. TABLES OF GRADIENTS AND DERIVATIVES 683

D.2.4 logarithmic determinant

x≻ 0, det X > 0 on some neighborhood of X , and det(X+ t Y )> 0 onsome open interval of t ; otherwise, log( ) would be discontinuous. [80, p.75]

ddx

log x = x−1 ∇X log det X = X−T

∇2X log det(X)kl =

∂X−T

∂Xkl

= −(

X−1ekeTl X−1

)T, confer (1780)(1827)

ddx

log x−1 = −x−1 ∇X log det X−1 = −X−T

ddx

log xµ = µx−1 ∇X log detµX = µX−T

∇X log det Xµ

= µX−T

∇X log det Xk = ∇X log detkX = kX−T

∇X log detµ(X+ t Y ) = µ(X+ t Y )−T

∇x log(aTx + b) = a 1aTx+b

∇X log det(AX+ B) = AT(AX+ B)−T

∇X log det(I ± ATXA) = ±A(I ± ATXA)−TAT

∇X log det(X+ t Y )k = ∇X log detk(X+ t Y ) = k(X+ t Y )−T

ddt

log det(X+ t Y ) = tr ((X+ t Y )−1 Y )

d2

dt2log det(X+ t Y ) = − tr ((X+ t Y )−1 Y (X+ t Y )−1 Y )

ddt

log det(X+ t Y )−1 = − tr ((X+ t Y )−1 Y )

d2

dt2log det(X+ t Y )−1 = tr ((X+ t Y )−1 Y (X+ t Y )−1 Y )

ddt

log det(δ(A(x + t y) + a)2 + µI)

= tr(

(δ(A(x + t y) + a)2 + µI)−12δ(A(x + t y) + a)δ(Ay))

Page 684: Convex Optimization & Euclidean Distance Geometry

684 APPENDIX D. MATRIX CALCULUS

D.2.5 determinant

∇X det X = ∇X det XT = det(X)X−T

∇X det X−1 = − det(X−1)X−T = − det(X)−1X−T

∇X detµX = µ detµ(X)X−T

∇X det Xµ

= µ det(Xµ)X−T

∇X det Xk = k detk−1(X)(

tr(X)I − XT)

, X∈ R2×2

∇X det Xk = ∇X detkX = k det(Xk)X−T = k detk(X)X−T

∇X detµ(X+ t Y ) = µ detµ(X+ t Y )(X+ t Y )−T

∇X det(X+ t Y )k = ∇X detk(X+ t Y ) = k detk(X+ t Y )(X+ t Y )−T

ddt

det(X+ t Y ) = det(X+ t Y ) tr((X+ t Y )−1 Y )

d2

dt2det(X+ t Y ) = det(X+ t Y )(tr2((X+ t Y )−1 Y ) − tr((X+ t Y )−1 Y (X+ t Y )−1 Y ))

ddt

det(X+ t Y )−1 = − det(X+ t Y )−1 tr((X+ t Y )−1 Y )

d2

dt2det(X+ t Y )−1 = det(X+ t Y )−1(tr2((X+ t Y )−1 Y ) + tr((X+ t Y )−1 Y (X+ t Y )−1 Y ))

ddt

detµ(X+ t Y ) = µ detµ(X+ t Y ) tr((X+ t Y )−1 Y )

D.2.6 logarithmic

Matrix logarithm.

ddt

log(X+ t Y )µ = µY (X+ t Y )−1 = µ(X+ t Y )−1Y , XY = Y X

ddt

log(I− t Y )µ = −µY (I− t Y )−1 = −µ(I− t Y )−1Y [200, p.493]

Page 685: Convex Optimization & Euclidean Distance Geometry

D.2. TABLES OF GRADIENTS AND DERIVATIVES 685

D.2.7 exponential

Matrix exponential. [75, §3.6, §4.5] [328, §5.4]

∇Xetr(Y TX) = ∇X det eY TX = etr(Y TX)Y (∀X,Y )

∇X tr eY X = eY TXTY T = Y TeXTY T

∇x1TeAx = ATeAx

∇x1Te|Ax| = ATδ(sgn(Ax))e|Ax| (Ax)i 6= 0

∇x log(1Tex) =1

1Texex

∇2x log(1Tex) =

1

1Tex

(

δ(ex) − 1

1TexexexT

)

∇x

k∏

i=1

x1k

i =1

k

(

k∏

i=1

x1k

i

)

1/x

∇2x

k∏

i=1

x1k

i = −1

k

(

k∏

i=1

x1k

i

)(

δ(x)−2 − 1

k(1/x)(1/x)T

)

ddt

etY = etY Y = Y etY

ddt

eX+ t Y = eX+ t Y Y = Y eX+ t Y , XY = Y X

d2

dt2eX+ t Y = eX+ t Y Y 2 = Y eX+ t Y Y = Y 2eX+ t Y , XY = Y X

d j

dt jetr(X+ t Y ) = etr(X+ t Y ) trj(Y )

Page 686: Convex Optimization & Euclidean Distance Geometry

686 APPENDIX D. MATRIX CALCULUS

Page 687: Convex Optimization & Euclidean Distance Geometry

Appendix E

Projection

For any A∈Rm×n, the pseudoinverse [199, §7.3, prob.9] [246, §6.12, prob.19][155, §5.5.4] [328, App.A]

A† , limt→0+

(ATA + t I )−1AT = limt→0+

AT(AAT + t I )−1 ∈ Rn×m (1832)

is a unique matrix from the optimal solution set to minimizeX

‖AX − I‖2F

(§3.7.0.0.2). For any t > 0

I − A(ATA + t I )−1AT = t(AAT + t I )−1 (1833)

Equivalently, pseudoinverse A† is that unique matrix satisfying theMoore-Penrose conditions : [201, §1.3] [370]

1. AA†A = A 3. (AA†)T = AA†

2. A†AA† = A† 4. (A†A)T = A†A

which are necessary and sufficient to establish the pseudoinverse whoseprincipal action is to injectively map R(A) onto R(AT) (Figure 159).Conditions 1 and 3 are necessary and sufficient for AA† to be the orthogonalprojector on R(A) , while conditions 2 and 4 hold iff A†A is the orthogonalprojector on R(AT).

Range and nullspace of the pseudoinverse [263] [325, §III.1, exer.1]

R(A†) = R(AT) , R(A†T) = R(A) (1834)

N (A†) = N (AT) , N (A†T) = N (A) (1835)

can be derived by singular value decomposition (§A.6).© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

687

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688 APPENDIX E. PROJECTION

x

Rn p = AA†b

x = A†p

x = A†bR(AT)

R(A)

N (A) N (AT)

0 = A†(b − p)b − p

b

p

Rm

b

Ax = p

00x

Figure 159: (confer Figure 16) Pseudoinverse A†∈ Rn×m action: [328, p.449]Component of vector b in N (AT) maps to origin, while component of b inR(A) maps to rowspace R(AT). For any A∈Rm×n, inversion is bijective∀ p ∈R(A).

The following relations reliably hold without qualification:

a. AT† = A†T

b. A†† = A

c. (AAT)† = A†TA†

d. (ATA)† = A†A†T

e. (AA†)† = AA†

f . (A†A)† = A†A

Yet for arbitrary A,B it is generally true that (AB)† 6= B†A† :

E.0.0.0.1 Theorem. Pseudoinverse of product. [165] [55] [237, exer.7.23]For A∈Rm×n and B∈Rn×k

(AB)† = B†A† (1836)

if and only if

R(ATAB) ⊆ R(B) and R(BBTAT) ⊆ R(AT) (1837)

Page 689: Convex Optimization & Euclidean Distance Geometry

689

UT = U † for orthonormal (including the orthogonal) matrices U . So, fororthonormal matrices U,Q and arbitrary A

(UAQT)† = QA†UT (1838)

E.0.0.0.2 Exercise. Kronecker inverse.Prove:

(A ⊗ B)† = A† ⊗ B† (1839)

H

E.0.1 Logical deductions

When A is invertible, A† = A−1 ; so A†A = AA† = I . Otherwise, forA∈Rm×n [148, §5.3.3.1] [237, §7] [292]

g. A†A = I , A† = (ATA)−1AT , rank A = n

h. AA† = I , A† = AT(AAT)−1 , rank A = m

i . A†Aω = ω , ω ∈ R(AT)

j. AA†υ = υ , υ ∈ R(A)

k. A†A = AA† , A normal

l. Ak† = A†k , A normal, k an integer

Equivalent to the corresponding Moore-Penrose condition:

1. AT = ATAA† or AT = A†AAT

2. A†T = A†TA†A or A†T = AA†A†T

When A is symmetric, A† is symmetric and (§A.6)

A º 0 ⇔ A† º 0 (1840)

E.0.1.0.1 Example. Solution to classical linear equation Ax = b .In §2.5.1.1, the solution set to matrix equation Ax = b was representedas an intersection of hyperplanes. Regardless of rank of A or its shape(fat or skinny), interpretation as a hyperplane intersection describing apossibly empty affine set generally holds. If matrix A is rank deficient orfat, there is an infinity of solutions x when b∈R(A). A unique solutionoccurs when the hyperplanes intersect at a single point.

Page 690: Convex Optimization & Euclidean Distance Geometry

690 APPENDIX E. PROJECTION

Given arbitrary matrix A (of any rank and dimension) and vector b notnecessarily in R(A) , we wish to find a best solution x⋆ to

Ax ≈ b (1841)

in a Euclidean sense by solving an algebraic expression for orthogonalprojection of b on R(A)

minimizex

‖Ax − b‖2 (1842)

Necessary and sufficient condition for optimal solution to this unconstrainedoptimization is the so-called normal equation that results from zeroing theconvex objective’s gradient: (§D.2.1)

ATAx = ATb (1843)

normal because error vector b−Ax is perpendicular to R(A) ; id est,AT(b−Ax)=0. Given any matrix A and any vector b , the normal equationis solvable exactly; always so, because R(ATA)=R(AT) and ATb∈R(AT).

When A is skinny-or-square full-rank, normal equation (1843) can besolved exactly by inversion:

x⋆ = (ATA)−1ATb ≡ A†b (1844)

For matrix A of arbitrary rank and shape, on the other hand, ATA mightnot be invertible. Yet the normal equation can always be solved exactly by:(1832)

x⋆ = limt→0+

(ATA + t I )−1ATb = A†b (1845)

invertible for any positive value of t by (1427). The exact inversion (1844)and this pseudoinverse solution (1845) each solve a limited regularization

limt→0+

minimizex

‖Ax − b‖2 + t ‖x‖2 ≡ limt→0+

minimizex

[

A√t I

]

x −[

b0

]∥

2

(1846)

simultaneously providing least squares solution to (1842) and the classicalleast norm solutionE.1 [328, App.A.4] (confer §3.3.0.0.1)

arg minimizex

‖x‖2

subject to Ax = AA†b(1847)

E.1This means: optimal solutions of lesser norm than the so-called least norm solution(1847) can be obtained (at expense of approximation Ax ≈ b hence, of perpendicularity)by ignoring the limiting operation and introducing finite positive values of t into (1846).

Page 691: Convex Optimization & Euclidean Distance Geometry

E.1. IDEMPOTENT MATRICES 691

where AA†b is the orthogonal projection of vector b on R(A). Least normsolution can be interpreted as orthogonal projection of the origin 0 on affinesubset A= x |Ax=AA†b ; (§E.5.0.0.5, §E.5.0.0.6)

arg minimizex

‖x − 0‖2

subject to x ∈ A (1848)

equivalently, maximization of the Euclidean ball until it kisses A ; rather,arg dist(0,A). 2

E.1 Idempotent matrices

Projection matrices are square and defined by idempotence, P 2 =P ;[328, §2.6] [201, §1.3] equivalent to the condition, P be diagonalizable[199, §3.3, prob.3] with eigenvalues φi ∈ 0, 1. [390, §4.1, thm.4.1]Idempotent matrices are not necessarily symmetric. The transpose of anidempotent matrix remains idempotent; PTPT = PT. Solely exceptingP = I , all projection matrices are neither orthogonal (§B.5) or invertible.[328, §3.4] The collection of all projection matrices of particular dimensiondoes not form a convex set.

Suppose we wish to project nonorthogonally (obliquely ) on the rangeof any particular matrix A∈Rm×n. All idempotent matrices projectingnonorthogonally on R(A) may be expressed:

P = A(A† + BZT) ∈ Rm×m (1849)

where R(P )=R(A) ,E.2 B∈Rn×k for k∈1 . . . m is otherwise arbitrary,and Z∈ Rm×k is any matrix whose range is in N (AT) ; id est,

ATZ = A†Z = 0 (1850)

Evidently, the collection of nonorthogonal projectors projecting on R(A) isan affine subset

Pk =

A(A† + BZT) | B∈Rn×k

(1851)

E.2Proof. R(P )⊆R(A) is obvious [328, §3.6]. By (139) and (140),

R(A† + BZT) = (A† + BZT)y | y ∈ Rm⊇ (A† + BZT)y | y ∈ R(A) = R(AT)

R(P ) = A(A† + BZT)y | y ∈ Rm⊇ A(A† + BZT)y | (A† + BZT)y ∈ R(AT) = R(A) ¨

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692 APPENDIX E. PROJECTION

When matrix A in (1849) is skinny full-rank (A†A = I ) or has orthonormalcolumns (ATA = I ), either property leads to a biorthogonal characterizationof nonorthogonal projection:

E.1.1 Biorthogonal characterization

Any nonorthogonal projector P 2 =P ∈Rm×m projecting on nontrivial R(U)can be defined by a biorthogonality condition QTU = I ; the biorthogonaldecomposition of P being (confer (1849))

P = UQT , QTU = I (1852)

whereE.3 (§B.1.1.1)

R(P )=R(U) , N (P )=N (QT) (1853)

and where generally (confer (1878))E.4

PT 6= P , P † 6= P , ‖P‖2 6= 1 , P ² 0 (1854)

and P is not nonexpansive (1879).

(⇐) To verify assertion (1852) we observe: because idempotent matricesare diagonalizable (§A.5), [199, §3.3, prob.3] they must have the form (1513)

P = SΦS−1 =m

i=1

φi siwTi =

k ≤m∑

i=1

siwTi (1855)

that is a sum of k = rankP independent projector dyads (idempotentdyads, §B.1.1, §E.6.2.1) where φi ∈ 0, 1 are the eigenvalues of P[390, §4.1, thm.4.1] in diagonal matrix Φ∈Rm×m arranged in nonincreasing

E.3Proof. Obviously, R(P ) ⊆ R(U). Because QTU = I

R(P ) = UQTx | x ∈ Rm⊇ UQTUy | y ∈ Rk = R(U) ¨

E.4Orthonormal decomposition (1875) (confer §E.3.4) is a special case of biorthogonaldecomposition (1852) characterized by (1878). So, these characteristics (1854) are notnecessary conditions for biorthogonality.

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E.1. IDEMPOTENT MATRICES 693

order, and where si , wi∈Rm are the right- and left-eigenvectors of P ,respectively, which are independent and real.E.5 Therefore

U , S(: , 1:k) = [ s1 · · · sk ] ∈ Rm×k (1856)

is the full-rank matrix S∈ Rm×m having m− k columns truncated(corresponding to 0 eigenvalues), while

QT , S−1(1 :k , :) =

wT1...

wTk

∈ Rk×m (1857)

is matrix S−1 having the corresponding m− k rows truncated. By the0 eigenvalues theorem (§A.7.3.0.1), R(U)=R(P ) , R(Q)=R(PT) , and

R(P ) = span si | φi = 1 ∀ iN (P ) = span si | φi = 0 ∀ iR(PT) = span wi | φi = 1 ∀ iN (PT) = span wi | φi = 0 ∀ i

(1858)

Thus biorthogonality QTU = I is a necessary condition for idempotence, andso the collection of nonorthogonal projectors projecting on R(U) is the affinesubset Pk =UQT

k where Qk = Q | QTU = I , Q∈ Rm×k.(⇒) Biorthogonality is a sufficient condition for idempotence;

P 2 =k

i=1

siwTi

k∑

j=1

sjwTj = P (1859)

id est, if the cross-products are annihilated, then P 2 =P . ¨

Nonorthogonal projection of x on R(P ) has expression like abiorthogonal expansion,

Px = UQTx =k

i=1

wTi x si (1860)

When the domain is restricted to range of P , say x=Uξ for ξ∈Rk, thenx = Px = UQTUξ = Uξ and expansion is unique because the eigenvectors

E.5Eigenvectors of a real matrix corresponding to real eigenvalues must be real.(§A.5.0.0.1)

Page 694: Convex Optimization & Euclidean Distance Geometry

694 APPENDIX E. PROJECTION

x

Px

cone(Q)

cone(U)

T

T ⊥ R(Q)

Figure 160: Nonorthogonal projection of x∈R3 on R(U)= R2 underbiorthogonality condition; id est, Px=UQTx such that QTU = I . Anypoint along imaginary line T connecting x to Px will be projectednonorthogonally on Px with respect to horizontal plane constitutingR2 = aff cone(U) in this example. Extreme directions of cone(U) correspondto two columns of U ; likewise for cone(Q). For purpose of illustration, wetruncate each conic hull by truncating coefficients of conic combination atunity. Conic hull cone(Q) is headed upward at an angle, out of plane ofpage. Nonorthogonal projection would fail were N (QT) in R(U) (were Tparallel to a line in R(U)).

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E.1. IDEMPOTENT MATRICES 695

are linearly independent. Otherwise, any component of x in N (P )=N (QT)will be annihilated. Direction of nonorthogonal projection is orthogonal toR(Q) ⇔ QTU = I ; id est, for Px∈R(U)

Px − x ⊥ R(Q) in Rm (1861)

E.1.1.0.1 Example. Illustration of nonorthogonal projector.Figure 160 shows cone(U) , the conic hull of the columns of

U =

1 1−0.5 0.3

0 0

(1862)

from nonorthogonal projector P = UQT. Matrix U has a limitless numberof left inverses because N (UT) is nontrivial. Similarly depicted is left inverseQT from (1849)

Q = U †T + ZBT =

0.3750 0.6250−1.2500 1.2500

0 0

+

001

[ 0.5 0.5 ]

=

0.3750 0.6250−1.2500 1.2500

0.5000 0.5000

(1863)

where Z∈N (UT) and matrix B is selected arbitrarily; id est, QTU = Ibecause U is full-rank.

Direction of projection on R(U) is orthogonal to R(Q). Any point alongline T in the figure, for example, will have the same projection. Were matrixZ instead equal to 0, then cone(Q) would become the relative dual tocone(U) (sharing the same affine hull; §2.13.8, confer Figure 55a). In thatcase, projection Px = UU †x of x on R(U) becomes orthogonal projection(and unique minimum-distance). 2

E.1.2 Idempotence summary

Nonorthogonal subspace-projector P is a (convex) linear operator defined byidempotence or biorthogonal decomposition (1852), but characterized not bysymmetry or positive semidefiniteness or nonexpansivity (1879).

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696 APPENDIX E. PROJECTION

E.2 I−P , Projection on algebraic complement

It follows from the diagonalizability of idempotent matrices that I − P mustalso be a projection matrix because it too is idempotent, and because it maybe expressed

I − P = S(I − Φ)S−1 =m

i=1

(1 − φi)siwTi (1864)

where (1 − φi) ∈ 1, 0 are the eigenvalues of I − P (1428) whoseeigenvectors si , wi are identical to those of P in (1855). A consequence ofthat complementary relationship of eigenvalues is the fact, [340, §2] [335, §2]for subspace projector P = P 2∈ Rm×m

R(P ) = span si | φi = 1 ∀ i = span si | (1 − φi) = 0 ∀ i = N (I − P )N (P ) = span si | φi = 0 ∀ i = span si | (1 − φi) = 1 ∀ i = R(I − P )R(PT) = span wi | φi = 1 ∀ i = span wi | (1 − φi) = 0 ∀ i = N (I − PT)N (PT) = span wi | φi = 0 ∀ i = span wi | (1 − φi) = 1 ∀ i = R(I − PT)

(1865)

that is easy to see from (1855) and (1864). Idempotent I−P thereforeprojects vectors on its range, N (P ). Because all eigenvectors of a realidempotent matrix are real and independent, the algebraic complement ofR(P ) [223, §3.3] is equivalent to N (P ) ;E.6 id est,

R(P )⊕N (P ) = R(PT)⊕N (PT) = R(PT)⊕N (P ) = R(P )⊕N (PT) = Rm

(1866)

because R(P ) ⊕ R(I−P )= Rm. For idempotent P ∈ Rm×m, consequently,

rank P + rank(I − P ) = m (1867)

E.2.0.0.1 Theorem. Rank/Trace. [390, §4.1, prob.9] (confer (1883))

P 2 = P⇔

rank P = tr P and rank(I − P ) = tr(I − P )

(1868)

⋄E.6The same phenomenon occurs with symmetric (nonidempotent) matrices, for example.

When the summands in A ⊕ B = Rm are orthogonal vector spaces, the algebraiccomplement is the orthogonal complement.

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E.3. SYMMETRIC IDEMPOTENT MATRICES 697

E.2.1 Universal projector characteristic

Although projection is not necessarily orthogonal and R(P ) 6⊥ R(I − P ) ingeneral, still for any projector P and any x∈Rm

Px + (I − P )x = x (1869)

must hold where R(I − P ) = N (P ) is the algebraic complement of R(P ).The algebraic complement of closed convex cone K , for example, is thenegative dual cone −K∗

. (1985)

E.3 Symmetric idempotent matrices

When idempotent matrix P is symmetric, P is an orthogonal projector. Inother words, the direction of projection of point x∈Rm on subspace R(P )is orthogonal to R(P ) ; id est, for P 2 =P ∈ Sm and projection Px∈R(P )

Px − x ⊥ R(P ) in Rm (1870)

(confer (1861)) Perpendicularity is a necessary and sufficient condition fororthogonal projection on a subspace. [106, §4.9]

A condition equivalent to (1870) is: Norm of direction x−Px is theinfimum over all nonorthogonal projections of x on R(P ) ; [246, §3.3] forP 2 =P ∈ Sm, R(P )=R(A) , matrices A ,B , Z and positive integer k asdefined for (1849), and given x∈Rm

‖x − Px‖2 = ‖x − AA†x‖2 = infB∈R

n×k‖x − A(A† + BZT)x‖2 = dist(x , R(P ))

(1871)

The infimum is attained for R(B)⊆N (A) over any affine subset ofnonorthogonal projectors (1851) indexed by k .

Proof is straightforward: The vector 2-norm is a convex function. Settinggradient of the norm-square to 0, applying §D.2,

(

ATABZT − AT(I − AA†))

xxTA = 0⇔

ATABZTxxTA = 0

(1872)

because AT = ATAA†. Projector P =AA† is therefore unique; theminimum-distance projector is the orthogonal projector, and vice versa.

¨

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698 APPENDIX E. PROJECTION

We get P =AA† so this projection matrix must be symmetric. Then forany matrix A∈Rm×n, symmetric idempotent P projects a given vector xin Rm orthogonally on R(A). Under either condition (1870) or (1871), theprojection Px is unique minimum-distance; for subspaces, perpendicularityand minimum-distance conditions are equivalent.

E.3.1 Four subspaces

We summarize the orthogonal projectors projecting on the four fundamentalsubspaces: for A∈Rm×n

A†A : Rn on R(A†A) = R(AT)AA† : Rm on R(AA†) = R(A)I−A†A : Rn on R(I−A†A) = N (A)I−AA† : Rm on R(I−AA†) = N (AT)

(1873)

For completeness:E.7 (1865)

N (A†A) = N (A)N (AA†) = N (AT)N (I−A†A) = R(AT)N (I−AA†) = R(A)

(1874)

E.3.2 Orthogonal characterization

Any symmetric projector P 2 =P ∈ Sm projecting on nontrivial R(Q) canbe defined by the orthonormality condition QTQ = I . When skinny matrixQ∈ Rm×k has orthonormal columns, then Q† = QT by the Moore-Penroseconditions. Hence, any P having an orthonormal decomposition (§E.3.4)

P = QQT , QTQ = I (1875)

where [328, §3.3] (1573)

R(P ) = R(Q) , N (P ) = N (QT) (1876)

E.7Proof is by singular value decomposition (§A.6.2): N (A†A)⊆N (A) is obvious.Conversely, suppose A†Ax=0. Then xTA†Ax=xTQQTx=‖QTx‖2 =0 where A=UΣQT

is the subcompact singular value decomposition. Because R(Q)=R(AT) , then x∈N (A)which implies N (A†A)⊇N (A). ∴ N (A†A)=N (A). ¨

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E.3. SYMMETRIC IDEMPOTENT MATRICES 699

is an orthogonal projector projecting on R(Q) having, for Px∈R(Q)(confer (1861))

Px − x ⊥ R(Q) in Rm (1877)

From (1875), orthogonal projector P is obviously positive semidefinite(§A.3.1.0.6); necessarily,

PT = P , P † = P , ‖P‖2 = 1 , P º 0 (1878)

and ‖Px‖= ‖QQTx‖= ‖QTx‖ because ‖Qy‖= ‖y‖ ∀ y∈Rk. All orthogonalprojectors are therefore nonexpansive because

〈Px , x〉 = ‖Px‖ = ‖QTx‖ ≤ ‖x‖ ∀x∈Rm (1879)

the Bessel inequality, [106] [223] with equality when x∈R(Q).From the diagonalization of idempotent matrices (1855) on page 692

P = SΦST =m

i=1

φi sisTi =

k ≤m∑

i=1

sisTi (1880)

orthogonal projection of point x on R(P ) has expression like an orthogonalexpansion [106, §4.10]

Px = QQTx =k

i=1

sTi x si (1881)

whereQ = S(: , 1:k) = [ s1 · · · sk ] ∈ Rm×k (1882)

and where the si [sic] are orthonormal eigenvectors of symmetricidempotent P . When the domain is restricted to range of P , say x=Qξ forξ∈Rk, then x = Px = QQTQξ = Qξ and expansion is unique. Otherwise,any component of x in N (QT) will be annihilated.

E.3.2.0.1 Theorem. Symmetric rank/trace. (confer (1868) (1432))

PT = P , P 2 = P⇔

rank P = tr P = ‖P‖2F and rank(I − P ) = tr(I − P ) = ‖I − P‖2

F

(1883)⋄

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700 APPENDIX E. PROJECTION

Proof. We take as given Theorem E.2.0.0.1 establishing idempotence.We have left only to show trP =‖P‖2

F ⇒ PT = P , established in [390, §7.1].¨

E.3.3 Summary, symmetric idempotent

(confer §E.1.2) Orthogonal projector P is a (convex) linear operatordefined [196, §A.3.1] by idempotence and symmetry, and characterized bypositive semidefiniteness and nonexpansivity. The algebraic complement(§E.2) to R(P ) becomes the orthogonal complement R(I − P ) ; id est,R(P )⊥ R(I − P ).

E.3.4 Orthonormal decomposition

When Z =0 in the general nonorthogonal projector A(A† + BZT) (1849),an orthogonal projector results (for any matrix A) characterized principallyby idempotence and symmetry. Any real orthogonal projector may, infact, be represented by an orthonormal decomposition such as (1875).[201, §1, prob.42]

To verify that assertion for the four fundamental subspaces (1873),we need only to express A by subcompact singular value decomposition(§A.6.2): From pseudoinverse (1542) of A = UΣQT∈ Rm×n

AA† = UΣΣ†UT = UUT, A†A = QΣ†ΣQT = QQT

I − AA† = I − UUT = U⊥U⊥T, I − A†A = I − QQT = Q⊥Q⊥T

(1884)

where U⊥∈ Rm×m−rank A holds columnar an orthonormal basis for theorthogonal complement of R(U) , and likewise for Q⊥∈ Rn×n−rank A.Existence of an orthonormal decomposition is sufficient to establishidempotence and symmetry of an orthogonal projector (1875). ¨

E.3.5 Unifying trait of all projectors: direction

Relation (1884) shows: orthogonal projectors simultaneously possessa biorthogonal decomposition (confer §E.1.1) (for example, AA† forskinny-or-square A full-rank) and an orthonormal decomposition (UUT

whence Px = UUTx).

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E.3. SYMMETRIC IDEMPOTENT MATRICES 701

E.3.5.1 orthogonal projector, orthonormal decomposition

Consider orthogonal expansion of x∈R(U) :

x = UUTx =n

i=1

uiuTi x (1885)

a sum of one-dimensional orthogonal projections (§E.6.3), where

U , [ u1 · · · un ] and UTU = I (1886)

and where the subspace projector has two expressions, (1884)

AA† , UUT (1887)

where A ∈ Rm×n has rank n . The direction of projection of x on uj forsome j∈1 . . . n , for example, is orthogonal to uj but parallel to a vectorin the span of all the remaining vectors constituting the columns of U ;

uTj (uju

Tj x − x) = 0

ujuTj x − x = uju

Tj x − UUTx ∈ R(ui | i=1 . . . n , i 6=j)

(1888)

E.3.5.2 orthogonal projector, biorthogonal decomposition

We get a similar result for the biorthogonal expansion of x∈R(A). Define

A , [ a1 a2 · · · an ] ∈ Rm×n (1889)

and the rows of the pseudoinverseE.8

A† ,

a∗T1

a∗T2...

a∗Tn

∈ Rn×m (1890)

under biorthogonality condition A†A=I . In biorthogonal expansion(§2.13.8)

x = AA†x =n

i=1

aia∗Ti x (1891)

E.8Notation * in this context connotes extreme direction of a dual cone; e.g., (385) orExample E.5.0.0.3.

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702 APPENDIX E. PROJECTION

the direction of projection of x on aj for some particular j∈1 . . . n , forexample, is orthogonal to a∗

j and parallel to a vector in the span of all theremaining vectors constituting the columns of A ;

a∗Tj (aj a

∗Tj x − x) = 0

aj a∗Tj x − x = aj a

∗Tj x − AA†x ∈ R(ai | i=1 . . . n , i 6=j)

(1892)

E.3.5.3 nonorthogonal projector, biorthogonal decomposition

Because the result in §E.3.5.2 is independent of matrix symmetryAA†=(AA†)T, we must get the same result for any nonorthogonal projectorcharacterized by a biorthogonality condition; namely, for nonorthogonalprojector P = UQT (1852) under biorthogonality condition QTU = I , inthe biorthogonal expansion of x∈R(U)

x = UQTx =k

i=1

uiqTi x (1893)

whereU , [ u1 · · · uk ] ∈ Rm×k

QT ,

qT1...

qTk

∈ Rk×m(1894)

the direction of projection of x on uj is orthogonal to qj and parallel to avector in the span of the remaining ui :

qTj (uj qT

j x − x) = 0

uj qTj x − x = uj qT

j x − UQTx ∈ R(ui | i=1 . . . k , i 6=j)(1895)

E.4 Algebra of projection on affine subsets

Let PAx denote projection of x on affine subset A,R + α where R is asubspace and α ∈A . Then, because R is parallel to A , it holds:

PAx = PR+αx = (I − PR)(α) + PRx

= PR(x − α) + α(1896)

Subspace projector PR is a linear operator (PA is not), and PR(x + y)=PRxwhenever y⊥R and PR is an orthogonal projector.

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E.5. PROJECTION EXAMPLES 703

E.4.0.0.1 Theorem. Orthogonal projection on affine subset. [106, §9.26]Let A=R + α be an affine subset where α ∈A , and let R⊥ be theorthogonal complement of subspace R . Then PAx is the orthogonalprojection of x∈Rn on A if and only if

PAx ∈ A , 〈PAx − x , a − α〉 = 0 ∀ a ∈ A (1897)

or if and only if

PAx ∈ A , PAx − x ∈ R⊥ (1898)

E.5 Projection examples

E.5.0.0.1 Example. Orthogonal projection on orthogonal basis.Orthogonal projection on a subspace can instead be accomplished byorthogonally projecting on the individual members of an orthogonal basis forthat subspace. Suppose, for example, matrix A∈Rm×n holds an orthonormalbasis for R(A) in its columns; A , [ a1 a2 · · · an ] . Then orthogonalprojection of vector x∈Rn on R(A) is a sum of one-dimensional orthogonalprojections

Px = AA†x = A(ATA)−1ATx = AATx =n

i=1

aiaTi x (1899)

where each symmetric dyad aiaTi is an orthogonal projector projecting on

R(ai). (§E.6.3) Because ‖x − Px‖ is minimized by orthogonal projection,Px is considered to be the best approximation (in the Euclidean sense) tox from the set R(A). [106, §4.9] 2

E.5.0.0.2 Example. Orthogonal projection on span of nonorthogonal basis.Orthogonal projection on a subspace can also be accomplished by projectingnonorthogonally on the individual members of any nonorthogonal basis forthat subspace. This interpretation is in fact the principal application of thepseudoinverse we discussed. Now suppose matrix A holds a nonorthogonalbasis for R(A) in its columns,

A = [ a1 a2 · · · an ] ∈ Rm×n (1889)

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704 APPENDIX E. PROJECTION

and define the rows a∗Ti of its pseudoinverse A† as in (1890). Then

orthogonal projection of vector x∈Rn on R(A) is a sum of one-dimensionalnonorthogonal projections

Px = AA†x =n

i=1

aia∗Ti x (1900)

where each nonsymmetric dyad aia∗Ti is a nonorthogonal projector projecting

on R(ai) , (§E.6.1) idempotent because of biorthogonality condition A†A = I .

The projection Px is regarded as the best approximation to x from theset R(A) , as it was in Example E.5.0.0.1. 2

E.5.0.0.3 Example. Biorthogonal expansion as nonorthogonal projection.Biorthogonal expansion can be viewed as a sum of components, each anonorthogonal projection on the range of an extreme direction of a pointedpolyhedral cone K ; e.g., Figure 161.

Suppose matrix A∈Rm×n holds a nonorthogonal basis for R(A) inits columns as in (1889), and the rows of pseudoinverse A† are definedas in (1890). Assuming the most general biorthogonality condition(A† + BZT)A = I with BZT defined as for (1849), then biorthogonalexpansion of vector x is a sum of one-dimensional nonorthogonal projections;for x∈R(A)

x = A(A† + BZT)x = AA†x =n

i=1

aia∗Ti x (1901)

where each dyad aia∗Ti is a nonorthogonal projector projecting on R(ai).

(§E.6.1) The extreme directions of K=cone(A) are a1 , . . . , an the linearlyindependent columns of A , while the extreme directions a∗

1 , . . . , a∗n

of relative dual cone K∗∩aff K=cone(A†T) (§2.13.9.4) correspond to thelinearly independent (§B.1.1.1) rows of A†. Directions of nonorthogonalprojection are determined by the pseudoinverse; id est, direction ofprojection aia

∗Ti x−x on R(ai) is orthogonal to a∗

i .E.9

E.9This remains true in high dimension although only a little more difficult to visualizein R3 ; confer , Figure 63.

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E.5. PROJECTION EXAMPLES 705

x

z

y

0

a1 ⊥ a∗2

a2 ⊥ a∗1

x = y + z = Pa1x + Pa2x

a1

a2

a∗1

a∗2

K

K∗

K∗

Figure 161: (confer Figure 62) Biorthogonal expansion of point x∈ aff K isfound by projecting x nonorthogonally on extreme directions of polyhedralcone K⊂R2. (Dotted lines of projection bound this translated negated cone.)Direction of projection on extreme direction a1 is orthogonal to extremedirection a∗

1 of dual cone K∗and parallel to a2 (§E.3.5); similarly, direction

of projection on a2 is orthogonal to a∗2 and parallel to a1 . Point x is

sum of nonorthogonal projections: x on R(a1) and x on R(a2). Expansionis unique because extreme directions of K are linearly independent. Werea1 orthogonal to a2 , then K would be identical to K∗

and nonorthogonalprojections would become orthogonal.

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706 APPENDIX E. PROJECTION

Because the extreme directions of this cone K are linearly independent,the component projections are unique in the sense: there is only one linear combination of extreme directions of K that

yields a particular point x∈R(A) whenever

R(A) = aff K = R(a1) ⊕ R(a2) ⊕ . . . ⊕ R(an) (1902)

2

E.5.0.0.4 Example. Nonorthogonal projection on elementary matrix.Suppose PY is a linear nonorthogonal projector projecting on subspace Y ,and suppose the range of a vector u is linearly independent of Y ; id est,for some other subspace M containing Y suppose

M = R(u) ⊕ Y (1903)

Assuming PMx = Pux + PYx holds, then it follows for vector x∈M

Pux = x − PYx , PYx = x − Pux (1904)

nonorthogonal projection of x on R(u) can be determined fromnonorthogonal projection of x on Y , and vice versa.

Such a scenario is realizable were there some arbitrary basis for Ypopulating a full-rank skinny-or-square matrix A

A , [ basisY u ] ∈ Rn+1 (1905)

Then PM =AA† fulfills the requirements, with Pu =A(: , n + 1)A†(n + 1 , :)and PY =A(: , 1 : n)A†(1 : n , :). Observe, PM is an orthogonal projectorwhereas PY and Pu are nonorthogonal projectors.

Now suppose, for example, PY is an elementary matrix (§B.3); inparticular,

PY = I − e11T =

[

0√

2VN]

∈ RN×N (1906)

where Y=N (1T). We have M= RN , A = [√

2VN e1 ] , and u = e1 .Thus Pu = e11

T is a nonorthogonal projector projecting on R(u) in adirection parallel to a vector in Y (§E.3.5), and PYx = x − e11

Tx is anonorthogonal projection of x on Y in a direction parallel to u . 2

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E.5.0.0.5 Example. Projecting origin on a hyperplane.(confer §2.4.2.0.2) Given the hyperplane representation having b∈R andnonzero normal a∈Rm

∂H = y | aTy = b ⊂ Rm (113)

orthogonal projection of the origin P0 on that hyperplane is the uniqueoptimal solution to a minimization problem: (1871)

‖P0 − 0‖2 = infy∈∂H

‖y − 0‖2

= infξ∈R

m−1‖Zξ + x‖2

(1907)

where x is any solution to aTy=b , and where the columns of Z∈ Rm×m−1

constitute a basis for N (aT) so that y = Zξ + x ∈ ∂H for all ξ∈Rm−1.The infimum can be found by setting the gradient (with respect to ξ) of

the strictly convex norm-square to 0. We find the minimizing argument

ξ⋆ = −(ZTZ)−1ZTx (1908)

soy⋆ =

(

I − Z(ZTZ)−1ZT)

x (1909)

and from (1873)

P0 = y⋆ = a(aTa)−1aTx =a

‖a‖aT

‖a‖ x =a

‖a‖2aTx , A†Ax = a

b

‖a‖2(1910)

In words, any point x in the hyperplane ∂H projected on its normal a(confer (1934)) yields that point y⋆ in the hyperplane closest to the origin.

2

E.5.0.0.6 Example. Projection on affine subset.The technique of Example E.5.0.0.5 is extensible. Given an intersection ofhyperplanes

A = y | Ay = b ⊂ Rm (1911)

where each row of A∈Rm×n is nonzero and b∈R(A) , then the orthogonalprojection Px of any point x∈Rn on A is the solution to a minimizationproblem:

‖Px − x‖2 = infy∈A

‖y − x‖2

= infξ∈R

n−rank A‖Zξ + yp − x‖2

(1912)

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708 APPENDIX E. PROJECTION

where yp is any solution to Ay = b , and where the columns ofZ∈ Rn×n−rank A constitute a basis for N (A) so that y = Zξ + yp ∈ A forall ξ∈Rn−rank A.

The infimum is found by setting the gradient of the strictly convexnorm-square to 0. The minimizing argument is

ξ⋆ = −(ZTZ)−1ZT(yp − x) (1913)

soy⋆ =

(

I − Z(ZTZ)−1ZT)

(yp − x) + x (1914)

and from (1873),

Px = y⋆ = x − A†(Ax − b)

= (I − A†A)x + A†Ayp

(1915)

which is a projection of x on N (A) then translated perpendicularly withrespect to the nullspace until it meets the affine subset A . 2

E.5.0.0.7 Example. Projection on affine subset, vertex-description.Suppose now we instead describe the affine subset A in terms of some givenminimal set of generators arranged columnar in X∈ Rn×N (76); id est,

A = aff X = Xa | aT1=1 ⊆ Rn (78)

Here minimal set means XVN = [ x2−x1 x3−x1 · · · xN−x1 ]/√

2 (936) isfull-rank (§2.4.2.2) where VN ∈RN×N−1 is the Schoenberg auxiliary matrix(§B.4.2). Then the orthogonal projection Px of any point x∈Rn on A isthe solution to a minimization problem:

‖Px − x‖2 = infaT1=1

‖Xa − x‖2

= infξ∈RN−1

‖X(VN ξ + ap) − x‖2(1916)

where ap is any solution to aT1=1. We find the minimizing argument

ξ⋆ = −(V TNXTXVN )−1V T

NXT(Xap − x) (1917)

and so the orthogonal projection is [202, §3]

Px = Xa⋆ = (I − XVN (XVN )†)Xap + XVN (XVN )†x (1918)

a projection of point x on R(XVN ) then translated perpendicularly withrespect to that range until it meets the affine subset A . 2

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E.6. VECTORIZATION INTERPRETATION, 709

E.5.0.0.8 Example. Projecting on hyperplane, halfspace, slab.Given the hyperplane representation having b ∈ R and nonzero normala∈Rm

∂H = y | aTy = b ⊂ Rm (113)

the orthogonal projection of any point x∈Rm on that hyperplane is

Px = x − a(aTa)−1(aTx − b) (1919)

Orthogonal projection of x on the halfspace parametrized by b ∈ R andnonzero normal a∈Rm

H− = y | aTy ≤ b ⊂ Rm (105)

is the pointPx = x − a(aTa)−1max0 , aTx − b (1920)

Orthogonal projection of x on the convex slab (Figure 11), for c < b

B , y | c ≤ aTy ≤ b ⊂ Rm (1921)

is the point [144, §5.1]

Px = x − a(aTa)−1(

max0 , aTx − b − max0 , c − aTx)

(1922)

2

E.6 Vectorization interpretation,

projection on a matrix

E.6.1 Nonorthogonal projection on a vector

Nonorthogonal projection of vector x on the range of vector y isaccomplished using a normalized dyad P0 (§B.1); videlicet,

〈z, x〉〈z, y〉 y =

zTx

zTyy =

yzT

zTyx , P0 x (1923)

where 〈z, x〉/〈z, y〉 is the coefficient of projection on y . Because P 20 =P0

and R(P0)=R(y) , rank-one matrix P0 is a nonorthogonal projector dyadprojecting on R(y). Direction of nonorthogonal projection is orthogonalto z ; id est,

P0 x − x ⊥ R(PT0 ) (1924)

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710 APPENDIX E. PROJECTION

E.6.2 Nonorthogonal projection on vectorized matrix

Formula (1923) is extensible. Given X,Y,Z∈ Rm×n, we have theone-dimensional nonorthogonal projection of X in isomorphic Rmn on therange of vectorized Y : (§2.2)

〈Z , X 〉〈Z , Y 〉 Y , 〈Z , Y 〉 6= 0 (1925)

where 〈Z , X 〉/〈Z , Y 〉 is the coefficient of projection. The inequalityaccounts for the fact: projection on R(vec Y ) is in a direction orthogonal tovec Z .

E.6.2.1 Nonorthogonal projection on dyad

Now suppose we have nonorthogonal projector dyad

P0 =yzT

zTy∈ Rm×m (1926)

Analogous to (1923), for X∈ Rm×m

P0XP0 =yzT

zTyX

yzT

zTy=

zTXy

(zTy)2yzT =

〈zyT, X 〉〈zyT, yzT〉 yzT (1927)

is a nonorthogonal projection of matrix X on the range of vectorizeddyad P0 ; from which it follows:

P0XP0 =zTXy

zTy

yzT

zTy=

zyT

zTy, X

yzT

zTy= 〈PT

0 , X 〉P0 =〈PT

0 , X 〉〈PT

0 , P0〉P0

(1928)

Yet this relationship between matrix product and vector inner-product onlyholds for a dyad projector. When nonsymmetric projector P0 is rank-one asin (1926), therefore,

R(vec P0XP0) = R(vec P0) in Rm2

(1929)

and

P0XP0 − X ⊥ PT0 in Rm2

(1930)

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E.6. VECTORIZATION INTERPRETATION, 711

E.6.2.1.1 Example. λ as coefficients of nonorthogonal projection.Any diagonalization (§A.5)

X = SΛS−1 =m

i=1

λi siwTi ∈ Rm×m (1513)

may be expressed as a sum of one-dimensional nonorthogonal projectionsof X , each on the range of a vectorized eigenmatrix Pj , sjw

Tj ;

X =m∑

i, j=1

〈(SeieTj S

−1)T, X 〉SeieTj S

−1

=m∑

j=1

〈(sjwTj )T, X 〉 sjw

Tj +

m∑

i, j=1

j 6= i

〈(SeieTj S

−1)T, SΛS−1〉SeieTj S

−1

=m∑

j=1

〈(sjwTj )T, X 〉 sjw

Tj

,m∑

j=1

〈PTj , X 〉Pj =

m∑

j=1

sjwTj Xsjw

Tj =

m∑

j=1

Pj XPj

=m∑

j=1

λj sjwTj

(1931)

This biorthogonal expansion of matrix X is a sum of nonorthogonalprojections because the term outside the projection coefficient 〈 〉 is notidentical to the inside-term. (§E.6.4) The eigenvalues λj are coefficients ofnonorthogonal projection of X , while the remaining M(M−1)/2 coefficients(for i 6=j) are zeroed by projection. When Pj is rank-one as in (1931),

R(vec PjXPj) = R(vec sjwTj ) = R(vec Pj) in Rm2

(1932)

and

PjXPj − X ⊥ PTj in Rm2

(1933)

Were matrix X symmetric, then its eigenmatrices would also be. So theone-dimensional projections would become orthogonal. (§E.6.4.1.1) 2

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712 APPENDIX E. PROJECTION

E.6.3 Orthogonal projection on a vector

The formula for orthogonal projection of vector x on the range of vector y(one-dimensional projection) is basic analytic geometry ; [13, §3.3] [328, §3.2][362, §2.2] [377, §1-8]

〈y, x〉〈y, y〉 y =

yTx

yTyy =

yyT

yTyx , P1 x (1934)

where 〈y, x〉/〈y, y〉 is the coefficient of projection on R(y). An equivalentdescription is: Vector P1 x is the orthogonal projection of vector x onR(P1)=R(y). Rank-one matrix P1 is a projection matrix because P 2

1 =P1 .The direction of projection is orthogonal

P1 x − x ⊥ R(P1) (1935)

because PT1 = P1 .

E.6.4 Orthogonal projection on a vectorized matrix

From (1934), given instead X, Y ∈ Rm×n, we have the one-dimensionalorthogonal projection of matrix X in isomorphic Rmn on the range ofvectorized Y : (§2.2)

〈Y , X 〉〈Y , Y 〉 Y (1936)

where 〈Y , X 〉/〈Y , Y 〉 is the coefficient of projection.For orthogonal projection, the term outside the vector inner-products 〈 〉

must be identical to the terms inside in three places.

E.6.4.1 Orthogonal projection on dyad

There is opportunity for insight when Y is a dyad yzT (§B.1): Instead givenX∈ Rm×n, y∈Rm, and z∈Rn

〈yzT, X 〉〈yzT, yzT〉 yzT =

yTXz

yTy zTzyzT (1937)

is the one-dimensional orthogonal projection of X in isomorphic Rmn onthe range of vectorized yzT. To reveal the obscured symmetric projection

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E.6. VECTORIZATION INTERPRETATION, 713

matrices P1 and P2 we rewrite (1937):

yTXz

yTy zTzyzT =

yyT

yTyX

zzT

zTz, P1XP2 (1938)

So for projector dyads, projection (1938) is the orthogonal projection in Rmn

if and only if projectors P1 and P2 are symmetric;E.10 in other words, for orthogonal projection on the range of a vectorized dyad yzT, theterm outside the vector inner-products 〈 〉 in (1937) must be identicalto the terms inside in three places.

When P1 and P2 are rank-one symmetric projectors as in (1938), (37)

R(vec P1XP2) = R(vec yzT) in Rmn (1939)

and

P1XP2 − X ⊥ yzT in Rmn (1940)

When y=z then P1 =P2 =PT2 and

P1XP1 = 〈P1 , X 〉P1 =〈P1 , X 〉〈P1 , P1〉

P1 (1941)

meaning, P1XP1 is equivalent to orthogonal projection of matrix X onthe range of vectorized projector dyad P1 . Yet this relationship betweenmatrix product and vector inner-product does not hold for general symmetricprojector matrices.

E.10For diagonalizable X∈ Rm×m (§A.5), its orthogonal projection in isomorphic Rm2

onthe range of vectorized yzT∈ Rm×m becomes:

P1XP2 =

m∑

i=1

λi P1 siwTi P2

When R(P1) = R(wj) and R(P2) = R(sj) , the j th dyad term from the diagonalizationis isolated but only, in general, to within a scale factor because neither set of left or righteigenvectors is necessarily orthonormal unless X is a normal matrix [390, §3.2]. Yet whenR(P2)=R(sk) , k 6=j∈1 . . . m , then P1XP2 =0.

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714 APPENDIX E. PROJECTION

E.6.4.1.1 Example. Eigenvalues λ as coefficients of orthogonal projection.Let C represent any convex subset of subspace SM , and let C1 be any elementof C . Then C1 can be expressed as the orthogonal expansion

C1 =M

i=1

M∑

j=1j ≥ i

〈Eij , C1〉Eij ∈ C ⊂ SM (1942)

where Eij ∈ SM is a member of the standard orthonormal basis for SM

(59). This expansion is a sum of one-dimensional orthogonal projectionsof C1 ; each projection on the range of a vectorized standard basis matrix.Vector inner-product 〈Eij , C1〉 is the coefficient of projection of svec C1 onR(svec Eij).

When C1 is any member of a convex set C whose dimension is L ,Caratheodory’s theorem [110] [303] [196] [40] [41] guarantees that no morethan L +1 affinely independent members from C are required to faithfullyrepresent C1 by their linear combination.E.11

Dimension of SM is L=M(M+1)/2 in isometrically isomorphicRM(M+1)/2. Yet because any symmetric matrix can be diagonalized, (§A.5.1)C1∈ SM is a linear combination of its M eigenmatrices qiq

Ti (§A.5.0.3)

weighted by its eigenvalues λi ;

C1 = QΛQT =M

i=1

λi qiqTi (1943)

where Λ∈ SM is a diagonal matrix having δ(Λ)i =λi , and Q=[ q1 · · · qM ]is an orthogonal matrix in RM×M containing corresponding eigenvectors.

To derive eigenvalue decomposition (1943) from expansion (1942), Mstandard basis matrices Eij are rotated (§B.5) into alignment with the Meigenmatrices qiq

Ti of C1 by applying a similarity transformation; [328, §5.6]

QEijQT =

qiqTi , i = j = 1 . . . M

1√2

(

qiqTj + qjq

Ti

)

, 1 ≤ i < j ≤ M

(1944)

E.11Caratheodory’s theorem guarantees existence of a biorthogonal expansion for anyelement in aff C when C is any pointed closed convex cone.

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E.6. VECTORIZATION INTERPRETATION, 715

which remains an orthonormal basis for SM . Then remarkably

C1 =M∑

i, j=1

j ≥ i

〈QEijQT, C1〉QEijQ

T

=M∑

i=1

〈qiqTi , C1〉 qiq

Ti +

M∑

i, j=1

j > i

〈QEijQT, QΛQT〉QEijQ

T

=M∑

i=1

〈qiqTi , C1〉 qiq

Ti

,M∑

i=1

〈Pi , C1〉Pi =M∑

i=1

qiqTi C1qiq

Ti =

M∑

i=1

Pi C1Pi

=M∑

i=1

λi qiqTi

(1945)

this orthogonal expansion becomes the diagonalization; still a sum ofone-dimensional orthogonal projections. The eigenvalues

λi = 〈qiqTi , C1〉 (1946)

are clearly coefficients of projection of C1 on the range of each vectorizedeigenmatrix. (confer §E.6.2.1.1) The remaining M(M−1)/2 coefficients(i 6=j) are zeroed by projection. When Pi is rank-one symmetric as in (1945),

R(svec Pi C1Pi) = R(svec qiqTi ) = R(svec Pi) in RM(M+1)/2 (1947)

andPi C1Pi − C1 ⊥ Pi in RM(M+1)/2 (1948)

2

E.6.4.2 Positive semidefiniteness test as orthogonal projection

For any given X∈ Rm×m the familiar quadratic construct yTXy≥ 0,over broad domain, is a fundamental test for positive semidefiniteness.(§A.2) It is a fact that yTXy is always proportional to a coefficient oforthogonal projection; letting z in formula (1937) become y∈Rm, thenP2 =P1 = yyT/yTy= yyT/‖yyT‖2 (confer (1576)) and formula (1938) becomes

〈yyT, X 〉〈yyT, yyT〉 yyT =

yTXy

yTy

yyT

yTy=

yyT

yTyX

yyT

yTy, P1XP1 (1949)

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716 APPENDIX E. PROJECTION

By (1936), product P1XP1 is the one-dimensional orthogonal projection of

X in isomorphic Rm2

on the range of vectorized P1 because, for rank P1 =1and P 2

1 =P1∈ Sm (confer (1928))

P1XP1 =yTXy

yTy

yyT

yTy=

yyT

yTy, X

yyT

yTy= 〈P1 , X 〉P1 =

〈P1 , X 〉〈P1 , P1〉

P1

(1950)

The coefficient of orthogonal projection 〈P1 , X 〉= yTXy/(yTy) is also knownas Rayleigh’s quotient.E.12 When P1 is rank-one symmetric as in (1949),

R(vec P1XP1) = R(vec P1) in Rm2

(1951)

and

P1XP1 − X ⊥ P1 in Rm2

(1952)

The test for positive semidefiniteness, then, is a test for nonnegativity ofthe coefficient of orthogonal projection of X on the range of each and everyvectorized extreme direction yyT (§2.8.1) from the positive semidefinite conein the ambient space of symmetric matrices.

E.12When y becomes the j th eigenvector sj of diagonalizable X , for example, 〈P1 , X 〉becomes the j th eigenvalue: [192, §III]

〈P1 , X 〉|y=sj=

sTj

(

m∑

i=1

λi siwTi

)

sj

sTj sj

= λj

Similarly for y = wj , the j th left-eigenvector,

〈P1 , X 〉|y=wj=

wTj

(

m∑

i=1

λi siwTi

)

wj

wTj wj

= λj

A quandary may arise regarding the potential annihilation of the antisymmetric part ofX when sT

jXsj is formed. Were annihilation to occur, it would imply the eigenvalue thusfound came instead from the symmetric part of X . The quandary is resolved recognizingthat diagonalization of real X admits complex eigenvectors; hence, annihilation could onlycome about by forming re(sH

j Xsj) = sHj (X +XT)sj/2 [199, §7.1] where (X +XT)/2 is

the symmetric part of X , and sHj denotes conjugate transpose.

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E.7. PROJECTION ON MATRIX SUBSPACES 717

E.6.4.3 PXP º 0

In some circumstances, it may be desirable to limit the domain of testyTXy≥ 0 for positive semidefiniteness; e.g., ‖y‖= 1. Another exampleof limiting domain-of-test is central to Euclidean distance geometry: ForR(V )=N (1T) , the test −V DV º 0 determines whether D∈ SN

h is aEuclidean distance matrix. The same test may be stated: For D∈ SN

h (andoptionally ‖y‖=1)

D ∈ EDMN ⇔ −yTDy = 〈yyT, −D〉 ≥ 0 ∀ y ∈ R(V ) (1953)

The test −V DV º 0 is therefore equivalent to a test for nonnegativity of thecoefficient of orthogonal projection of −D on the range of each and everyvectorized extreme direction yyT from the positive semidefinite cone SN

+ suchthat R(yyT)=R(y)⊆R(V ). (The validity of this result is independent ofwhether V is itself a projection matrix.)

E.7 Projection on matrix subspaces

E.7.1 PXP misinterpretation for higher-rank P

For a projection matrix P of rank greater than 1, PXP is generally notcommensurate with 〈P,X 〉

〈P,P 〉P as is the case for projector dyads (1950). Yetfor a symmetric idempotent matrix P of any rank we are tempted to say“ PXP is the orthogonal projection of X∈ Sm on R(vec P ) ”. The fallacyis: vec PXP does not necessarily belong to the range of vectorized P ; themost basic requirement for projection on R(vec P ).

E.7.2 Orthogonal projection on matrix subspaces

With A1∈ Rm×n, B1∈Rn×k, Z1∈Rm×k, A2∈ Rp×n, B2∈Rn×k, Z2∈Rp×k asdefined for nonorthogonal projector (1849), and defining

P1 , A1A†1 ∈ Sm , P2 , A2A

†2 ∈ Sp (1954)

then, given compatible X

‖X−P1XP2‖F = infB1 , B2∈R

n×k‖X−A1(A

†1+B1Z

T1 )X(A†T

2 +Z2BT2 )AT

2 ‖F (1955)

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718 APPENDIX E. PROJECTION

As for all subspace projectors, range of the projector is the subspace on whichprojection is made: P1Y P2 | Y ∈ Rm×p. For projectors P1 and P2 of anyrank, altogether, this means projection P1XP2 is unique minimum-distance,orthogonal

P1XP2 − X ⊥ P1Y P2 | Y ∈ Rm×p in Rmp (1956)

and P1 and P2 must each be symmetric (confer (1938)) to attain the infimum.

E.7.2.0.1 Proof. Minimum Frobenius norm (1955).Defining P , A1(A

†1 + B1Z

T1 ) ,

infB1 , B2

‖X − A1(A†1 + B1Z

T1 )X(A†T

2 + Z2BT2 )AT

2 ‖2F

= infB1 , B2

‖X − PX(A†T2 + Z2B

T2 )AT

2 ‖2F

= infB1 , B2

tr(

(XT− A2(A†2 + B2Z

T2 )XTPT)(X − PX(A†T

2 + Z2BT2 )AT

2 ))

= infB1 , B2

tr(

XTX−XTPX(A†T2 +Z2B

T2 )AT

2−A2(A†2+B2Z

T2 )XTPTX

+A2(A†2+B2Z

T2 )XTPTPX(A†T

2 +Z2BT2 )AT

2

)

(1957)

Necessary conditions for a global minimum are ∇B1=0 and ∇B2=0. Termsnot containing B2 in (1957) will vanish from gradient ∇B2 ; (§D.2.3)

∇B2 tr(

−XTPXZ2BT2A

T2−A2B2Z

T2X

TPTX+A2A†2X

TPTPXZ2BT2A

T2

+A2B2ZT2X

TPTPXA†T2 AT

2+A2B2ZT2X

TPTPXZ2BT2A

T2

)

= −2AT2X

TPXZ2 + 2AT2A2A

†2X

TPTPXZ2 + 2AT2A2B2Z

T2X

TPTPXZ2

= AT2

(

−XT + A2A†2X

TPT + A2B2ZT2X

TPT)

PXZ2

= 0 ⇔R(B1)⊆ N (A1) and R(B2)⊆ N (A2)

(1958)

(or Z2 = 0) because AT = ATAA†. Symmetry requirement (1954) is implicit.Were instead PT , (A†T

2 + Z2BT2 )AT

2 and the gradient with respect to B1

observed, then similar results are obtained. The projector is unique.Perpendicularity (1956) establishes uniqueness [106, §4.9] of projectionP1XP2 on a matrix subspace. The minimum-distance projector is theorthogonal projector, and vice versa. ¨

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E.7. PROJECTION ON MATRIX SUBSPACES 719

E.7.2.0.2 Example. PXP redux & N (V).Suppose we define a subspace of m×n matrices, each elemental matrixhaving columns constituting a list whose geometric center (§5.5.1.0.1) is theorigin in Rm :

Rm×nc , Y ∈ Rm×n | Y 1 = 0

= Y ∈ Rm×n | N (Y ) ⊇ 1 = Y ∈ Rm×n | R(Y T) ⊆ N (1T)= XV | X∈ Rm×n ⊂ Rm×n (1959)

the nonsymmetric geometric center subspace. Further suppose V ∈ Sn isa projection matrix having N (V )=R(1) and R(V ) = N (1T). Then linearmapping T (X)=XV is the orthogonal projection of any X∈ Rm×n on Rm×n

c

in the Euclidean (vectorization) sense because V is symmetric, N (XV )⊇1,and R(VXT)⊆N (1T).

Now suppose we define a subspace of symmetric n×n matrices each ofwhose columns constitute a list having the origin in Rn as geometric center,

Snc , Y ∈ Sn | Y 1 = 0

= Y ∈ Sn | N (Y ) ⊇ 1 = Y ∈ Sn | R(Y ) ⊆ N (1T)(1960)

the geometric center subspace. Further suppose V ∈ Sn is a projectionmatrix, the same as before. Then V XV is the orthogonal projection ofany X∈ Sn on Sn

c in the Euclidean sense (1956) because V is symmetric,V XV 1=0, and R(V XV )⊆N (1T). Two-sided projection is necessary onlyto remain in the ambient symmetric matrix subspace. Then

Snc = V XV | X∈ Sn ⊂ Sn (1961)

has dim Snc = n(n−1)/2 in isomorphic Rn(n+1)/2. We find its orthogonal

complement as the aggregate of all negative directions of orthogonalprojection on Sn

c : the translation-invariant subspace (§5.5.1.1)

Sn⊥c , X − V XV | X∈ Sn ⊂ Sn

= u1T + 1uT | u∈Rn(1962)

characterized by the doublet u1T + 1uT (§B.2).E.13 Defining thegeometric center mapping V(X) = −V XV 1

2consistently with (972), then

E.13Proof.

X − V X V | X∈ Sn = X − (I − 1n11T)X(I − 11T 1

n) | X∈ Sn= 1

n11TX + X11T 1n − 1

n11TX 11T 1n | X∈ Sn

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720 APPENDIX E. PROJECTION

N (V)=R(I − V) on domain Sn analogously to vector projectors (§E.2);id est,

N (V) = Sn⊥c (1963)

a subspace of Sn whose dimension is dim Sn⊥c = n in isomorphic Rn(n+1)/2.

Intuitively, operator V is an orthogonal projector; any argumentduplicitously in its range is a fixed point. So, this symmetric operator’snullspace must be orthogonal to its range.

Now compare the subspace of symmetric matrices having all zeros in thefirst row and column

Sn1 , Y ∈ Sn | Y e1 = 0

=

[

0 0T

0 I

]

X

[

0 0T

0 I

]

| X∈ Sn

=

[

0√

2VN]T

Z[

0√

2VN]

| Z ∈ SN

(1964)

where P =

[

0 0T

0 I

]

is an orthogonal projector. Then, similarly, PXP is

the orthogonal projection of any X∈ Sn on Sn1 in the Euclidean sense (1956),

and

Sn⊥1 ,

[

0 0T

0 I

]

X

[

0 0T

0 I

]

− X | X∈ Sn

⊂ Sn

=

ueT1 + e1u

T | u∈Rn

(1965)

Obviously, Sn1 ⊕ Sn⊥

1 = Sn. 2

Because X1 | X∈ Sn= Rn,

X − V X V | X∈ Sn = 1ζT + ζ 1T − 11T(1Tζ 1n) | ζ∈Rn

= 1ζT(I − 11T 12n) + (I − 1

2n11T)ζ 1T | ζ∈Rn

where I− 12n11T is invertible. ¨

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E.8. RANGE/ROWSPACE INTERPRETATION 721

E.8 Range/Rowspace interpretation

For idempotent matrices P1 and P2 of any rank, P1XPT2 is a projection

of R(X) on R(P1) and a projection of R(XT) on R(P2) : For any given

X = UΣQT =η

i=1

σi uiqTi ∈ Rm×p, as in compact SVD (1529),

P1XPT2 =

η∑

i=1

σi P1 uiqTi PT

2 =

η∑

i=1

σi P1ui (P2 qi)T (1966)

where η , minm , p. Recall ui∈R(X) and qi∈R(XT) when thecorresponding singular value σi is nonzero. (§A.6.1) So P1 projects ui onR(P1) while P2 projects qi on R(P2) ; id est, the range and rowspace of anyX are respectively projected on the ranges of P1 and P2 .E.14

E.9 Projection on convex set

Thus far we have discussed only projection on subspaces. Now wegeneralize, considering projection on arbitrary convex sets in Euclidean space;convex because projection is, then, unique minimum-distance and a convexoptimization problem:

For projection PCx of point x on any closed set C⊆Rn it is obvious:

C ≡ PCx | x∈Rn (1967)

where PC is a projection operator that is convex when C is convex. [59, p.88]If C⊆Rn is a closed convex set, then for each and every x∈Rn there exists

a unique point PCx belonging to C that is closest to x in the Euclideansense. Like (1871), unique projection Px (or PCx) of a point x on convexset C is that point in C closest to x ; [246, §3.12]

‖x − Px‖2 = infy∈C

‖x − y‖2 = dist(x , C) (1968)

There exists a converse (in finite-dimensional Euclidean space):

E.14When P1 and P2 are symmetric and R(P1)=R(uj) and R(P2)=R(qj) , then the j th

dyad term from the singular value decomposition of X is isolated by the projection. Yetif R(P2)=R(qℓ) , ℓ 6=j∈1 . . . η , then P1XP2 =0.

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722 APPENDIX E. PROJECTION

E.9.0.0.1 Theorem. (Bunt-Motzkin) Convex set if projections unique.[368, §7.5] [193] If C⊆ Rn is a nonempty closed set and if for each and everyx in Rn there is a unique Euclidean projection Px of x on C belonging to C ,then C is convex. ⋄

Borwein & Lewis propose, for closed convex set C [53, §3.3 exer.12(d)]

∇‖x − Px‖22 = 2(x − Px) (1969)

for any point x whereas, for x /∈C∇‖x − Px‖2 = (x − Px) ‖x − Px‖−1

2 (1970)

E.9.0.0.2 Exercise. Norm gradient.Prove (1969) and (1970). (Not proved in [53].) H

A well-known equivalent characterization of projection on a convex set isa generalization of the perpendicularity condition (1870) for projection on asubspace:

E.9.1 Dual interpretation of projection on convex set

E.9.1.0.1 Definition. Normal vector. [303, p.15]Vector z is normal to convex set C at point Px∈ C if

〈z , y−Px〉 ≤ 0 ∀ y ∈ C (1971)

A convex set has at least one nonzero normal at each of its boundarypoints. [303, p.100] (Figure 66) Hence, the normal or dual interpretation ofprojection:

E.9.1.0.2 Theorem. Unique minimum-distance projection. [196, §A.3.1][246, §3.12] [106, §4.1] [76] (Figure 166b p.739) Given a closed convex setC⊆Rn, point Px is the unique projection of a given point x∈Rn on C(Px is that point in C nearest x) if and only if

Px ∈ C , 〈x − Px , y − Px〉 ≤ 0 ∀ y ∈ C (1972)

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E.9. PROJECTION ON CONVEX SET 723

As for subspace projection, convex operator P is idempotent in the sense:for each and every x∈Rn, P (Px)=Px . Yet operator P is nonlinear; Projector P is a linear operator if and only if convex set C (on which

projection is made) is a subspace. (§E.4)

E.9.1.0.3 Theorem. Unique projection via normal cone.E.15 [106, §4.3]Given closed convex set C⊆Rn, point Px is the unique projection of agiven point x∈Rn on C if and only if

Px ∈ C , Px − x ∈ (C − Px)∗ (1973)

In other words, Px is that point in C nearest x if and only if Px − x belongsto that cone dual to translate C − Px . ⋄

E.9.1.1 Dual interpretation as optimization

Deutsch [108, thm.2.3] [109, §2] and Luenberger [246, p.134] carry forwardNirenberg’s dual interpretation of projection [275] as solution to amaximization problem: Minimum distance from a point x∈Rn to a convexset C⊂ Rn can be found by maximizing distance from x to hyperplane ∂Hover the set of all hyperplanes separating x from C . Existence of aseparating hyperplane (§2.4.2.7) presumes point x lies on the boundary orexterior to set C .

The optimal separating hyperplane is characterized by the fact it alsosupports C . Any hyperplane supporting C (Figure 29a) has form

∂H− =

y∈Rn | aTy = σC(a)

(128)

where the support function is convex, defined

σC(a) = supz∈C

aTz (531)

When point x is finite and set C contains finite points, under this projectioninterpretation, if the supporting hyperplane is a separating hyperplane thenthe support function is finite. From Example E.5.0.0.8, projection P∂H−

x ofx on any given supporting hyperplane ∂H− is

P∂H−x = x − a(aTa)−1

(

aTx − σC(a))

(1974)

E.15 −(C − Px)∗ is the normal cone to set C at point Px . (§E.10.3.2)

Page 724: Convex Optimization & Euclidean Distance Geometry

724 APPENDIX E. PROJECTION

C

κa

∂H−

x

PCx

(a)

(b)

H−

H+

P∂H−x

Figure 162: Dual interpretation of projection of point x on convex set Cin R2. (a) κ = (aTa)−1

(

aTx − σC(a))

(b) Minimum distance from x toC is found by maximizing distance to all hyperplanes supporting C andseparating it from x . A convex problem for any convex set, distance ofmaximization is unique.

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E.9. PROJECTION ON CONVEX SET 725

With reference to Figure 162, identifying

H+ = y∈Rn | aTy ≥ σC(a) (106)

then

‖x − PCx‖ = sup∂H− | x∈H+

‖x − P∂H−x‖ = sup

a |x∈H+

‖a(aTa)−1(aTx − σC(a))‖

= supa |x∈H+

1‖a‖ |aTx − σC(a)|

(1975)

which can be expressed as a convex optimization, for arbitrary positiveconstant τ

‖x − PCx‖ =1

τmaximize

aaTx − σC(a)

subject to ‖a‖ ≤ τ(1976)

The unique minimum-distance projection on convex set C is therefore

PCx = x − a⋆(

a⋆Tx − σC(a⋆)

) 1

τ 2(1977)

where optimally ‖a⋆‖= τ .

E.9.1.1.1 Exercise. Dual projection technique on polyhedron.Test that projection paradigm from Figure 162 on any convex polyhedralset. H

E.9.1.2 Dual interpretation of projection on cone

In the circumstance set C is a closed convex cone K and there exists ahyperplane separating given point x from K , then optimal σK(a⋆) takesvalue 0 [196, §C.2.3.1]. So problem (1976) for projection of x on K becomes

‖x − PKx‖ =1

τmaximize

aaTx

subject to ‖a‖ ≤ τ

a ∈ K

(1978)

The norm inequality in (1978) can be handled by Schur complement (§3.6.2).Normals a to all hyperplanes supporting K belong to the polar coneK

=−K∗by definition: (297)

a ∈ K ⇔ 〈a , x〉 ≤ 0 for all x ∈ K (1979)

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726 APPENDIX E. PROJECTION

Projection on cone K is

PKx = (I − 1

τ 2a⋆a⋆T)x (1980)

whereas projection on the polar cone −K∗is (§E.9.2.2.1)

PKx = x − PKx =1

τ 2a⋆a⋆Tx (1981)

Negating vector a , this maximization problem (1978) becomes aminimization (the same problem) and the polar cone becomes the dual cone:

‖x − PKx‖ = −1

τminimize

aaTx

subject to ‖a‖ ≤ τ

a ∈ K∗

(1982)

E.9.2 Projection on cone

When convex set C is a cone, there is a finer statement of optimalityconditions:

E.9.2.0.1 Theorem. Unique projection on cone. [196, §A.3.2]Let K⊆ Rn be a closed convex cone, and K∗

its dual (§2.13.1). Then Px isthe unique minimum-distance projection of x∈Rn on K if and only if

Px ∈ K , 〈Px − x , Px〉 = 0 , Px − x ∈ K∗(1983)

In words, Px is the unique minimum-distance projection of x on K ifand only if

1) projection Px lies in K

2) direction Px−x is orthogonal to the projection Px

3) direction Px−x lies in the dual cone K∗.

As the theorem is stated, it admits projection on K not full-dimensional;id est, on closed convex cones in a proper subspace of Rn.

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E.9. PROJECTION ON CONVEX SET 727

Projection on K of any point x∈−K∗, belonging to the negative dual

cone, is the origin. By (1983): the set of all points reaching the origin, whenprojecting on K , constitutes the negative dual cone; a.k.a, the polar cone

K= −K∗

= x∈Rn | Px = 0 (1984)

E.9.2.1 Relationship to subspace projection

Conditions 1 and 2 of the theorem are common with orthogonal projection ona subspace R(P ) : Condition 1 corresponds to the most basic requirement;namely, the projection Px∈R(P ) belongs to the subspace. Recall theperpendicularity requirement for projection on a subspace;

Px − x ⊥ R(P ) or Px − x ∈ R(P )⊥ (1870)

which corresponds to condition 2.Yet condition 3 is a generalization of subspace projection; id est, for

unique minimum-distance projection on a closed convex cone K , polar cone−K∗

plays the role R(P )⊥ plays for subspace projection (PRx = x − PR⊥ x).Indeed, −K∗

is the algebraic complement in the orthogonal vector sum (p.759)[265] [196, §A.3.2.5]

K ⊞ −K∗= Rn ⇔ cone K is closed and convex (1985)

Also, given unique minimum-distance projection Px on K satisfyingTheorem E.9.2.0.1, then by projection on the algebraic complement via I−Pin §E.2 we have

−K∗= x − Px | x∈Rn = x∈Rn | Px = 0 (1986)

consequent to Moreau (1989). Recalling any subspace is a closed convexconeE.16

K = R(P ) ⇔ −K∗= R(P )⊥ (1987)

meaning, when a cone is a subspace R(P ) , then the dual cone becomes itsorthogonal complement R(P )⊥. [59, §2.6.1] In this circumstance, condition 3becomes coincident with condition 2.

The properties of projection on cones following in §E.9.2.2 furthergeneralize to subspaces by: (4)

K = R(P ) ⇔ −K = R(P ) (1988)

E.16 but a proper subspace is not a proper cone (§2.7.2.2.1).

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728 APPENDIX E. PROJECTION

E.9.2.2 Salient properties: Projection Px on closed convex cone K[196, §A.3.2] [106, §5.6] For x , x1 , x2∈ Rn

1. PK(αx) = α PKx ∀α≥0 (nonnegative homogeneity)

2. ‖PKx‖ ≤ ‖x‖

3. PKx = 0 ⇔ x ∈ −K∗

4. PK(−x) = −P−Kx

5. (Jean-Jacques Moreau (1962)) [265]

x = x1 + x2 , x1∈K , x2∈−K∗, x1⊥ x2

⇔x1 = PKx , x2 = P−K∗x

(1989)

6. K = x − P−K∗x | x∈Rn = x∈Rn | P−K∗x = 0

7. −K∗= x − PKx | x∈Rn = x∈Rn | PKx = 0 (1986)

E.9.2.2.1 Corollary. I−P for cones. (confer §E.2)Denote by K⊆ Rn a closed convex cone, and call K∗

its dual. Thenx−P−K∗x is the unique minimum-distance projection of x∈Rn on K if andonly if P−K∗x is the unique minimum-distance projection of x on −K∗

thepolar cone. ⋄

Proof. Assume x1 = PKx . Then by Theorem E.9.2.0.1 we have

x1∈ K , x1− x ⊥ x1 , x1− x ∈ K∗(1990)

Now assume x − x1 = P−K∗x . Then we have

x − x1 ∈ −K∗, −x1 ⊥ x − x1 , −x1 ∈ −K (1991)

But these two assumptions are apparently identical. We must therefore have

x−P−K∗x = x1 = PKx (1992)

¨

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E.9. PROJECTION ON CONVEX SET 729

E.9.2.2.2 Corollary. Unique projection via dual or normal cone.[106, §4.7] (§E.10.3.2, confer Theorem E.9.1.0.3) Given point x∈Rn andclosed convex cone K⊆Rn, the following are equivalent statements:

1. point Px is the unique minimum-distance projection of x on K

2. Px ∈ K , x − Px ∈ −(K − Px)∗ = −K∗∩ (Px)⊥

3. Px ∈ K , 〈x − Px , Px〉 = 0 , 〈x − Px , y〉 ≤ 0 ∀ y ∈ K⋄

E.9.2.2.3 Example. Unique projection on nonnegative orthant.(confer (1296)) From Theorem E.9.2.0.1, to project matrix H∈ Rm×n onthe selfdual orthant (§2.13.5.1) of nonnegative matrices Rm×n

+ in isomorphicRmn, the necessary and sufficient conditions are:

H⋆ ≥ 0tr

(

(H⋆− H)TH⋆)

= 0H⋆− H ≥ 0

(1993)

where the inequalities denote entrywise comparison. The optimal solutionH⋆ is simply H having all its negative entries zeroed;

H⋆ij = maxHij , 0 , i, j∈1 . . . m × 1 . . . n (1994)

Now suppose the nonnegative orthant is translated by T ∈ Rm×n ; id est,consider Rm×n

+ + T . Then projection on the translated orthant is [106, §4.8]

H⋆ij = maxHij , Tij (1995)

2

E.9.2.2.4 Example. Unique projection on truncated convex cone.Consider the problem of projecting a point x on a closed convex cone thatis artificially bounded; really, a bounded convex polyhedron having a vertexat the origin:

minimizey∈RN

‖x − Ay‖2

subject to y º 0

‖y‖∞ ≤ 1

(1996)

Page 730: Convex Optimization & Euclidean Distance Geometry

730 APPENDIX E. PROJECTION

where the convex cone has vertex-description (§2.12.2.0.1), for A∈Rn×N

K = Ay | y º 0 (1997)

and where ‖y‖∞ ≤ 1 is the artificial bound. This is a convex optimizationproblem having no known closed-form solution, in general. It arises, forexample, in the fitting of hearing aids designed around a programmablegraphic equalizer (a filter bank whose only adjustable parameters are gainper band each bounded above by unity). [94] The problem is equivalent to aSchur-form semidefinite program (§3.6.2)

minimizey∈RN , t∈R

t

subject to

[

tI x − Ay(x − Ay)T t

]

º 0

0 ¹ y ¹ 1

(1998)

2

E.9.3 nonexpansivity

E.9.3.0.1 Theorem. Nonexpansivity. [171, §2] [106, §5.3]When C ⊂ Rn is an arbitrary closed convex set, projector P projecting on Cis nonexpansive in the sense: for any vectors x, y∈Rn

‖Px − Py‖ ≤ ‖x − y‖ (1999)

with equality when x−Px = y−Py .E.17 ⋄

Proof. [52]

‖x − y‖2 = ‖Px − Py‖2 + ‖(I − P )x − (I − P )y‖2

+ 2〈x − Px , Px − Py〉 + 2〈y − Py , Py − Px〉(2000)

Nonnegativity of the last two terms follows directly from the uniqueminimum-distance projection theorem (§E.9.1.0.2). ¨

E.17This condition for equality corrects an error in [76] (where the norm is applied to eachside of the condition given here) easily revealed by counter-example.

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E.9. PROJECTION ON CONVEX SET 731

The foregoing proof reveals another flavor of nonexpansivity; for each andevery x, y∈Rn

‖Px − Py‖2 + ‖(I − P )x − (I − P )y‖2 ≤ ‖x − y‖2 (2001)

Deutsch shows yet another: [106, §5.5]

‖Px − Py‖2 ≤ 〈x − y , Px − Py〉 (2002)

E.9.4 Easy projections To project any matrix H∈ Rn×n orthogonally in Euclidean/Frobenius

sense on subspace of symmetric matrices Sn in isomorphic Rn2

, takesymmetric part of H ; (§2.2.2.0.1) id est, (H+HT)/2 is the projection. To project any H∈ Rn×n orthogonally on symmetric hollow subspaceSn

h in isomorphic Rn2

(§2.2.3.0.1, §7.0.1), take symmetric part then zeroall entries along main diagonal or vice versa (because this is projectionon intersection of two subspaces); id est, (H + HT)/2 − δ2(H). To project a matrix on nonnegative orthant Rm×n

+ , simply clip allnegative entries to 0. Likewise, projection on nonpositive orthant Rm×n

−sees all positive entries clipped to 0. Projection on other orthants isequally simple with appropriate clipping. Projecting on hyperplane, halfspace, slab: §E.5.0.0.8. Projection of y∈Rn on Euclidean ball B = x∈Rn | ‖x − a‖ ≤ c :for y 6= a , PBy = c

‖y−a‖(y − a) + a . Clipping in excess of |1| each entry of point x∈Rn is equivalent tounique minimum-distance projection of x on a hypercube centered atthe origin. (confer §E.10.3.2) Projection of x∈Rn on a (rectangular) hyperbox : [59, §8.1.1]

C = y∈Rn | l ¹ y ¹ u , l ≺ u (2003)

P (x)k=0...n =

lk , xk ≤ lkxk , lk ≤ xk ≤ uk

uk , xk ≥ uk

(2004)

Page 732: Convex Optimization & Euclidean Distance Geometry

732 APPENDIX E. PROJECTION Orthogonal projection of x on a Cartesian subspace, whose basis issome given subset of the Cartesian axes, zeroes entries correspondingto the remaining (complementary) axes. Projection of x on set of all cardinality-k vectors y | card y≤k keepsk entries of greatest magnitude and clips to 0 those remaining. Unique minimum-distance projection of H∈ Sn on positive semidefinitecone Sn

+ in Euclidean/Frobenius sense is accomplished by eigenvaluedecomposition (diagonalization) followed by clipping all negativeeigenvalues to 0. Unique minimum-distance projection on generally nonconvex subset ofall matrices belonging to Sn

+ having rank not exceeding ρ (§2.9.2.1)is accomplished by clipping all negative eigenvalues to 0 and zeroingsmallest nonnegative eigenvalues keeping only ρ largest. (§7.1.2) Unique minimum-distance projection, in Euclidean/Frobenius sense,of H∈ Rm×n on generally nonconvex subset of all m×n matrices ofrank no greater than k is singular value decomposition (§A.6) of Hhaving all singular values beyond kth zeroed. This is also a solution toprojection in sense of spectral norm. [325, p.79, p.208] Projection on monotone nonnegative cone KM+⊂ Rn in less than onecycle (in sense of alternating projections §E.10): [Wıκımization]. Fast projection on a simplicial cone: [Wıκımization]. Projection on closed convex cone K of any point x∈−K∗

, belonging topolar cone, is equivalent to projection on origin. (§E.9.2) PSN

+∩ SNc

= PSN+PSN

c(1240) P

RN×N+ ∩ SN

h= P

RN×N+

PSNh

(§7.0.1.1) PR

N×N+ ∩ SN = P

RN×N+

PSN (§E.9.5)

E.9.4.0.1 Exercise. Projection on spectral norm ball.Find the unique minimum-distance projection on the convex set of allm×n matrices whose largest singular value does not exceed 1 ; id est, onX∈ Rm×n | ‖X‖2 ≤ 1 the spectral norm ball (§2.3.2.0.5). H

Page 733: Convex Optimization & Euclidean Distance Geometry

E.9. PROJECTION ON CONVEX SET 733

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

""

""

""

""

""

""

""

""

""

""

""

""

""

""

CRn

Rn⊥

x− z ⊥ z− y

sx

s

z

BBBBBQ

QQ

Q

SS

SS

SS

Ss

y

BBBBBBBB

BBBBB

Figure 163: Closed convex set C belongs to subspace Rn (shown boundedin sketch and drawn without proper perspective). Point y is uniqueminimum-distance projection of x on C ; equivalent to product of orthogonalprojection of x on Rn and minimum-distance projection of result z on C .

E.9.4.1 notes

Projection on Lorentz (second-order) cone: [59, exer.8.3(c)].Deutsch [109] provides an algorithm for projection on polyhedral cones.Youla [389, §2.5] lists eleven “useful projections”, of square-integrable

uni- and bivariate real functions on various convex sets, in closed form.Unique minimum-distance projection on an ellipsoid: Example 4.6.0.0.1.Numerical algorithms for projection on the nonnegative simplex and

1-norm ball are in [123].

E.9.5 Projection on convex set in subspace

Suppose a convex set C is contained in some subspace Rn. Then uniqueminimum-distance projection of any point in Rn⊕ Rn⊥ on C can beaccomplished by first projecting orthogonally on that subspace, and thenprojecting the result on C ; [106, §5.14] id est, the ordered product of twoindividual projections that is not commutable.

Page 734: Convex Optimization & Euclidean Distance Geometry

734 APPENDIX E. PROJECTION

Proof. (⇐) To show that, suppose unique minimum-distance projectionPCx on C⊂Rn is y as illustrated in Figure 163;

‖x − y‖ ≤ ‖x − q‖ ∀ q ∈ C (2005)

Further suppose PRnx equals z . By the Pythagorean theorem

‖x − y‖2 = ‖x − z‖2 + ‖z − y‖2 (2006)

because x− z ⊥ z− y . (1870) [246, §3.3] Then point y = PCx is the sameas PCz because

‖z − y‖2 = ‖x− y‖2 − ‖x− z‖2 ≤ ‖z − q‖2 = ‖x− q‖2 − ‖x− z‖2 ∀ q ∈ C(2007)

which holds by assumption (2005).(⇒) Now suppose z = PR

nx and

‖z − y‖ ≤ ‖z − q‖ ∀ q ∈ C (2008)

meaning y = PCz . Then point y is identical to PCx because

‖x− y‖2 = ‖x− z‖2 + ‖z − y‖2 ≤ ‖x− q‖2 = ‖x− z‖2 + ‖z − q‖2 ∀ q ∈ C(2009)

by assumption (2008). ¨

This proof is extensible via translation argument. (§E.4) Uniqueminimum-distance projection on a convex set contained in an affine subsetis, therefore, similarly accomplished.

Projecting matrix H∈ Rn×n on convex cone K= Sn∩ Rn×n+ in isomorphic

Rn2

can be accomplished, for example, by first projecting on Sn and only thenprojecting the result on Rn×n

+ (confer §7.0.1). This is because that projectionproduct is equivalent to projection on the subset of the nonnegative orthantin the symmetric matrix subspace.

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E.10. ALTERNATING PROJECTION 735

E.10 Alternating projection

Alternating projection is an iterative technique for finding a point in theintersection of a number of arbitrary closed convex sets Ck , or for findingthe distance between two nonintersecting closed convex sets. Because it cansometimes be difficult or inefficient to compute the intersection or expressit analytically, one naturally asks whether it is possible to instead project(unique minimum-distance) alternately on the individual Ck ; often easierand what motivates adoption of this technique. Once a cycle of alternatingprojections (an iteration) is complete, we then iterate (repeat the cycle)until convergence. If the intersection of two closed convex sets is empty, thenby convergence we mean the iterate (the result after a cycle of alternatingprojections) settles to a point of minimum distance separating the sets.

While alternating projection can find the point in the nonemptyintersection closest to a given point b , it does not necessarily find theclosest point. Finding that closest point is made dependable by an elegantlysimple enhancement via correction to the alternating projection technique:this Dykstra algorithm (2047) for projection on the intersection is one ofthe most beautiful projection algorithms ever discovered. It is accuratelyinterpreted as the discovery of what alternating projection originally soughtto accomplish: unique minimum-distance projection on the nonemptyintersection of a number of arbitrary closed convex sets Ck . Alternatingprojection is, in fact, a special case of the Dykstra algorithm whose discussionwe defer until §E.10.3.

E.10.0.1 commutative projectors

Given two arbitrary convex sets C1 and C2 and their respectiveminimum-distance projection operators P1 and P2 , if projectors commutefor each and every x∈Rn then it is easy to show P1P2x∈ C1∩ C2 andP2P1x∈ C1∩ C2 . When projectors commute (P1P2 =P2P1), a point in theintersection can be found in a finite number of steps; while commutativity isa sufficient condition, it is not necessary (§6.8.1.1.1 for example).

When C1 and C2 are subspaces, in particular, projectors P1 and P2

commute if and only if P1P2 = PC1∩C2 or iff P2P1 = PC1∩C2 or iff P1P2 isthe orthogonal projection on a Euclidean subspace. [106, lem.9.2] Subspaceprojectors will commute, for example, when P1(C2)⊂ C2 or P2(C1)⊂ C1 orC1⊂ C2 or C2⊂ C1 or C1⊥ C2 . When subspace projectors commute, this

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736 APPENDIX E. PROJECTION

C1

C2

∂K⊥C1∩C2

(Pb) + Pb

b

Figure 164: First several alternating projections in von Neumann-styleprojection (2020), of point b , converging on closest point Pb in intersectionof two closed convex sets in R2 : C1 and C2 are partially drawn in vicinity oftheir intersection. Pointed normal cone K⊥ (2049) is translated to Pb , theunique minimum-distance projection of b on intersection. For this particularexample, it is possible to start anywhere in a large neighborhood of b and stillconverge to Pb . Alternating projections are themselves robust with respectto significant noise because they belong to translated normal cone.

means we can find a point in the intersection of those subspaces in a finitenumber of steps; we find, in fact, the closest point.

E.10.0.1.1 Theorem. Kronecker projector. [324, §2.7]Given any projection matrices P1 and P2 (subspace projectors), then

P1 ⊗ P2 and P1 ⊗ I (2010)

are projection matrices. The product preserves symmetry if present. ⋄

E.10.0.2 noncommutative projectors

Typically, one considers the method of alternating projection when projectorsdo not commute; id est, when P1P2 6=P2P1 .

The iconic example for noncommutative projectors illustrated inFigure 164 shows the iterates converging to the closest point in theintersection of two arbitrary convex sets. Yet simple examples likeFigure 165 reveal that noncommutative alternating projection does not

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E.10. ALTERNATING PROJECTION 737

H1

H2

b

Pb

P2b

P1P2bH1 ∩ H2

Figure 165: The sets Ck in this example comprise two halfspaces H1 andH2 . The von Neumann-style alternating projection in R2 quickly convergesto P1P2b (feasibility). The unique minimum-distance projection on theintersection is, of course, Pb .

always yield the closest point, although we shall show it always yields somepoint in the intersection or a point that attains the distance between twoconvex sets.

Alternating projection is also known as successive projection [174] [171][61], cyclic projection [144] [256, §3.2], successive approximation [76], orprojection on convex sets [322] [323, §6.4]. It is traced back to von Neumann(1933) [366] and later Wiener [373] who showed that higher iterates of aproduct of two orthogonal projections on subspaces converge at each pointin the ambient space to the unique minimum-distance projection on theintersection of the two subspaces. More precisely, if R1 and R2 are closedsubspaces of a Euclidean space and P1 and P2 respectively denote orthogonalprojection on R1 and R2 , then for each vector b in that space,

limi→∞

(P1P2)ib = PR1∩R2b (2011)

Deutsch [106, thm.9.8, thm.9.35] shows rate of convergence for subspaces tobe geometric [388, §1.4.4]; bounded above by κ2i+1‖b‖ , i=0, 1, 2 . . . , where

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738 APPENDIX E. PROJECTION

0≤κ<1 :‖ (P1P2)

ib − PR1∩R2b ‖ ≤ κ2i+1‖b‖ (2012)

This means convergence can be slow when κ is close to 1. Rate ofconvergence on intersecting halfspaces is also geometric. [107] [293]

This von Neumann sense of alternating projection may be applied toconvex sets that are not subspaces, although convergence is not necessarilyto the unique minimum-distance projection on the intersection. Figure 164illustrates one application where convergence is reasonably geometric and theresult is the unique minimum-distance projection. Figure 165, in contrast,demonstrates convergence in one iteration to a fixed point (of the projectionproduct)E.18 in the intersection of two halfspaces; a.k.a, feasibility problem.It was Dykstra who in 1983 [124] (§E.10.3) first solved this projectionproblem.

E.10.0.3 the bullets

Alternating projection has, therefore, various meaning dependent on theapplication or field of study; it may be interpreted to be: a distance problem,a feasibility problem (von Neumann), or a projection problem (Dykstra): Distance. Figure 166a-b. Find a unique point of projection P1b∈C1

that attains the distance between any two closed convex sets C1 and C2 ;

‖P1b − b‖ = dist(C1 , C2) , infz∈C2

‖P1z − z‖ (2013) Feasibility. Figure 166c,⋂ Ck 6= ∅ . Given a number L of indexed

closed convex sets Ck⊂ Rn, find any fixed point in their intersectionby iterating (i) a projection product starting from b ;

( ∞∏

i=1

L∏

k=1

Pk

)

b ∈L⋂

k=1

Ck (2014) Optimization. Figure 166c,⋂ Ck 6= ∅ . Given a number of indexed

closed convex sets Ck⊂ Rn, uniquely project a given point b on⋂ Ck ;

‖Pb − b‖ = infx∈⋂ Ck

‖x − b‖ (2015)

E.18Fixed point of a mapping T : Rn→Rn is a point x whose image is identical underthe map; id est, Tx = x .

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E.10. ALTERNATING PROJECTION 739

C2

C2

C2

C1

C1

C1

Pb

P1b

P1P2b

y | (b −P1b)T(y −P1b)=0

y

(a)

(b)

(c)

b

b

a

Figure 166:(a) (distance) Intersection of two convex sets in R2 is empty. Method ofalternating projection would be applied to find that point in C1 nearest C2 .(b) (distance) Given b ∈ C2 , then P1b ∈ C1 is nearest b iff(y−P1b)

T(b−P1b)≤ 0 ∀ y∈C1 by the unique minimum-distance projectiontheorem (§E.9.1.0.2). When P1b attains the distance between the two sets,hyperplane y | (b −P1b)

T(y −P1b)=0 separates C1 from C2 . [59, §2.5.1](c) (0 distance) Intersection is nonempty.(optimization) We may want the point Pb in

⋂ Ck nearest point b .(feasibility) We may instead be satisfied with a fixed point of the projectionproduct P1P2b in

⋂ Ck .

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740 APPENDIX E. PROJECTION

E.10.1 Distance and existence

Existence of a fixed point is established:

E.10.1.0.1 Theorem. Distance. [76]Given any two closed convex sets C1 and C2 in Rn, then P1b∈C1 is a fixedpoint of the projection product P1P2 if and only if P1b is a point of C1

nearest C2 . ⋄

Proof. (⇒) Given fixed point a =P1P2a∈C1 with b ,P2a∈C2 intandem so that a =P1b , then by the unique minimum-distance projectiontheorem (§E.9.1.0.2)

(b − a)T(u − a) ≤ 0 ∀u∈C1

(a − b)T(v − b) ≤ 0 ∀ v∈C2

⇔‖a − b‖ ≤ ‖u − v‖ ∀u∈C1 and ∀ v∈C2

(2016)

by Cauchy-Schwarz inequality |〈x, y〉| ≤ ‖x‖ ‖y‖ [303] (with equality iffx=κy where κ∈R [223, p.137]).(⇐) Suppose a∈C1 and ‖a − P2a‖ ≤ ‖u − P2u‖ ∀u∈C1 and we chooseu =P1P2a . Then

‖u − P2u‖ = ‖P1P2a − P2P1P2a‖ ≤ ‖a − P2a‖ ⇔ a = P1P2a (2017)

Thus a = P1b (with b =P2a∈C2) is a fixed point in C1 of the projectionproduct P1P2 .E.19 ¨

E.10.2 Feasibility and convergence

The set of all fixed points of any nonexpansive mapping is a closed convexset. [152, lem.3.4] [28, §1] The projection product P1P2 is nonexpansive byTheorem E.9.3.0.1 because, for any vectors x, a∈Rn

‖P1P2 x − P1P2 a‖ ≤ ‖P2x − P2a‖ ≤ ‖x − a‖ (2018)

If the intersection of two closed convex sets C1 ∩ C2 is empty, then the iteratesconverge to a point of minimum distance, a fixed point of the projectionproduct. Otherwise, convergence is to some fixed point in their intersection (a

E.19Point b=P2a can be shown, similarly, to be a fixed point of product P2P1 .

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E.10. ALTERNATING PROJECTION 741

feasible solution) whose existence is guaranteed by virtue of the fact that eachand every point in the convex intersection is in one-to-one correspondencewith fixed points of the nonexpansive projection product.

Bauschke & Borwein [28, §2] argue that any sequence monotonic in thesense of Fejer is convergent:E.20

E.10.2.0.1 Definition. Fejer monotonicity. [266]Given closed convex set C 6= ∅ , then a sequence xi∈ Rn, i=0, 1, 2 . . . , ismonotonic in the sense of Fejer with respect to C iff

‖xi+1 − c‖ ≤ ‖xi − c‖ for all i≥0 and each and every c ∈ C (2019)

Given x0 , b , if we express each iteration of alternating projection by

xi+1 = P1P2 xi , i=0, 1, 2 . . . (2020)

and define any fixed point a =P1P2 a , then sequence xi is Fejer monotonewith respect to fixed point a because

‖P1P2 xi − a‖ ≤ ‖xi − a‖ ∀ i ≥ 0 (2021)

by nonexpansivity. The nonincreasing sequence ‖P1P2 xi − a‖ is boundedbelow hence convergent because any bounded monotonic sequence in Ris convergent; [254, §1.2] [41, §1.1] P1P2 xi+1 = P1P2 xi = xi+1 . Sequencexi therefore converges to some fixed point. If the intersection C1 ∩ C2

is nonempty, convergence is to some point there by the distance theorem.Otherwise, xi converges to a point in C1 of minimum distance to C2 .

E.10.2.0.2 Example. Hyperplane/orthant intersection.Find a feasible solution (2014) belonging to the nonempty intersection of twoconvex sets: given A∈Rm×n, β∈R(A)

C1 ∩ C2 = Rn+∩ A = y | y º 0 ∩ y | Ay = β ⊂ Rn (2022)

the nonnegative orthant with affine subset A an intersection of hyperplanes.Projection of an iterate xi∈Rn on A is calculated

P2 xi = xi − AT(AAT)−1(Axi − β) (1915)

E.20Other authors prove convergence by different means; e.g., [171] [61].

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742 APPENDIX E. PROJECTION

C2 = A = y | [ 1 1 ] y = 1

C1 = R2

+

b

θ

Pb = (∞∏

j=1

2∏

k=1

Pk)b

y1

y2

Figure 167: From Example E.10.2.0.2 in R2, showing von Neumann-stylealternating projection to find feasible solution belonging to intersection ofnonnegative orthant with hyperplane A . Point Pb lies at intersection ofhyperplane with ordinate axis. In this particular example, feasible solutionfound is coincidentally optimal. Rate of convergence depends upon angle θ ;as it becomes more acute, convergence slows. [171, §3]

0 5 10 15 20 25 30 35 40 450

2

4

6

8

10

12

14

16

18

20

i

xi − (∞∏

j=1

2∏

k=1

Pk)b

Figure 168: Example E.10.2.0.2 in R1000 ; geometric convergence of iteratesin norm.

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E.10. ALTERNATING PROJECTION 743

while, thereafter, projection of the result on the orthant is simply

xi+1 = P1P2 xi = max0, P2 xi (2023)

where the maximum is entrywise (§E.9.2.2.3).One realization of this problem in R2 is illustrated in Figure 167: For

A = [ 1 1 ] , β =1, and x0 = b = [−3 1/2 ]T, iterates converge to a feasiblesolution Pb = [ 0 1 ]T.

To give a more palpable sense of convergence in higher dimension, wedo this example again but now we compute an alternating projection forthe case A∈R400×1000, β∈R400, and b∈R1000, all of whose entries areindependently and randomly set to a uniformly distributed real number inthe interval [−1, 1] . Convergence is illustrated in Figure 168. 2

This application of alternating projection to feasibility is extensible toany finite number of closed convex sets.

E.10.2.0.3 Example. Under- and over-projection. [56, §3]Consider the following variation of alternating projection: We begin withsome point x0∈ Rn then project that point on convex set C and thenproject that same point x0 on convex set D . To the first iterate we assignx1 = (PC(x0) + PD(x0))

12. More generally,

xi+1 = (PC(xi) + PD(xi))1

2, i=0, 1, 2 . . . (2024)

Because the Cartesian product of convex sets remains convex, (§2.1.8) wecan reformulate this problem.

Consider the convex set

S ,

[

CD

]

(2025)

representing Cartesian product C ×D . Now, those two projections PC andPD are equivalent to one projection on the Cartesian product; id est,

PS

([

xi

xi

])

=

[

PC(xi)PD(xi)

]

(2026)

Define the subspace

R ,

v ∈[

Rn

Rn

] ∣

[ I −I ] v = 0

(2027)

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744 APPENDIX E. PROJECTION

By the results in Example E.5.0.0.6

PRS

([

xi

xi

])

= PR

([

PC(xi)PD(xi)

])

=

[

PC(xi) + PD(xi)PC(xi) + PD(xi)

]

1

2(2028)

This means the proposed variation of alternating projection is equivalent toan alternation of projection on convex sets S and R . If S and R intersect,these iterations will converge to a point in their intersection; hence, to a pointin the intersection of C and D .

We need not apply equal weighting to the projections, as supposed in(2024). In that case, definition of R would change accordingly. 2

E.10.2.1 Relative measure of convergence

Inspired by Fejer monotonicity, the alternating projection algorithm fromthe example of convergence illustrated by Figure 168 employs a redundant

sequence: The first sequence (indexed by j) estimates point (∞∏

j=1

L∏

k=1

Pk)b in

the presumably nonempty intersection of L convex sets, then the quantity∥

xi −( ∞

j=1

L∏

k=1

Pk

)

b

(2029)

in second sequence xi is observed per iteration i for convergence. A prioriknowledge of a feasible solution (2014) is both impractical and antithetical.We need another measure:

Nonexpansivity implies∥

(

L∏

ℓ=1

Pℓ

)

xk,i−1 −(

L∏

ℓ=1

Pℓ

)

xki

= ‖xki − xk,i+1‖ ≤ ‖xk,i−1 − xki‖ (2030)

wherexki , Pkxk+1,i ∈ Rn, xL+1,i , x1,i−1 (2031)

xki represents unique minimum-distance projection of xk+1,i on convex set kat iteration i . So a good convergence measure is total monotonic sequence

εi ,

L∑

k=1

‖xki − xk,i+1‖ , i=0, 1, 2 . . . (2032)

where limi→∞

εi = 0 whether or not the intersection is nonempty.

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E.10. ALTERNATING PROJECTION 745

E.10.2.1.1 Example. Affine subset ∩ positive semidefinite cone.Consider the problem of finding X∈ Sn that satisfies

X º 0 , 〈Aj , X 〉 = bj , j =1 . . . m (2033)

given nonzero Aj ∈ Sn and real bj . Here we take C1 to be the positivesemidefinite cone Sn

+ while C2 is the affine subset of Sn

C2 = A , X | tr(Aj X)= bj , j =1 . . . m ⊆ Sn

= X |

svec(A1)T

...svec(Am)T

svec X = b

, X | A svec X = b

(2034)

where b = [bj]∈Rm, A∈ Rm×n(n+1)/2, and symmetric vectorization svec isdefined by (56). Projection of iterate Xi∈ Sn on A is: (§E.5.0.0.6)

P2 svec Xi = svec Xi − A†(A svec Xi − b) (2035)

Euclidean distance from Xi to A is therefore

dist(Xi , A) = ‖Xi − P2Xi‖F = ‖A†(A svec Xi − b)‖2 (2036)

Projection of P2Xi ,∑

j λj qjqTj on the positive semidefinite cone (§7.1.2) is

found from its eigenvalue decomposition (§A.5.1);

P1P2Xi =n

j=1

max0 , λj qjqTj (2037)

Distance from P2Xi to the positive semidefinite cone is therefore

dist(P2Xi , Sn+) = ‖P2Xi − P1P2Xi‖F =

n∑

j=1

(min0 , λj)2 (2038)

When the intersection is empty A ∩ Sn+ =∅ , the iterates converge to that

positive semidefinite matrix closest to A in the Euclidean sense. Otherwise,convergence is to some point in the nonempty intersection.

Barvinok (§2.9.3.0.1) shows that if a solution to (2033) exists, then thereexists an X∈A ∩ Sn

+ such that

rank X ≤⌊√

8m + 1 − 1

2

(252)

2

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746 APPENDIX E. PROJECTION

E.10.2.1.2 Example. Semidefinite matrix completion.Continuing Example E.10.2.1.1: When m≤n(n + 1)/2 and the Aj matricesare distinct members of the standard orthonormal basis Eℓq∈ Sn (59)

Aj ∈ Sn, j =1 . . . m ⊆ Eℓq =

eℓeTℓ , ℓ = q = 1 . . . n

1√2(eℓe

Tq + eqe

Tℓ ) , 1 ≤ ℓ < q ≤ n

(2039)

and when the constants bj are set to constrained entries of variableX , [Xℓq]∈ Sn

bj , j =1 . . . m ⊆

Xℓq , ℓ = q = 1 . . . n

Xℓq

√2 , 1 ≤ ℓ < q ≤ n

= 〈X,Eℓq〉 (2040)

then the equality constraints in (2033) fix individual entries of X∈ Sn. Thusthe feasibility problem becomes a positive semidefinite matrix completionproblem. Projection of iterate Xi∈ Sn on A simplifies to (confer (2035))

P2 svec Xi = svec Xi − AT(A svec Xi − b) (2041)

From this we can see that orthogonal projection is achieved simply by settingcorresponding entries of P2Xi to the known entries of X , while the entriesof P2Xi remaining are set to corresponding entries of the current iterate Xi .

Using this technique, we find a positive semidefinite completion for

4 3 ? 23 4 3 ?? 3 4 32 ? 3 4

(2042)

Initializing the unknown entries to 0, they all converge geometrically to1.5858 (rounded) after about 42 iterations.

Laurent gives a problem for which no positive semidefinite completionexists: [232]

1 1 ? 01 1 1 ?? 1 1 10 ? 1 1

(2043)

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E.10. ALTERNATING PROJECTION 747

0 2 4 6 8 10 12 14 16 1810

−1

100

i

dist(P2Xi , Sn+)

Figure 169: Distance (confer (2038)) between positive semidefinite cone anditerate (2041) in affine subset A (2034) for Laurent’s completion problem;initially, decreasing geometrically.

Initializing unknowns to 0, by alternating projection we find the constrainedmatrix closest to the positive semidefinite cone,

1 1 0.5454 01 1 1 0.5454

0.5454 1 1 10 0.5454 1 1

(2044)

and we find the positive semidefinite matrix closest to the affine subset A(2034):

1.0521 0.9409 0.5454 0.02920.9409 1.0980 0.9451 0.54540.5454 0.9451 1.0980 0.94090.0292 0.5454 0.9409 1.0521

(2045)

These matrices (2044) and (2045) attain the Euclidean distance dist(A , Sn+).

Convergence is illustrated in Figure 169. 2

Page 748: Convex Optimization & Euclidean Distance Geometry

748 APPENDIX E. PROJECTION

H1

H2

b

x12

x21

x11

x22

H1 ∩ H2

Figure 170: H1 and H2 are the same halfspaces as in Figure 165.Dykstra’s alternating projection algorithm generates the alternationsb , x21 , x11 , x22 , x12 , x12 . . . x12 . The path illustrated from b to x12 in R2

terminates at the desired result, Pb in Figure 165. The yki correspondto the first two difference vectors drawn (in the first iteration i=1), thenoscillate between zero and a negative vector thereafter. These alternationsare not so robust in presence of noise as for the example in Figure 164.

E.10.3 Optimization and projection

Unique projection on the nonempty intersection of arbitrary convex sets tofind the closest point therein is a convex optimization problem. The firstsuccessful application of alternating projection to this problem is attributedto Dykstra [124] [60] who in 1983 provided an elegant algorithm that prevailstoday. In 1988, Han [174] rediscovered the algorithm and provided aprimal−dual convergence proof. A synopsis of the history of alternatingprojectionE.21 can be found in [62] where it becomes apparent that Dykstra’swork is seminal.

E.21For a synopsis of alternating projection applied to distance geometry, see [350, §3.1].

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E.10. ALTERNATING PROJECTION 749

E.10.3.1 Dykstra’s algorithm

Assume we are given some point b ∈ Rn and closed convex setsCk⊂ Rn | k=1 . . . L. Let xki∈ Rn and yki∈ Rn respectively denote aprimal and dual vector (whose meaning can be deduced from Figure 170and Figure 171) associated with set k at iteration i . Initialize

yk0 = 0 ∀ k=1 . . . L and x1,0 = b (2046)

Denoting by Pkt the unique minimum-distance projection of t on Ck , andfor convenience xL+1,i = x1,i−1 (2031), iterate x1i calculation proceeds:E.22

for i=1, 2, . . . until convergence for k=L . . . 1

t = xk+1,i − yk,i−1

xki = Pktyki = Pkt − t

(2047)

Assuming a nonempty intersection, then the iterates converge to the uniqueminimum-distance projection of point b on that intersection; [106, §9.24]

Pb = limi→∞

x1i (2048)

In the case all the Ck are affine, then calculation of yki is superfluousand the algorithm becomes identical to alternating projection. [106, §9.26][144, §1] Dykstra’s algorithm is so simple, elegant, and represents such a tinyincrement in computational intensity over alternating projection, it is nearlyalways arguably cost effective.

E.10.3.2 Normal cone

Glunt [151, §4] observes that the overall effect of Dykstra’s iterativeprocedure is to drive t toward the translated normal cone to

⋂ Ck atthe solution Pb (translated to Pb). The normal cone gets its namefrom its graphical construction; which is, loosely speaking, to draw theoutward-normals at Pb (Definition E.9.1.0.1) to all the convex sets Ck

touching Pb . The relative interior of the normal cone subtends these normalvectors.E.22We reverse order of projection (k=L . . . 1) in the algorithm for continuity of exposition.

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750 APPENDIX E. PROJECTION

0

H1

H2

Pb

b

K⊥H1∩H2

(0)

K⊥H1∩H2

(Pb) + Pb

K , H1 ∩ H2

Figure 171: Two examples (truncated): Normal cone to H1 ∩ H2 at theorigin, and at point Pb on the boundary. H1 and H2 are the same halfspacesfrom Figure 170. The normal cone at the origin K⊥

H1∩H2(0) is simply −K∗

.

E.10.3.2.1 Definition. Normal cone. [264] [41, p.261] [196, §A.5.2][53, §2.1] [302, §3] [303, p.15] The normal cone to any set S⊆Rn at anyparticular point a∈Rn is defined as the closed cone

K⊥S (a) , z∈Rn | zT(y−a)≤ 0 ∀ y∈S = −(S − a)∗ (2049)

an intersection of halfspaces about the origin in Rn, hence convex regardlessof convexity of S ; it is the negative dual cone to the translate S − a ; theset of all vectors normal to S at a .

Examples of normal cone construction are illustrated in Figure 66 andFigure 171: The normal cone at the origin in Figure 171 is the vector sum(§2.1.8) of two normal cones; [53, §3.3, exer.10] for H1∩ intH2 6= ∅

K⊥H1∩H2

(0) = K⊥H1

(0) + K⊥H2

(0) (2050)

This formula applies more generally to other points in the intersection.The normal cone to any affine set A at α∈A , for example, is the

orthogonal complement of A− α . Projection of any point in the translatednormal cone K⊥

C (a∈C) + a on convex set C is identical to a ; in other words,

Page 751: Convex Optimization & Euclidean Distance Geometry

E.10. ALTERNATING PROJECTION 751

E3

K⊥E3(11T) + 11T

Figure 172: A few renderings (including next page) of normal cone K⊥E3 to

elliptope E3 (Figure 128) at point 11T, projected on R3. In [233, fig.2],normal cone is claimed circular in this dimension (severe numerical artifactscorrupt boundary and make interior corporeal, drawn truncated).

Page 752: Convex Optimization & Euclidean Distance Geometry

752 APPENDIX E. PROJECTION

point a is that point in C closest to any point belonging to the translatednormal cone K⊥

C (a) + a ; e.g., Theorem E.4.0.0.1.When set S is a convex cone K , then the normal cone to K at the origin

K⊥K(0) = −K∗

(2051)

is the negative dual cone. Any point belonging to −K∗, projected on K ,

projects on the origin. More generally, [106, §4.5]

K⊥K(a) = −(K − a)∗ (2052)

K⊥K(a∈K) = −K∗∩ a⊥ (2053)

The normal cone to⋂ Ck at Pb in Figure 165 is ray ξ(b−Pb) | ξ≥0

illustrated in Figure 171. Applying Dykstra’s algorithm to that example,convergence to the desired result is achieved in two iterations as illustrated inFigure 170. Yet applying Dykstra’s algorithm to the example in Figure 164does not improve rate of convergence, unfortunately, because the givenpoint b and all the alternating projections already belong to the translatednormal cone at the vertex of intersection.

E.10.3.3 speculation

Dykstra’s algorithm always converges at least as quickly as classicalalternating projection, never slower [106], and it succeeds where alternating

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E.10. ALTERNATING PROJECTION 753

projection fails. Rate of convergence is wholly dependent on particulargeometry of a given problem. From these few examples we surmise, uniqueminimum-distance projection on blunt (not sharp or acute, informally)full-dimensional polyhedral cones may be found by Dykstra’s algorithm infew iterations. But total number of alternating projections, constitutingthose iterations, can never be less than number of convex sets.

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754 APPENDIX E. PROJECTION

Page 755: Convex Optimization & Euclidean Distance Geometry

Appendix F

Notation and a few definitions

b (italic abcdefghijklmnopqrstuvwxyz) column vector, scalar, logicalcondition

bi ith entry of vector b=[bi , i=1 . . . n] or ith b vector from a set or listbj , j =1 . . . n or ith iterate of vector b

bi:j or b(i :j) , truncated vector comprising ith through j th entry of vector b

bk(i :j) or bi:j,k , truncated vector comprising ith through j th entry of vector bk

bT vector transpose

bH Hermitian (complex conjugate) transpose b∗T

AT Matrix transpose. Regarding A as a linear operator, AT is its adjoint.

A−2T matrix transpose of squared inverse

AT1 first of various transpositions of a cubix or quartix A (p.666, p.671)

A matrix, scalar, or logical condition(italic ABCDEFGHIJKLMNOPQRSTUV WXY Z)

skinny a skinny matrix; meaning, more rows than columns:

. When

there are more equations than unknowns, we say that the system Ax = bis overdetermined. [155, §5.3]© 2001 Jon Dattorro. co&edg version 2010.02.04. All rights reserved.

citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.02.04.

755

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756 APPENDIX F. NOTATION AND A FEW DEFINITIONS

fat a fat matrix; meaning, more columns than rows:[ ]

underdetermined

A some set (calligraphic ABCDEFGHIJKLMNOPQRST UVWXYZ)

F(C ∋A) smallest face (162) that contains element A of set C

G(K) generators (§2.13.4.2.1) of set K ; any collection of points and directionswhose hull constructs K

Lνν level set (533)

Lν sublevel set (537)

Lν superlevel set (620)

A−1 inverse of matrix A

A† Moore-Penrose pseudoinverse of matrix A

√positive square root

ℓ√

positive ℓ th root

A1/2 and√

A A1/2 is any matrix such that A1/2A1/2 =A .For A ∈ Sn

+ ,√

A ∈ Sn+ is unique and

√A√

A =A . [53, §1.2] (§A.5.1.3)

D = [√

dij ] . (1323) Hadamard positive square root: D = √

D √

D

A Euler Fraktur ABC D EFGH I J KL MNOPQRS T U VWXYZ

L Lagrangian (485)

E member of elliptope Et (1073) parametrized by scalar t

E elliptope (1052)

E elementary matrix

Eij member of standard orthonormal basis for symmetric (59) or symmetrichollow (75) matrices

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757

Aij or A(i , j) , ij th entry of matrix A =

1 2 3

· · ·· · ·· · ·

1

2

3

or rank-one matrix aiaTj (§4.7)

A(i , j) A is a function of i and j

Ai ith matrix from a set or ith principal submatrix or ith iterate of A

A(i , :) ith row of matrix A

A(: , j) j th column of matrix A [155, §1.1.8]

Ai:j,k:ℓ or A(i :j , k :ℓ) , submatrix taken from ith through j th row andkth through ℓth column

e.g. exempli gratia, from the Latin meaning for sake of example

videlicet from the Latin meaning it is permitted to see

no. number, from the Latin numero

a.i. affinely independent (§2.4.2.3)

c.i. conically independent (§2.10)

l.i. linearly independent

w.r.t with respect to

a.k.a also known as

re real part

im imaginary part

ı or √−1

⊂ ⊃ ∩ ∪ standard set theory, subset, superset, intersection, union

∈ membership, element belongs to, or element is a member of

∋ membership, contains as in C ∋ y (C contains element y)

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758 APPENDIX F. NOTATION AND A FEW DEFINITIONS

Ä such that

∃ there exists

∴ therefore

∀ for all, or over all

∝ proportional to

∞ infinity

≡ equivalent to

, defined equal to, equal by definition

≈ approximately equal to

≃ isomorphic to or with

∼= congruent to or with

Hadamard quotient as in, for x, y∈Rn,x

y, [ xi/yi , i=1 . . . n ]∈ Rn

Hadamard product of matrices: x y , [ xi yi , i=1 . . . n ]∈ Rn

⊗ Kronecker product of matrices (§D.1.2.1)

⊕ vector sum of sets X =Y ⊕ Z where every element x∈X hasunique expression x = y + z where y∈Y and z∈Z ; [303, p.19] thensummands are algebraic complements. X =Y ⊕ Z ⇒ X =Y + Z .Now assume Y and Z are nontrivial subspaces. X =Y + Z ⇒X =Y ⊕ Z ⇔ Y ∩ Z=0 [304, §1.2] [106, §5.8]. Each element froma vector sum (+) of subspaces has unique expression (⊕) when a basisfrom each subspace is linearly independent of bases from all the othersubspaces.

⊖ likewise, the vector difference of sets

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759

⊞ orthogonal vector sum of sets X =Y ⊞ Z where every element x∈Xhas unique orthogonal expression x = y + z where y∈Y , z∈Z ,and y ⊥ z . [327, p.51] X =Y ⊞ Z ⇒ X =Y + Z . If Z⊆Y⊥ thenX =Y ⊞ Z ⇔ X =Y ⊕ Z . [106, §5.8] If Z= Y⊥ then summands areorthogonal complements.

± plus or minus or plus and minus

⊥ as in A⊥B meaning A is orthogonal to B (and vice versa), whereA and B are sets, vectors, or matrices. When A and B arevectors (or matrices under Frobenius’ norm), A ⊥ B ⇔ 〈A ,B〉= 0⇔ ‖A + B‖2 = ‖A‖2 + ‖B‖2

\ as in \A means logical not A , or relative complement of set A ;id est, \A = x /∈A ; e.g., B\A , x∈B | x /∈A ≡ B ∩\A

⇒ or ⇐ sufficient or necessary, implies, or is implied by ; e.g.,A is sufficient: A ⇒ B , A is necessary: A ⇐ B ,A ⇒ B ⇔ \A ⇐ \B , A ⇐ B ⇔ \A ⇒ \B ,if A then B , if B then A ,A only if B . B only if A .

⇔ if and only if (iff) or corresponds with or necessary and sufficientor logical equivalence

is as in A is B means A ⇒ B ; conventional usage of English languageimposed by logicians

; or : insufficient or unnecessary, does not imply, or is not implied by ; e.g.,A ; B ⇔ \A : \B . A : B ⇔ \A ; \B .

← is replaced with; substitution, assignment

→ goes to, or approaches, or maps to

t → 0+ t goes to 0 from above; meaning, from the positive [196, p.2]

.... . . · · · as in 1 · · · 1 meaning continuation, e.g., a sequence of ones in a row

. . . as in i=1 . . . N meaning, i is a sequence of successive integersbeginning with 1 and ending with N

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760 APPENDIX F. NOTATION AND A FEW DEFINITIONS

: as in f : Rn→ Rm meaning f is a mapping,or sequence of successive integers specified by bounds as in i :j(if j < i then sequence is descending)

f : M→R meaning f is a mapping from ambient space M to ambient R , notnecessarily denoting either domain or range

| as in f(x) | x∈ C means with the condition(s) or such that orevaluated for, or as in f(x) | x∈ C means evaluated for each andevery x belonging to set C

g|xpexpression g evaluated at xp

A , B as in, for example, A , B ∈ SN means A ∈ SN and B ∈ SN

(A , B) open interval between A and B in R ,or variable pair perhaps of disparate dimension

[A , B ] closed interval or line segment between A and B in R

( ) hierarchal, parenthetical, optional

curly braces denote a set or list, e.g., Xa | aº0 the set comprisingXa evaluated for each and every aº0 where membership of a tosome space is implicit, a union

〈 〉 angle brackets denote vector inner-product (33) (38)

[ ] matrix or vector, or quote insertion, or citation

[dij] matrix whose ij th entry is dij

[xi] vector whose ith entry is xi

xp particular value of x

x0 particular instance of x , or initial value of a sequence xi

x1 first entry of vector x , or first element of a set or list xi

xε extreme point

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761

x+ vector x whose negative entries are replaced with 0 ; x+ = 12(x + |x|)

(513) or clipped vector x or nonnegative part of x

x− x− , 12(x − |x|) or nonpositive part of x = x+ + x−

x known data

x⋆ optimal value of variable x

x∗ complex conjugate or dual variable or extreme direction of dual cone

f ∗ convex conjugate function f ∗(s)= sup〈s , x〉 − f(x) | x∈dom f

PCx or Px projection of point x on set C , P is operator or idempotent matrix

Pkx projection of point x on set Ck or on range of implicit vector

δ(A) (§A.1) vector made from the main diagonal of A if A is a matrix;otherwise, diagonal matrix made from vector A

δ2(A) ≡ δ(δ(A)). For vector or diagonal matrix Λ , δ2(Λ) = Λ

δ(A)2 = δ(A)δ(A) where A is a vector

λi(X) ith entry of vector λ is function of X

λ(X)i ith entry of vector-valued function of X

λ(A) vector of eigenvalues of matrix A , (1437) typically arranged innonincreasing order

σ(A) vector of singular values of matrix A (always arranged in nonincreasingorder), or support function in direction A

Σ diagonal matrix of singular values, not necessarily square

sum

π(γ) nonlinear permutation operator (or presorting function) arrangesvector γ into nonincreasing order (§7.1.3)

Ξ permutation matrix

Π doublet or permutation operator or matrix

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762 APPENDIX F. NOTATION AND A FEW DEFINITIONS

product

ψ(Z) signum-like step function that returns a scalar for matrix argument(684), it returns a vector for vector argument (1545)

D symmetric hollow matrix of distance-square,or Euclidean distance matrix

D Euclidean distance matrix operator

DT(X) adjoint operator

D(X)T transpose of D(X)

D−1(X) inverse operator

D(X)−1 inverse of D(X)

D⋆ optimal value of variable D

D∗ dual to variable D

D polar variable D

∂ partial derivative or partial differential or matrix of distance-squaresquared (1363) or boundary of set K as in ∂K

dij (absolute) distance scalar

dij distance-square scalar, EDM entry

V geometric centering operator, V(D)=−V DV 12

(972)

VN VN (D)=−V TNDVN (986)

V N×N symmetric elementary, auxiliary, projector, geometric centeringmatrix, R(V )=N (1T) , N (V )=R(1) , V 2 =V (§B.4.1)

VN N×N−1 Schoenberg auxiliary matrix, R(VN )=N (1T) ,N (V T

N )=R(1) (§B.4.2)

VX VXV TX ≡ V TXTXV (1166)

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763

X point list ((76) having cardinality N) arranged columnar in Rn×N ,or set of generators, or extreme directions, or matrix variable

G Gram matrix XTX

r affine dimension

k number of conically independent generators or raw-data domain ofMagnetic Resonance Imaging machine as in k-space

n Euclidean (ambient spatial) dimension of list X∈ Rn×N , or integer

N cardinality of list X∈ Rn×N , or integer

on function f(x) on A means A is dom f , or projection of x on Ameans A is Euclidean body on which projection of x is made

onto function f(x) maps onto M means f over its domain is a surjectionwith respect to M

one-to-one injective map or unique correspondence between sets

epi function epigraph

dom function domain

Rf function range

R(A) the subspace: range of A , or span basisR(A) ; R(A) ⊥ N (AT)

span as in span A = R(A) = Ax | x∈Rn when A is a matrix

basisR(A) columnar basis for range of A , or a minimal set constituting generatorsfor the vertex-description of R(A) , or a linearly independent set ofvectors spanning R(A)

N (A) the subspace: nullspace of A ; N (A) ⊥ R(AT)

Rn Euclidean n-dimensional real vector space (nonnegative integer n).R0 = 0, R = R1 or vector space of unspecified dimension.

Rm×n Euclidean vector space of m by n dimensional real matrices

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764 APPENDIX F. NOTATION AND A FEW DEFINITIONS

× Cartesian product. Rm×n−m , Rm×(n−m). K1×K2 =

[

K1

K2

]

[

Rm

Rn

]

Rm× Rn = Rm+n

Cn or Cn×n Euclidean complex vector space of respective dimension n and n×n

Rn+ or Rn×n

+ nonnegative orthant in Euclidean vector space of respective dimensionn and n×n

Rn− or Rn×n

− nonpositive orthant in Euclidean vector space of respective dimensionn and n×n

Sn subspace of real symmetric n×n matrices; the symmetric matrixsubspace. S = S1 or symmetric subspace of unspecified dimension.

Sn⊥ orthogonal complement of Sn in Rn×n, the antisymmetric matrices

Sn+ convex cone comprising all (real) symmetric positive semidefinite n×n

matrices, the positive semidefinite cone

int Sn+ interior of convex cone comprising all (real) symmetric positive

semidefinite n×n matrices; id est, positive definite matrices

Sn+(ρ) = X∈ Sn

+ | rank X ≥ ρ (240) convex set of all positive semidefiniten×n symmetric matrices whose rank equals or exceeds ρ

EDMN cone of N×N Euclidean distance matrices in the symmetric hollowsubspace

EDMN nonconvex cone of N×N Euclidean absolute distance matrices in thesymmetric hollow subspace (§6.3)

PSD positive semidefinite

SDP semidefinite program

SVD singular value decomposition

SNR signal to noise ratio

dB decibel

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765

EDM Euclidean distance matrix

Sn1 subspace comprising all symmetric n×n matrices having all zeros in

first row and column (1964)

Snh subspace comprising all symmetric hollow n×n matrices (0 main

diagonal), the symmetric hollow subspace (66)

Sn⊥h orthogonal complement of Sn

h in Sn (67), the set of all diagonal matrices

Snc subspace comprising all geometrically centered symmetric n×n

matrices; geometric center subspace SNc = Y ∈ SN | Y 1=0 (1960)

Sn⊥c orthogonal complement of Sn

c in Sn (1962)

Rm×nc subspace comprising all geometrically centered m×n matrices

X⊥ basisN (XT)

x⊥ N (xT) , y∈ Rn | xTy = 0

R(P )⊥ N (PT)

R⊥ = y∈Rn | 〈x, y〉=0 ∀x∈R (351);for R a subspace, orthogonal complement of R in Rn

K⊥ normal cone (2049)

K cone

K∗dual cone

Kpolar cone; K∗

= −K

KM+ monotone nonnegative cone

KM monotone cone

Kλ spectral cone

K∗λδ cone of majorization

H halfspace

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766 APPENDIX F. NOTATION AND A FEW DEFINITIONS

H− halfspace described using an outward-normal (105) to the hyperplanepartially bounding it

H+ halfspace described using an inward-normal (106) to the hyperplanepartially bounding it

∂H hyperplane; id est, partial boundary of halfspace

∂H supporting hyperplane

∂H− a supporting hyperplane having outward-normal with respect to set itsupports

∂H+ a supporting hyperplane having inward-normal with respect to set itsupports

d vector of distance-square

dij lower bound on distance-square dij

dij upper bound on distance-square dij

AB closed line segment between points A and B

AB matrix multiplication of A and B

C closure of set C

decomposition orthonormal (1875) page 698, biorthogonal (1852) page 692

expansion orthogonal (1885) page 701, biorthogonal (381) page 183

vector column vector in Rn

entry scalar element or real variable constituting a vector or matrix

cubix member of RM×N×L

quartix member of RM×N×L×K

feasible set most simply, the set of all variable values satisfying all constraints ofan optimization problem

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767

solution set most simply, the set of all optimal solutions to an optimization problem;a subset of the feasible set and not necessarily a single point

optimal solution solution to an optimization problem

feasible solution satisfies (“subject to”) the constraints of an optimization problem, mayor may not be optimal

natural order with reference to stacking columns in a vectorization means a vectormade from superposing column 1 on top of column 2 then superposingthe result on column 3 and so on; as in a vector made from entries of themain diagonal δ(A) means taken from left to right and top to bottom

operator mapping to a vector space (a multidimensional function)

tight with reference to a bound means a bound that can be met,with reference to an inequality means equality is achievable

g′ first derivative of possibly multidimensional function with respect toreal argument

g′′ second derivative with respect to real argument

→Y

dg first directional derivative of possibly multidimensional function g indirection Y∈RK×L (maintains dimensions of g)

→Y

dg2 second directional derivative of g in direction Y

∇ gradient from calculus, ∇f is shorthand for ∇xf(x). ∇f(y) means∇yf(y) or gradient ∇xf(y) of f(x) with respect to x evaluated at y

∇2 second-order gradient

∆ distance scalar (Figure 25), or first-order difference matrix (827),or infinitesimal difference operator (§D.1.4)

ijk triangle made by vertices i , j , and k

I Roman numeral one

I identity matrix

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768 APPENDIX F. NOTATION AND A FEW DEFINITIONS

I index set, a discrete set of indices

∅ empty set, an implicit member of every set

0 real zero

0 origin or vector or matrix of zeros

O sort-index matrix

O order of magnitude information required, or computational intensity :O(N ) is first order, O(N 2) is second, and so on

1 real one

1 vector of ones

ei vector whose ith entry is 1 (otherwise 0),ith member of the standard basis for Rn (60)

max maximum [196, §0.1.1] or largest element of a totally ordered set

maximizex

find the maximum of a function with respect to variable x

arg argument of operator or function, or variable of optimization

supX supremum of totally ordered set X , least upper bound, may or maynot belong to set [196, §0.1.1]

arg sup f(x) argument x at supremum of function f ; not necessarily unique or amember of function domain

subject to specifies constraints of an optimization problem

min minimum [196, §0.1.1] or smallest element of a totally ordered set

minimizex

find the function minimum with respect to variable x

inf X infimum of totally ordered set X , greatest lower bound, may or maynot belong to set [196, §0.1.1]

arg inf f(x) argument x at infimum of function f ; not necessarily unique or amember of function domain

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769

iff if and only if , necessary and sufficient ; also the meaningindiscriminately attached to appearance of the word “if ” in thestatement of a mathematical definition, [125, p.106] [254, p.4] anesoteric practice worthy of abolition because of ambiguity thusconferred

rel relative

int interior

lim limit

sgn signum function or sign

round round to nearest integer

mod modulus function

tr matrix trace

rank as in rankA , rank of matrix A ; dimR(A)

dim dimension, dim Rn = n , dim(x∈Rn) = n , dimR(x∈Rn) = 1,dimR(A∈Rm×n)= rank(A)

aff affine hull

dim aff affine dimension

card cardinality, number of nonzero entries card x , ‖x‖0

or N is cardinality of list X∈ Rn×N (p.315)

conv convex hull (§2.3.2)

cone conic hull (§2.3.3)

cenv convex envelope (§7.2.2.1)

content of high-dimensional bounded polyhedron, volume in 3 dimensions, areain 2, and so on

cof matrix of cofactors corresponding to matrix argument

dist distance between point or set arguments; e.g., dist(x , B)

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770 APPENDIX F. NOTATION AND A FEW DEFINITIONS

vec columnar vectorization of m×n matrix, Euclidean dimension mn (37)

svec columnar vectorization of symmetric n×n matrix, Euclideandimension n(n + 1)/2 (56)

dvec columnar vectorization of symmetric hollow n×n matrix, Euclideandimension n(n − 1)/2 (73)

Á(x, y) angle between vectors x and y , or dihedral angle between affinesubsets

º generalized inequality; e.g., Aº 0 means: vector or matrix A must beexpressible in a biorthogonal expansion having nonnegative coordinateswith respect to extreme directions of some implicit pointed closedconvex cone K (§2.13.2.0.1, §2.13.7.1.1), or comparison to the originwith respect to some implicit pointed closed convex cone (2.7.2.2),or (when K= Sn

+) matrix A belongs to the positive semidefinite coneof symmetric matrices (§2.9.0.1), or (when K= Rn

+) vector A belongsto the nonnegative orthant (each vector entry is nonnegative, §2.3.1.1)

≻ strict generalized inequality, membership to cone interior

⊁ not positive definite

≥ scalar inequality, greater than or equal to; comparison of scalars,or entrywise comparison of vectors or matrices with respect to R+

nonnegative for α∈Rn, α º 0

> greater than

positive for α∈Rn, α ≻ 0 ; id est, no zero entries when with respect tononnegative orthant, no vector on boundary with respect to pointedclosed convex cone K

⌊ ⌋ floor function, ⌊x⌋ is greatest integer not exceeding x

| | entrywise absolute value of scalars, vectors, and matrices

log natural (or Napierian) logarithm

det matrix determinant

Page 771: Convex Optimization & Euclidean Distance Geometry

771

‖x‖ vector 2-norm or Euclidean norm ‖x‖2

‖x‖ℓ

= ℓ

n∑

j=1

|xj|ℓ vector ℓ-norm for ℓ ≥ 1 (convex)

,n∑

j=1

|xj|ℓ vector ℓ-norm for 0 ≤ ℓ < 1 (violates §3.3 no.3)

‖x‖∞ = max|xj| ∀ j infinity-norm

‖x‖22 = xTx = 〈x , x〉 Euclidean norm square

‖x‖1 = 1T|x| 1-norm, dual infinity-norm

‖x‖0 = 1T|x|0 (00 , 0) 0-norm (§4.5.1), cardinality of vector x (card x)

‖x‖nk

k-largest norm (§3.3.2.1)

‖X‖2 = sup‖a‖=1

‖Xa‖2 = σ1 =√

λ(XTX)1 (615) matrix 2-norm (spectral norm),

largest singular value, ‖Xa‖2 ≤ ‖X‖2‖a‖2 [155, p.56], ‖δ(x)‖2 = ‖x‖∞‖X‖∗2 = 1Tσ(X) nuclear norm, dual spectral norm

‖X‖ = ‖X‖F Frobenius’ matrix norm

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772 APPENDIX F. NOTATION AND A FEW DEFINITIONS

Page 773: Convex Optimization & Euclidean Distance Geometry

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802

Page 803: Convex Optimization & Euclidean Distance Geometry

Index

0-norm, 221, 225, 284, 325, 330, 337, 338,340, 360, 371, 771

Boolean, 2841-norm, 217–221, 223–225, 254, 343, 458,

562, 771ball, see ballBoolean, 284gradient, 225heuristic, 326, 327, 562nonnegative, 221, 222, 330, 337, 338,

3402-norm, 50, 213, 214, 217, 218, 344, 679, 771

ball, see ballBoolean, 284matrix, 70, 258, 542, 555, 612, 624, 631,

642, 732, 771dual, 643, 771inequality, 70, 771Schur-form, 258, 642

Ax = b , 49, 86, 219, 220, 225, 288, 297, 325,328, 330, 341–344, 378, 688, 689

Boolean, 284nonnegative, 88, 89, 221, 330, 337, 338,

340, 343V , 632δ , 581∞-norm, 217, 218, 224, 771

ball, see balldual, 771

λ , 582, 761π , 223, 325, 549, 553, 575, 761ψ , 293, 614σ , 582, 761√

EDM , 493≻ , 98º , 770

× , 47, 155, 743, 764k-largest norm, 224, 225, 328

gradient, 225, 226

active, 299adaptive, 366, 371, 374additivity, 102, 113, 596adjacency, 53adjoint

operator, 50, 51, 157, 527, 755, 762self, 399, 581

affine, 38, 229algebra, 38, 40, 62, 79, 93, 702boundary, 38, 46combination, 67, 72, 78, 143dimension, 23, 38, 61, 435, 437, 438,

447, 527complementary, 529cone, 134low, 411minimization, 560, 572Precis, 438reduction, 484spectral projection, 573

face, 93hull, see hullindependence, 71, 77, 78, 134, 138

preservation, 78interior, 40map, see transformationmembership relation, 216normal cone, 750set, 38, 61, 62

parallel, 38subset, 62, 72

hyperplane intersection, 86nullspace basis span, 87

803

Page 804: Convex Optimization & Euclidean Distance Geometry

804 INDEX

projector, 691, 702transformation, see transformation

algebraaffine, 38, 40, 62, 79, 93, 702arg, 329, 641, 642cone, 157face, 93function

linear, 212support, 232

fundamental theorem, 607linear, 85

infimum, 641, 642interior, 40, 46, 175linear, 85, 581projection, 702subspace, 36, 85

algebraiccomplement, 94, 148, 195, 696, 700multiplicity, 607

Alizadeh, 291alternation, 475alternative, see theoremambient, 519, 532

vector space, 36anchor, 307, 308, 407, 411angle, 84

acute, 72alternating projection, 742brackets, 50, 51constraint, 398, 399, 402, 418dihedral, 29, 84, 419, 481, 639EDM cone, 121halfspace, 73inequality, 390, 479

matrix, 423interpoint, 395obtuse, 72positive semidefinite cone, 121relative, 420, 422right, 123, 151, 173, 174torsion, 29, 418

antidiagonal, 427antisymmetry, 100, 596arg, 768

algebra, 329, 641, 642artificial intelligence, 25asymmetry, 35, 72axioms of the metric, 387axis of revolution, 122, 123, 639

ball1-norm, 219, 220

nonnegative, 338∞-norm, 219Euclidean, 39, 219, 220, 403, 691, 731

smallest, 64nuclear norm, 67packing, 403spectral norm, 70, 732

Barvinok, 35, 110, 132, 135, 138, 384base, 94

station, 415basis, 58, 182, 763

discrete cosine transform, 332nonorthogonal, 182, 703

projection on, 703orthogonal

projection, 703orthonormal, 58, 89, 700pursuit, 363standard

matrix, 58, 60, 756vector, 58, 145, 396, 768

bees, 30, 403Bellman, 268bijective, 50, 53, 116, 653

Gram form, 429, 432inner-product form, 433, 434invertible, 53, 688isometry, 54, 475linear, 52, 54, 57, 60, 137, 176, 553, 637

biorthogonaldecomposition, 692, 700expansion, 148, 161, 178, 182, 693

EDM, 507projection, 704unique, 180, 182, 185, 508, 628, 705

biorthogonality, 180, 183condition, 178, 608, 628

Birkhoff, 69

Page 805: Convex Optimization & Euclidean Distance Geometry

INDEX 805

boundlower, 542polyhedral, 491

boundary, 39, 40, 46, 64, 93-point, 92extreme, 108of affine, 38, 46of cone, 103, 170, 176, 177

EDM, 489, 504, 511membership, 162, 170, 279PSD, 40, 111, 116, 128, 129, 131, 279,

516ray, 104

of halfspace, 72of hypercube slice, 374of open set, 41of point, 46of ray, 94relative, 39, 40, 46, 64, 93

bounded, 64, 84bowl, 668, 669box, 82, 731bridge, 419, 448, 498, 543, 559

calculusmatrix, 657of inequalities, 7of variations, 166

Caratheodory’s theorem, 161, 714cardinality, 325, 392, 769, 771

1-norm heuristic, 326, 327, 562-one, 219, 338convex envelope, see convexgeometry, 219, 338minimization, 219, 220, 330, 360, 371

Boolean, 284nonnegative, 221, 225, 324, 328, 330,

337, 338, 340rank connection, 302signed, 342

problem, see cardinality minimizationquasiconcavity, 225, 325regularization, 333, 361

Cartesianaxes, 38, 63, 80, 96, 105, 146, 158, 327,

732

cone, see orthantcoordinates, 470product, 47, 276, 743, 764

cone, 99, 100, 155function, 216

subspace, 327, 732Cayley-Menger

determinant, 481, 483form, 439, 460, 461, 549, 574

cellular, 416telephone network, 415

centergeometric, 397, 401, 425, 468gravity, 425mass, 425

central path, 275certificate, 209, 279, 280, 301, 382

null, 162, 171chain rule, 664

two argument, 664Chu, 7clipping, 226, 538, 731, 732, 761closed, see setcoefficient

binomial, 224, 341projection, 180, 514, 709, 716

cofactor, 482compaction, 310, 323, 398, 402, 496, 563comparable, 102, 179complement

algebraic, 94, 148, 195, 696, 700projection on, 696, 727

orthogonal, 56, 84, 148, 302, 696, 700,727, 750, 759, 765

relative, 90, 759complementarity, 195, 217, 283, 301, 326

linear, 199maximal, 270, 275, 299problem, 198

linear, 199strict, 283

complementaryaffine dimension, 529dimension, 85eigenvalues, 696

Page 806: Convex Optimization & Euclidean Distance Geometry

806 INDEX

halfspace, 74inertia, 462, 602slackness, 282subspace, 56

compressed sensing, 216, 219–221, 225, 328,330, 362, 363, 365, 372, 376

nonnegative, 221, 222, 330, 337, 338,340

problem, 221, 222, 330, 337, 338, 340Procrustes, 350

theorem, 330, 363compressive sampling, see compressedconcave, 211, 212, 226condition

biorthogonality, 178, 608, 628convexity

first order, 248, 251, 253, 255second order, 252, 257

necessary, 759optimality

dual, 154first order, 166, 167, 195–198, 200KKT, 198, 200, 547, 656semidefinite program, 282Slater, 154, 208, 277, 281unconstrained, 166, 242, 243, 718

sufficient, 759cone, 70, 94, 97, 769

algebra, 157blade, 160boundary, see boundaryCartesian, see orthant

product, 99, 155circular, 45, 98, 122–125, 534

Lorentz, see cone, Lorentzright, 122

closed, see set, closedconvex, 70, 97–99, 106difference, 47, 156dimension, 134dual, 148–152, 155–159, 200, 203, 204,

465algorithm, 200construction, 150, 151dual, 155, 168

facet, 178, 205formula, 167, 168, 186, 193, 194halfspace-description, 167in subspace, 157, 162, 183, 187, 188Lorentz, 159, 173orthant, see orthantpointed, 144, 156product, Cartesian, 155properties, 155self, 162, 172–174unique, 149, 178vertex-description, 186, 200

EDM, 393, 487, 489, 490, 492√EDM , 490, 493, 556

boundary, 489, 504, 511circular, 122construction, 505convexity, 493decomposition, 528dual, 519, 520, 530, 532–534extreme direction, 507face, 506, 508interior, 489, 504one-dimensional, 108positive semidefinite, 508, 516, 525projection, 556, 568, 571

empty set, 97face, 103

intersection, 170pointed, 100PSD, 139PSD, smallest, 119, 515, 521, 531smallest, 99, 132, 170–172, 222, 273,

341, 506facet, 178, 205halfline, 94, 98, 100, 106, 108, 135, 144halfspace, 144, 152hull, 156ice cream, see cone, circularinterior, see interiorintersection, 99, 156

pointed, 100invariance, see invarianceLorentz

circular, 98, 123

Page 807: Convex Optimization & Euclidean Distance Geometry

INDEX 807

dual, 159, 173polyhedral, 141, 159

majorization, 585membership, see membership relationmonotone, 191–193, 462, 576, 655

nonnegative, 189–191, 464, 465, 471,473, 474, 550–552, 576, 585, 732

negative, 179, 705nonconvex, 94–97normal, 195, 196, 198, 216, 513, 723,

729, 749–752affine, 750elliptope, 751membership relation, 195orthant, 195

one-dimensional, 94, 98, 100, 135, 144EDM, 108PSD, 106

origin, 70, 97, 99, 100, 104, 106, 135orthant, see orthantorthogonal complement, 158, 727, 750pointed, 99, 100, 103–106, 143, 161,

180, 193closed convex, 100, 137dual, 144, 156extreme, 100, 106, 144, 152face, 100polyhedral, 105, 143, 177

polar, 94, 149, 513, 727, 765polyhedral, 70, 135, 140–144, 152

dual, 136, 157, 162, 168, 174, 177,178

equilateral, 174facet, 178, 205halfspace-description, 141majorization, 576pointed, 105, 143, 177proper, 136vertex-description, 70, 143, 162, 177

positive semidefinite, 43, 110, 112, 113,523

boundary, 111, 116, 128, 279, 516circular, 122–125, 227dimension, 119dual, 172

EDM, 525extreme direction, 106, 112, 120face, 139face, smallest, 119, 515, 521, 531hyperplane, supporting, 118, 120inscription, 126interior, 111, 278inverse image, 114, 115one-dimensional, 106optimization, 274polyhedral, 125rank, 119, 271visualization, 272

productCartesian, 99, 155

proper, 103, 200dual, 157

quadratic, 98ray, 94, 98, 100, 135, 144

EDM, 108positive semidefinite, 106

recession, 152second order, see cone, Lorentzselfdual, 162, 172–174shrouded, 174, 533simplicial, 71, 145–147, 179, 732

dual, 157, 179, 187spectral, 460, 462, 549

dual, 465orthant, 552

subspace, 98, 141, 144sum, 99, 156supporting hyperplane, 99, 100, 178tangent, 513transformation, see transformationtwo-dimensional, 135union, 95, 97, 156, 201unique, 111, 149, 489vertex, 100

congruence, 428, 459conic

combination, 72, 97, 143constraint, 197, 198, 200, 227coordinates, 148, 182, 185, 205hull, see hull

Page 808: Convex Optimization & Euclidean Distance Geometry

808 INDEX

independence, 27, 71, 133, 134,136–139, 290

versus dimension, 135dual cone, 188extreme direction, 137preservation, 135

problem, 154, 197, 198, 200, 216, 267dual, 154, 267

section, 122circular cone, 124

conjugatecomplex, 367, 586, 617, 637, 716, 755,

761convex, 155, 159, 161, 162, 168, 178,

187, 195function, 560, 761gradient, 372

conservationdimension, 438, 616, 631

constellation, 22constraint

angle, 398, 399, 402, 418binary, 284, 353cardinality, 330, 360, 371

nonnegative, 221, 225, 324, 328, 330,337, 338, 340

signed, 342conic, 197, 198, 200, 227equality, 167, 200

affine, 266inequality, 198, 299matrix

Gram, 398, 399orthogonal, see Procrustes, 382orthonormal, 346

nonnegative, 88, 89, 198norm, 344, 354, 615

spectral, 70polynomial, 227, 351, 352rank, see rankSchur-form, 70sort, 474tractable, 266

content, 483, 769contour plot, 196, 250

contraction, 70, 199, 372, 374convergence, 303, 475, 735

geometric, 737global, 301, 303, 305, 326, 382local, 305, 327, 331, 343, 382, 384measure, 287

convex, 21, 36, 212combination, 36, 72, 143envelope, 560–562

bilinear, 261cardinality, 289, 562rank, 317, 414, 560–562

equivalent, 317form, 7, 265geometry, 35

fundamental, 74, 79, 84, 385, 397hull, see hulliteration

cardinality, 221, 324–331, 342, 343convergence, 301, 303, 305, 326, 327,

343, 384rank, 300–305, 565rank one, 346, 380, 577rank, indefinite, 375stall, 305, 328, 331, 332, 345, 354,

380, 384log, 262optimization, 7, 21, 214, 266, 407

art, 8, 290, 310, 579fundamental, 266

polyhedron, see polyhedronproblem, 154, 167, 197, 200, 218, 231,

238, 240, 270, 276, 554, 721definition, 266geometry, 21, 27nonlinear, 265statement as solution, 7, 265tractable, 266

program, see problemquadratic, see function

convexity, 211, 226condition

first order, 248, 251, 253, 255second order, 252, 257

K-, 212

Page 809: Convex Optimization & Euclidean Distance Geometry

INDEX 809

log, 262norm, 70, 213, 214, 217, 218, 247, 258,

643, 771strict, see function, convex

coordinatesconic, 148, 182, 185, 205

Crippen & Havel, 418criterion

EDM, 514, 518dual, 526

Finsler, 518metric versus matrix, 440Schoenberg, 30, 397, 461, 531, 717

cubix, 658, 661, 766curvature, 252, 260

Dantzig, 268decomposition, 766

biorthogonal, 692, 700completely PSD, 98, 306dyad, 620eigenvalue, 606–609nonnegative matrix factorization, 306orthonormal, 698, 700singular value, see singular value

deconvolution, 363deLeeuw, 466, 557, 567Delsarte, 404dense, 91derivative, 242

directional, 251, 665, 668, 669, 673–675dimension, 668, 678gradient, 669second, 670

gradient, 675partial, 657, 762table, 678

descriptionconversion, 148halfspace, 71, 74

affine, 86dual cone, 167polyhedral cone, 141

vertex, 71, 72, 107affine, 87of halfspace, 138

polyhedral cone, 70, 143, 162, 177polyhedron, 64

determinant, 259, 478, 592, 605, 678, 684Cayley-Menger, 481, 483inequality, 597Kronecker, 584product, 594

Deza, 293, 513diagonal, 581

δ , 581binary, 692dominance, 128inequality, 597matrix, 606

commutative, 595diagonalization, 609

valueseigen, 606singular, 610

diagonalizable, 181, 606simultaneously, 120, 283, 593, 595, 618

diagonalization, 42, 606–609diagonal matrix, 609expansion by, 181symmetric, 608

diamond, 427differential

Frechet, 666partial, 762

diffusion, 496dilation, 471dimension, 38

affine, see affinecomplementary, 85conservation, 438, 616, 631domain, 55embedding, 61Euclidean, 763invariance, 55positive semidefinite cone

face, 119Precis, 438range, 55rank, 134, 437, 438, 769

direction, 94, 106, 665

Page 810: Convex Optimization & Euclidean Distance Geometry

810 INDEX

matrix, 300, 317, 322, 361, 405, 566interpretation, 301

Newton, 242steepest descent, 242, 668vector, 328

interpretation, 326, 327, 562disc, 65discretization, 167, 213, 252, 531dist, 738, 769distance, 738, 739

comparative, see distance, ordinalEuclidean, 392geometry, 22, 415matrix, see EDMordinal, 469–476origin to affine, 76, 219, 220, 691, 707

nonnegative, 221, 222, 337property, 387taxicab, 218–220, 254, 255

Donoho, 376doublet, see matrixdual

affine dimension, 529dual, 155, 162, 168, 282, 286, 358feasible set, 276norm, 159, 643, 771problem, 152, 153, 208, 223, 224, 267,

280, 282, 283, 358, 400, 500projection, 722, 724

on cone, 725on convex set, 722, 724on EDM cone, 571optimization, 723

strong, 153, 154, 281, 653duality, 154, 400

gap, 153, 155, 277, 281, 282, 358strong, 154, 208, 230, 282weak, 280

dvec, 60, 473, 770dyad, see matrixDykstra, 738, 749

algorithm, 735, 748, 749, 753

Eckart, 546, 548, 579edge, 90EDM, 28, 385, 386, 393, 534

√EDM , 493

closest, 547composition, 455–458, 490, 491, 493construction, 502criterion, 514, 518

dual, 526definition, 392, 500

Gram form, 395, 511inner-product form, 420interpoint angle, 395relative-angle form, 422

dual, 526, 527exponential entry, 455graph, 389, 404, 408, 411, 412, 496indefinite, 458membership relation, 531projection, 517range, 503test, numerical, 390, 397unique, 454, 467, 468, 548

eigen, 606matrix, 113, 181, 608spectrum, 460, 549

ordered, 462unordered, 464

value, 591, 606, 642, 644λ , 582, 761coefficients, 711, 714decomposition, 606–609EDM, 439, 459, 472, 501interlaced, 445, 459, 599intertwined, 445inverse, 608, 609, 614Kronecker, 584largest, 447, 598, 644, 645, 647left, 606of sum, 598principal, 360product, 593, 595, 596, 649real, 586, 606, 609repeated, 130, 606, 607Schur, 603singular, 610, 614smallest, 447, 456, 500, 598, 644sum of, 53, 592, 646, 648

Page 811: Convex Optimization & Euclidean Distance Geometry

INDEX 811

symmetric matrix, 609, 615transpose, 607unique, 122, 607zero, 615, 625

vector, 644, 645distinct, 607EDM, 501Kronecker, 584left, 606normal, 608principal, 360, 362real, 606, 609, 617symmetric, 608unique, 607

elbow, 541, 542ellipsoid, 36, 41, 43, 344

invariance, 49elliptope, 286, 288, 398, 407, 449, 450, 458,

511–513smooth, 449vertex, 287, 450, 451

embedding, 435dimension, 61

emptyinterior, 39set, 38, 39, 46, 641

affine, 38closed, 41cone, 97face, 90hull, 62, 65, 70open, 41

entry, 37, 212, 755, 760–762, 766largest, 223smallest, 223

epigraph, 233, 256, 262convex function, 233form, 239, 570, 571intersection, 232, 258, 263

equalityconstraint, 167, 200

affine, 266EDM, 525normal equation, 690

errata, 204, 230, 307, 432, 648, 730

Euclideanball, 39, 219, 220, 403, 691, 731

smallest, 64distance, 386, 392

geometry, 22, 408, 415, 717matrix, see EDM

metric, 387, 442fifth property, 387, 390, 391, 476,

479, 485norm, see 2-normprojection, 130space, 36, 763

exclusivemutually, 165

expansion, 183, 766biorthogonal, 148, 161, 178, 182, 693

EDM, 507projection, 704unique, 180, 182, 185, 508, 628, 705

implied by diagonalization, 181orthogonal, 58, 148, 181, 699, 701w.r.t orthant, 182

exponential, 685exposed, 90

closed, 508direction, 108extreme, 92, 100, 139face, 91, 139point, 91, 92, 108

density, 91extreme, 90

boundary, 108direction, 104, 109, 137

distinct, 106dual cone, 136, 178, 183, 185, 188,

202–204EDM cone, 108, 507, 525one-dimensional, 106, 108origin, 106pointed cone, 100, 106, 144, 152PSD cone, 106, 112, 120shared, 204supporting hyperplane, 178unique, 106

exposed, 92, 100, 139

Page 812: Convex Optimization & Euclidean Distance Geometry

812 INDEX

point, 69, 90, 92, 93, 275cone, 100, 132Fantope, 302hypercube slice, 327

ray, 104, 511

face, 42, 90affine, 93algebra, 93exposed, see exposedintersection, 170isomorphic, 119, 400, 506polyhedron, 139, 140recognition, 25smallest, 93, 170–172, see conetransitivity, 93

facet, 91, 178, 205factorization

matrix, 643nonnegative matrix, 306

Fan, 65, 268, 593, 646Fantope, 65, 66, 122, 237, 301

extreme point, 302Farkas’ lemma, 159, 164

positive definite, 278not, 279

positive semidefinite, 276fat, 86, 756

full-rank, 86Fejer, 172, 741Fenchel, 211Fermat point, 400, 401Fiacco, 268finitely generated, 141floor, 770Forsgren & Gill, 265Fourier

transform, discrete, 54, 366Frisch, 268Frobenius, 52, see normfull, 39

-dimensional, 39, 103rank, 86

function, see operatoraffine, 47, 229–231, 244

supremum, 232, 643

bilinear, 261composition, 226, 246, 664

affine, 247, 262concave, 211, 226, 262, 564, 644conjugate, 560, 761continuity, 212, 253, 262convex, 211, 212, 394, 423

matrix, 253nonlinear, 212, 231simultaneously, 235, 236, 256strictly, 213, 214, 246, 252, 257, 258,

557, 563, 569, 668, 669sum, 214, 247, 263supremum, 258, 263trivial, 214vector, 212

differentiable, 231, 257non, 212, 213, 218, 243, 262, 372, 373

distanceEuclidean, 392, 393taxicab, 254, 255

fractional, 235, 247, 252, 256inverted, 226, 603, 678, 685power, 227, 685projector, 236root, 226, 597, 609, 756

inverted, see function, fractionalinvertible, 49, 53, 55linear, see operatormonotonic, 211, 223, 224, 246, 261, 262multidimensional, 211, 241, 257, 767

affine, 229, 230negative, 226objective, see objectivepresorting, 223, 325, 549, 553, 575, 761product, 235, 247, 261projection, see projectorpseudofractional, 234quadratic, 214, 239, 249, 258, 351, 421,

602, 656, 715distance, 392nonnegative, 602strictly, 214, 252, 257, 258, 569, 668,

669quasiconcave, 260, 261, 293

Page 813: Convex Optimization & Euclidean Distance Geometry

INDEX 813

cardinality, 225, 325rank, 129

quasiconvex, 129, 233, 260, 262continuity, 261, 262

quasilinear, 252, 262, 614quotient, see function, fractionalratio, see function, fractionalsmooth, 212, 218sorting, 223, 325, 549, 553, 575, 761step, 614

matrix, 293stress, 252, 544, 566support, 82, 231

algebra, 232fundamental

convexgeometry, 74, 79, 84, 385, 397optimization, 266

metric property, 387, 442semidefiniteness test, 173, 586subspace, 49, 84, 85, 688, 698theorem algebra, 607

linear, 85

Gateaux differential, 666Gaffke, 520generators, 62, 72, 107, 168

c.i., 137hull, 142

affine, 62, 77, 708conic, 72, 137, 138convex, 64

l.i., 138minimal set, 64, 77, 137, 763

affine, 708extreme, 107, 178halfspace, 138hyperplane, 138orthant, 213positive semidefinite cone, 112, 173,

254geometric

center, 397, 401, 425, 468operator, 431subspace, 118, 425, 429, 430, 488,

498, 522–524, 560, 619, 719

Hahn-Banach theorem, 74, 79, 84mean, 685multiplicity, 607

Gersgorin, 125gimbal, 638, 639global positioning system, 24, 308Glunt, 570, 749Gower, 385gradient, 166, 196, 231, 241, 251, 657, 676

composition, 664conjugate, 372derivative, 675

directional, 669dimension, 241first order, 675image, 369monotonic, 246norm, 225, 226, 679, 681, 697, 707, 708,

718, 722product, 661second order, 676, 677sparsity, 364table, 678zero, 166, 242, 243, 718

Gram matrix, 315, 395, 407constraint, 398, 399uniqueness, 425, 429

Hadamardproduct, 51, 556, 582, 583, 664, 758quotient, 678, 685, 758square root, 756

halfline, 94halfspace, 72–74, 152

boundary, 72description, 71, 74

affine, 86dual cone, 167polyhedral cone, 141vertex-, 138

interior, 46intersection, 74, 156, 173polarity, 72

Hayden & Wells, 239, 487, 501, 508, 568Herbst, 614Hessian, 242, 657

Page 814: Convex Optimization & Euclidean Distance Geometry

814 INDEX

hexagon, 402Hiriart-Urruty, 82homogeneity, 102, 217, 262, 393honeycomb, 30Hong, 570Horn & Johnson, 587, 588horn, flared, 98hull, 61

affine, 38, 61–65, 79, 143EDM cone, 431, 519empty set, 62point, 39, 62positive semidefinite cone, 62rank-1 correlation matrices, 62unique, 61

conic, 63, 70, 143empty set, 70

convex, 61, 63, 64, 143, 386cone, 156dyads, 67empty set, 65extreme directions, 107orthogonal matrices, 70orthonormal matrices, 70outer product, 65permutation matrices, 69positive semidefinite cone, 121projection matrices, 65rank-one matrices, 67, 68rank-one symmetric matrices, 121unique, 64

hyperboloid, 620hyperbox, 82, 731hypercube, 52, 82, 219, 731

nonnegative, 289slice, nonnegative, 327, 374

hyperdimensional, 662hyperdisc, 669hyperplane, 72–77, 244

hypersphere radius, 75independence, 86intersection, 86, 171

convex set, 80, 492movement, 75, 88, 222normal, 74

separating, 84, 157, 162, 164strictly, 84

supporting, 79, 81, 196, 248–251cone, 118, 120, 149, 150, 171, 178,

519polarity, 79strictly, 82, 99, 100, 249trivially, 79, 172unique, 248, 249, 251

vertex-description, 77hypersphere, 65, 75, 398, 403, 451

circumhypersphere, 439hypograph, 234, 262, 680

idempotent, see matrixiff, 769image, 47

inverse, 47–49, 175cone, 114, 115, 157, 176

magnetic resonance, 364independence

affine, 71, 77, 78, 134, 138preservation, 78

conic, 27, 71, 133, 134, 136–139, 290versus dimension, 135dual cone, 188extreme direction, 137preservation, 135

linear, 37, 71, 134, 138, 339, 627matrix, 593of affine, 86, 87preservation, 37subspace, 37, 758

inequalityangle, 390, 479

matrix, 423Bessel, 699Cauchy-Schwarz, 740constraint, 198, 299determinant, 597diagonal, 597generalized, 102, 148, 159

dual, 27, 161, 172partial order, 100

Hardy-Littlewood-Polya, 550Jensen, 247

Page 815: Convex Optimization & Euclidean Distance Geometry

INDEX 815

linear, 35, 72, 162matrix, 27, 173, 175, 176, 276, 277,

279log, 680norm, 285, 771

triangle, 217rank, 597singular value, 70spectral, 70, 460trace, 597triangle, 387, 388, 421, 442–444, 446,

454, 477, 478, 480–482, 491, 493norm, 217strict, 444unique, 477

variation, 166inertia, 459, 602

complementary, 462, 602Sylvester’s law, 594

infimum, 263, 641, 644, 697, 718, 768algebra, 641, 642

inflection, 252injective, 53, 54, 103, 207, 429, 763

Gram form, 425, 429, 432inner-product form, 433, 434invertible, 53, 687, 688non, 55nullspace, 53

innovationrate, 363

interior, 39, 40, 46, 93-point, 39

method, 141, 265, 268, 270, 287, 324,404, 406, 529, 567

algebra, 40, 46, 175empty, 39of affine, 40of cone, 175, 177

EDM, 489, 504membership, 161positive semidefinite, 111, 278

of halfspace, 46of point, 39, 41of ray, 94relative, 39, 40, 46, 93

transformation, 175intersection, 46

cone, 99, 156pointed, 100

epigraph, 232, 258, 263face, 170halfspace, 74, 156, 173hyperplane, 86, 171

convex set, 80, 492line with boundary, 42nullspace, 89planes, 87positive semidefinite cone

affine, 132, 279geometric center subspace, 619line, 409

subspace, 89tangential, 43

invarianceclosure, 39, 140, 175cone, 99, 100, 137, 156, 489

pointed, 103, 144convexity, 46dimension, 55ellipsoid, 49Gram form, 425inner-product form, 428isometric, 23, 410, 424, 428, 466, 467orthogonal, 54, 475, 542, 637, 640reflection, 426rotation, 122, 426scaling, 262, 393set, 457translation, 424–426, 431, 452, 500, 719

inversionconic coordinates, 207Gram form, 431injective, 53matrix, see matrix, inversenullspace, 49, 53operator, 49, 53, 55, 175

is, 759isedm(), 390, 397isometry, 54, 475, 544isomorphic, 52, 56, 59, 758

Page 816: Convex Optimization & Euclidean Distance Geometry

816 INDEX

face, 119, 400, 506isometrically, 42, 53, 115, 172non, 619range, 53

isomorphism, 52, 176, 432, 434, 475, 488isometric, 53, 54, 57, 60, 475

symmetric hollow subspace, 60symmetric matrix subspace, 57

isotonic, 471iterate, 735, 745, 746, 749iteration

alternating projection, 570, 735, 741,749

convex, see convex

Jacobian, 242

Karhunen-Loeve transform, 466kissing, 153, 219, 220, 337–339, 403, 691

number, 403KKT conditions, 198, 200, 547, 656Klanfer, 398Klee, 107Krein-Rutman, 157Kronecker product, 115, 582–584, 653, 662,

663, 680eigen, 584inverse, 367positive semidefinite, 392, 596projector, 736pseudoinverse, 689trace, 584vector, 664vectorization, 583

Lagrange multiplier, 167, 656sign, 208

Lagrangian, 208, 216, 358, 370, 400Lasso, 363lattice, 311–314, 318–320

regular, 30, 311Laurent, 291, 293, 450, 455, 513, 518, 746,

747law of cosines, 420least

1-norm, 219, 220, 285, 338

energy, 398, 563norm, 219, 285, 690squares, 307, 309, 690

Legendre-Fenchel transform, 560Lemarechal, 21, 82line, 244, 257

tangential, 44linear

algebra, 85, 581bijection, see bijectivecomplementarity, 199function, see operatorindependence, 37, 71, 134, 138, 339, 627

matrix, 593of affine, 86, 87preservation, 37subspace, 37, 758

inequality, 35, 72, 162matrix, 27, 173, 175, 176, 276, 277,

279injective, see injectiveoperator, 47, 59, 212, 229, 231, 395,

431, 433, 475, 509, 581, 720projector, 695, 700, 702, 723

program, 83, 88, 89, 155, 217, 231, 265,267–270, 404

dual, 155, 267prototypical, 155, 267

regression, 363surjective, see surjectivetransformation, see transformation

list, 28Liu, 487, 501, 508, 568localization, 24, 409

sensor network, 307, 308standardized test, 311unique, 23, 308, 310, 408, 411wireless, 415

log, 680, 684-convex, 262barrier, 404det, 259, 415, 564, 675, 683

Lowner, 113, 596

machine learning, 25Magaia, 614

Page 817: Convex Optimization & Euclidean Distance Geometry

INDEX 817

majorization, 585cone, 585symmetric hollow, 585

manifold, 25, 26, 69, 70, 118, 494, 495, 542,577, 637, 649

map, see transformationisotonic, 469, 472USA, 28, 469, 470

Mardia, 546Markov process, 500mater, 658Mathar, 520Matlab, 33, 134, 285, 323, 331, 336, 345,

353, 360, 405, 470, 473, 612MRI, 362, 367, 372, 374

matrix, 658angle

interpoint, 395, 398relative, 422

antisymmetric, 56, 586, 764antihollow, 59, 60, 539

auxiliary, 397, 632, 636orthonormal, 635projector, 632Schoenberg, 394, 634table, 636

binary, see matrix, BooleanBoolean, 287, 451bordered, 445, 459calculus, 657circulant, 259, 608, 633

permutation, 367symmetric, 367

commutative, see productcompletion, see problemcorrelation, 62, 450, 451covariance, 397determinant, see determinantdiagonal, see diagonaldiagonalizable, see diagonalizabledirection, 300, 317, 322, 361, 405, 566

interpretation, 301distance

Euclidean, see EDMtaxicab, 254, 458

doublet, 425, 629, 719nullspace, 629, 630range, 629, 630

dyad, 621, 624-decomposition, 620independence, 89, 181, 467, 503,

626–628negative, 624nullspace, 624, 625projection on, 710, 712projector, 625, 692, 709, 710range, 624, 625sum, 606, 610, 628symmetric, 120, 626

EDM, see EDMelementary, 457, 630, 632

nullspace, 630, 631range, 630, 631

exponential, 259, 685fat, 55, 86, 756Fourier, 54, 366Gale, 411geometric centering, 397, 425, 489, 498,

632Gram, 315, 395, 407

constraint, 398, 399uniqueness, 425, 429

Hermitian, 586hollow, 56, 59Householder, 631

auxiliary, 633idempotent, 695

nonsymmetric, 691range, 691, 692symmetric, 697, 700

indefinite, 375, 458inverse, 257, 597, 608, 626, 674, 675

injective, 688symmetric, 609

Jordan form, 591, 607measurement, 538nonnegative factorization, 306nonsingular, 608, 609normal, 53, 592, 608, 610normalization, 288

Page 818: Convex Optimization & Euclidean Distance Geometry

818 INDEX

nullspace, see nullspaceorthogonal, 347, 348, 426, 609, 637

manifold, 637product, 637symmetric, 638

orthonormal, 54, 65, 70, 301, 346, 426,611, 635, 646, 689, 699

manifold, 70partitioned, 601–605permutation, 69, 347, 348, 441, 584,

609, 637circulant, 367extreme point, 69product, 637symmetric, 367, 632, 652

positive definiteinverse, 597positive real numbers, 253

positive factorization, 306positive semidefinite, 586, 589, 591, 599

completely, 98, 306difference, 131from extreme directions, 121Kronecker, 392, 596nonsymmetric, 591pseudoinverse, 689square root, 597, 609, 756

product, see productprojection, 116, 633, 698

Kronecker, 736nonorthogonal, 691orthogonal, 697positive semidefinite, 600, 699product, 735

pseudoinverse, 115, 183, 243, 285, 601,687

Kronecker, 689nullspace, 687, 688positive semidefinite, 689product, 688range, 687, 688SVD, 614symmetric, 614, 689transpose, 180unique, 687

quotientHadamard, 678, 685, 758

range, see rangerank, see rankreflection, 426, 427, 638, 652rotation, 426, 637–639, 651

of range, 638rotation of, 640Schur-form, 601simple, 623skinny, 54, 183, 755sort index, 471square root, 756, see matrix, positive

semidefinitesquared, 258standard basis, 58, 60, 756Stiefel, see matrix, orthonormalstochastic, 69, 347sum

eigenvalues, 598rank, 128, 129, 593

symmetric, 56, 609inverse, 609pseudoinverse, 614, 689real numbers, 253subspace of, 56

symmetrized, 587trace, see traceunitary, 637

symmetric, 366, 369max cut problem, 356maximum variance unfolding, 494, 500McCormick, 268membership relation, 64, 100, 113, 159, 187,

277, 770affine, 216boundary, 162, 170, 279discretized, 167, 168, 531EDM, 531in subspace, 187interior, 161, 278normal cone, 195orthant, 64, 169, 170

metric, 54, 254, 386postulate, 387

Page 819: Convex Optimization & Euclidean Distance Geometry

INDEX 819

property, 387, 442fifth, 387, 390, 391, 476, 479, 485

space, 387minimal

element, 101, 102, 214, 215set, see set

minimization, 166, 195, 310, 669affine function, 230cardinality, 219, 220, 330, 360, 371

Boolean, 284nonnegative, 221, 225, 324, 328, 330,

337, 338, 340rank connection, 302signed, 342

fractional, inverted, see functionnorm

1-, 218, see problem2-, 218, 400, 401∞-, 218Frobenius, 218, 239, 240, 243, 543,

718nuclear, 375, 562, 643

on hypercube, 82rank, 300, 346, 360, 378, 380, 565, 573

cardinality connection, 302indefinite, 375

trace, 302–304, 375, 409, 560, 562, 563,603, 643

minimumcardinality, see minimizationelement, 101, 102, 214, 215global

unique, 211, 669value

unique, 214Minkowski, 141MIT, 376molecular conformation, 24, 418monotonic, 238, 246, 305, 741, 744

Fejer, 741function, 211, 223, 224, 246, 262gradient, 246

Moore-Penrose conditions, 687Moreau, 198, 727, 728Motzkin, 722

MRI, 362, 364Muller, 614multidimensional

function, 211, 241, 257affine, 229, 230

objective, 101, 102, 214, 215, 324, 371scaling, 24, 466

ordinal, 471multilateration, 308, 415multiobjective optimization, 101, 102,

214–216, 324, 371multipath, 415multiplicity, 593, 607Musin, 404, 406

nearest neighbors, 496neighborhood graph, 495, 496Nemirovski, 537nesting, 444Neumann, 570, 654, 736, 737Newton, 404, 657

direction, 242nondegeneracy, 387nonexpansive, 699, 730nonnegative, 440, 770

constraint, 88, 89, 198matrix factorization, 306part, 226, 538, 731, 732, 761

nonnegativity, 217, 387nonvertical, 244norm, 214, 216–218, 221, 325, 771

0, 6150-, see 0-norm1-, see 1-norm2-, see 2-norm∞-, see ∞-normk-largest, 224, 225, 328

gradient, 225, 226ball, see ballconstraint, 344, 354, 615

spectral, 70convexity, 70, 213, 214, 217, 218, 247,

258, 643, 771dual, 159, 643, 771Euclidean, see 2-normFrobenius, 53, 214, 218, 542

Page 820: Convex Optimization & Euclidean Distance Geometry

820 INDEX

minimization, 218, 239, 240, 243,543, 718

Schur-form, 239gradient, 225, 226, 679, 681, 697, 707,

708, 718, 722inequality, 285, 771

triangle, 217Ky Fan, 642least, 219, 285, 690nuclear, 375, 562, 642, 643, 771

ball, 67orthogonally invariant, 54, 475, 542,

637, 640property, 216regularization, 355, 690spectral, 70, 258, 542, 555, 612, 624,

631, 642, 732, 771ball, 70, 732dual, 643, 771inequality, 70, 771Schur-form, 258, 642

square, 214, 681, 707, 708, 718, 771vs. square root, 218, 247, 542, 697

normal, 74, 196, 722cone, see coneequation, 690facet, 178inward, 72, 519outward, 72vector, 722

Notation, 755NP-hard, 566nuclear

magnetic resonance, 418norm, 375, 562, 642, 643, 771

ball, 67nullspace, 85, 632, 720

basis, 87doublet, 629, 630dyad, 624, 625elementary matrix, 630, 631form, 84hyperplane, 73injective, 53intersection, 89

orthogonal complement, 763product, 594, 624, 628, 698pseudoinverse, 687, 688

objective, 83convex, 266

strictly, 557, 563function, 83, 154, 166, 195, 197linear, 231, 317, 409multidimensional, 101, 102, 214, 215,

324, 371nonlinear, 231polynomial, 351quadratic, 239, 656real, 214value, 280, 296

unique, 267offset, 424on, 763one-to-one, 763, see injectiveonly if, 759onto, 763, see surjectiveopen, see setoperator, 767

adjoint, 50, 51, 157, 527, 755, 762self, 399, 581

affine, see functioninjective, see injectiveinvertible, 49, 53, 55, 175linear, 47, 59, 212, 229, 231, 395, 431,

433, 475, 509, 581, 720projector, 695, 700, 702, 723

nonlinear, 212, 231nullspace, 425, 429, 431, 433, 720permutation, 223, 325, 549, 553, 575,

761projection, see projectorsurjective, see surjectiveunitary, 54

optimalitycondition

directional derivative, 669dual, 154first order, 166, 167, 195–198, 200KKT, 198, 200, 547, 656semidefinite program, 282

Page 821: Convex Optimization & Euclidean Distance Geometry

INDEX 821

Slater, 154, 208, 277, 281unconstrained, 166, 242, 243, 718

constraintconic, 197, 198, 200equality, 167, 200inequality, 198nonnegative, 198

global, 8, 214, 238, 266perturbation, 296

optimization, 21, 265combinatorial, 69, 83, 222, 284, 347,

353, 356, 451, 578conic, see problemconvex, 7, 21, 214, 266, 407

art, 8, 290, 310, 579fundamental, 266

multiobjective, 101, 102, 214–216, 324,371

tractable, 266vector, 101, 102, 214–216, 324, 371

ordernatural, 50, 581, 767nonincreasing, 551, 609, 610, 761of projection, 538partial, 64, 98, 100–102, 113, 169, 170,

178, 179, 596, 770generalized inequality, 100orthant, 169property, 100, 102

total, 100, 101, 768origin, 36

cone, 70, 97, 99, 100, 104, 106, 135projection of, 219, 220, 691, 707projection on, 727, 732translation to, 436

Orion nebula, 22orthant, 37, 145, 148, 159, 169, 182, 195

dual, 173nonnegative, 162, 525

orthogonal, 50complement, 56, 84, 148, 302, 696, 700,

727, 750, 759, 765equivalence, 640expansion, 58, 148, 181, 699, 701invariance, 54, 475, 542, 637, 640

sum, 759orthonormal, 50

decomposition, 698matrix, see matrix

orthonormality condition, 181overdetermined, 755

PageRank, 500parallel, 38, 701pattern recognition, 24Penrose conditions, 687pentahedron, 482pentatope, 145, 482perpendicular, 50, 697, 718perturbation, 291, 293

optimality, 296phantom, 364plane, 72

segment, 104point

boundary, 92of hypercube slice, 374

exposed, see exposedfeasible, see solutionfixed, 305, 372, 720, 738–741inflection, 252

Polya, 550Polyak, 641polychoron, 46, 140, 145, 481, 482polyhedron, 42, 61, 83, 140, 448

bounded, 64, 69, 222face, 139, 140halfspace-description, 140norm ball, 220permutation, 69, 347, 350stochastic, 69, 347, 350transformation

linear, 137, 140, 162unbounded, 38, 42, 142vertex, 69, 142, 220vertex-description, 64, 142

polynomialconstraint, 227, 351, 352nonnegative, 602objective, 351quadratic, see function

Page 822: Convex Optimization & Euclidean Distance Geometry

822 INDEX

sum of squares, 602polytope, 140positive, 770

semidefinite, see matrixstrictly, 441

power, see function, fractionalPrecis, 438precision, numerical, 141, 268, 270, 287,

324, 370, 404, 406, 537, 567presolver, 339–342primal

feasible set, 270, 276problem, 154, 267

via dual, 155, 283principal

component analysis, 360, 466eigenvalue, 360eigenvector, 360, 362submatrix, 407, 477, 482, 506, 599

leading, 442, 444theorem, 599

principlehalfspace-description, 74separating hyperplane, 84supporting hyperplane, 79

problem1-norm, 218–222, 285, 343

nonnegative, 221, 222, 326, 327, 330,337, 338, 340, 562

Boolean, 69, 83, 284, 353, 451cardinality, 330, 360, 371

Boolean, 284nonnegative, 221, 225, 324, 328, 330,

337, 338, 340signed, 342

combinatorial, 69, 83, 222, 284, 347,353, 356, 451, 578

complementarity, 198linear, 199

completion, 311–314, 322, 367, 388,389, 408, 411, 412, 447, 454, 455,484, 485, 492, 494, 495, 579

geometry, 494semidefinite, 746, 747

compressed sensing, see compressed

concave, 266conic, 154, 197, 198, 200, 216, 267

dual, 154, 267convex, 154, 167, 197, 200, 218, 231,

238, 240, 270, 276, 554, 721definition, 266geometry, 21, 27nonlinear, 265statement as solution, 7, 265tractable, 266

dual, see dualepigraph form, 239, 570, 571equivalent, 301feasibility, 143, 170, 171

semidefinite, 280, 300, 301, 381fractional, inverted, see functionmax cut, 356minimax, 64, 154, 155nonlinear, 407

convex, 265open, 30, 128, 305, 465, 508, 578primal, see primalProcrustes, see Procrustesproximity, 537, 539, 542, 545, 556, 568

EDM, nonconvex, 552in spectral norm, 555rank heuristic, 562, 565SDP, Gram form, 559semidefinite program, 548, 558, 569

quadratic, 656same, 301stress, 252, 544, 566tractable, 266

procedurerank reduction, 292

Procrustescombinatorial, 347compressed sensing, 350diagonal, 655linear program, 653maximization, 652orthogonal, 649

two sided, 651, 654orthonormal, 346symmetric, 654

Page 823: Convex Optimization & Euclidean Distance Geometry

INDEX 823

translation, 650unique solution, 650

product, 261, 663Cartesian, 47, 276, 743, 764

cone, 99, 100, 155function, 216

commutative, 259, 283, 593–595, 618,735

determinant, 594direct, 662eigenvalue, 593, 595, 596, 649

matrix, 595, 596fractional, see functionfunction, see functiongradient, 661Hadamard, 51, 556, 582, 583, 664, 758inner

matrix, 395vector, 50, 261, 268, 363, 421, 710,

760vectorized matrix, 50, 591

Kronecker, 115, 582–584, 653, 662, 663,680

eigen, 584inverse, 367positive semidefinite, 392, 596projector, 736pseudoinverse, 689trace, 584vector, 664vectorization, 583

nullspace, 594, 624, 628, 698orthogonal, 637outer, 65permutation, 637positive definite, 588

nonsymmetric, 588positive semidefinite, 595, 596, 599, 600

nonsymmetric, 595projector, 524, 710, 735pseudofractional, see functionpseudoinverse, 688

Kronecker, 689range, 594, 624, 628, 698rank, 593, 594

tensor, 662trace, see tracezero, 618

program, 83, 265convex, see problemgeometric, 8, 269integer, 69, 83, 451linear, 83, 88, 89, 155, 217, 231, 265,

267–270, 404dual, 155, 267prototypical, 155, 267

nonlinearconvex, 265

quadratic, 269, 270, 474second-order cone, 269, 270semidefinite, 155, 217, 231, 265–267,

269, 270, 349dual, 155, 267equality constraint, 113Gram form, 571prototypical, 155, 267, 281, 299Schur-form, 227, 239, 240, 258, 475,

558, 570, 571, 642, 730projection, 53, 687

I − P , 696, 727algebra, 702alternating, 403, 735, 736

angle, 742convergence, 740, 742, 744, 747distance, 738, 739Dykstra algorithm, 735, 748, 749feasibility, 738–740iteration, 735, 749on affine/psd cone, 745on EDM cone, 570on halfspaces, 737on orthant/hyperplane, 741, 742optimization, 738, 739, 748over/under, 743

biorthogonal expansion, 704coefficient, 180, 514, 709, 716cyclic, 737direction, 697, 700, 712

nonorthogonal, 694, 695, 709dual, see dual, projection

Page 824: Convex Optimization & Euclidean Distance Geometry

824 INDEX

easy, 731Euclidean, 130, 539, 542, 546, 548, 556,

568, 722matrix, see matrixminimum-distance, 166, 695, 698, 718,

721nonorthogonal, 396, 555, 691, 694, 695,

709eigenvalue coefficients, 711

oblique, 219, 220, 544, see projection,nonorthogonal

of convex set, 50of hypercube, 52of matrix, 717, 719of origin, 219, 220, 691, 707of PSD cone, 117on 1-norm ball, 733on affine, 50, 702, 703, 707

hyperplane, 707, 709of origin, 219, 220, 691, 707vertex-description, 708

on basisnonorthogonal, 703orthogonal, 703

on box, 731on cardinality k , 732on complement, 696, 727on cone, 726, 728, 732

dual, 94, 148, 727, 728EDM, 556, 557, 568, 573, 574Lorentz (second-order), 733monotone nonnegative, 464, 474,

550–552, 732polyhedral, 733simplicial, 732truncated, 729

on cone, PSD, 130, 545, 547, 548, 732geometric center subspace, 524rank constrained, 242, 548, 554, 645

on convex set, 166, 721–723in affine subset, 733in subspace, 733

on convex sets, 737on dyad, 710, 712on ellipsoid, 344

on Euclidean ball, 731on function domain, 234on geometric center subspace, 524, 719on halfspace, 709on hyperbox, 731on hypercube, 731on hyperplane, 707, 709, see projection,

on affineon hypersphere, 731on intersection, 733on matrices

rank constrained, 732on matrix, 709, 710, 712on origin, 727, 732on orthant, 551, 729, 731

in subspace, 551on range, 690on rowspace, 55on simplex, 733on slab, 709on spectral norm ball, 732on subspace, 50, 94, 148, 698, 727

Cartesian, 732elementary, 706hollow symmetric, 546, 731matrix, 717symmetric matrices, 731

on vectorcardinality k , 732nonorthogonal, 709orthogonal, 712

one-dimensional, 712order of, 538, 540, 733orthogonal, 690, 697, 712

eigenvalue coefficients, 714semidefiniteness test as, 715

range/rowspace, 721spectral, 498, 549, 552

norm, 555, 732unique, 552

successive, 737two sided, 717, 719, 721umbral, 50unique, 695, 698, 718, 721–723vectorization, 514, 719

Page 825: Convex Optimization & Euclidean Distance Geometry

INDEX 825

projectorbiorthogonal, 701characteristic, 692, 695, 696, 699, 700commutative, 735

non, 524, 733, 736complement, 697convexity, 695, 700, 721, 723direction, 697, 700, 712

nonorthogonal, 694, 695, 709dyad, 625, 692, 709, 710Kronecker, 736linear operator, 55, 695, 700, 702, 723nonorthogonal, 396, 695, 702, 709orthogonal, 698orthonormal, 701product, 524, 710, 735range, 698semidefinite, 600, 699

prototypicalcomplementarity problem, 199compressed sensing, 338coordinate system, 148linear program, 267SDP, 155, 267, 281, 299signal, 363

pyramid, 483, 484

quadrant, 37quadratic, see functionquadrature, 426quartix, 660, 766quasi-, see functionquotient

Hadamard, 678, 685, 758Rayleigh, 716

range, 63, 85, 718, 763dimension, 55doublet, 629, 630dyad, 624, 625elementary matrix, 630, 631form, 84idempotent, 691, 692orthogonal complement, 763product, 594, 624, 628, 698pseudoinverse, 687, 688

rotation, 638rank, 593, 594, 610–612, 769

-ρ subset, 118, 130, 546, 548–555, 573,579

constraint, 300, 346, 351, 352, 360, 378,380, 565, 573

indefinite, 375projection on matrices, 732projection on PSD cone, 242, 548,

554, 645convex envelope, see convexconvex iteration, see convexdiagonalizable, 593dimension, 134, 437, 438, 769

affine, 439, 565full, 86heuristic, 562, 565inequality, 597log det, 564, 565lowest, 301minimization, 300, 346, 360, 378, 380,

565, 573cardinality connection, 302indefinite, 375

one, 600, 621, 624, 626Boolean, 287, 451convex iteration, 346, 380, 577hull, see hull, convexmodification, 626, 630subset, 242symmetric, 120

partitioned matrix, 604positive semidefinite cone, 271

face, 119Precis, 438product, 593, 594projection matrix, 696, 699quasiconcavity, 129reduction, 270, 291, 292, 297

procedure, 292regularization, 324, 346, 356, 379, 380,

565, 577Schur-form, 604sum, 129

positive semidefinite, 128, 129, 593

Page 826: Convex Optimization & Euclidean Distance Geometry

826 INDEX

trace heuristic, 302–304, 375, 560–563,643

ray, 94boundary of, 94cone, 94, 97, 98, 100, 135, 144

boundary, 104EDM, 108positive semidefinite, 106

extreme, 104, 511interior, 94

Rayleighquotient, 716

realizable, 22, 402, 544, 545reconstruction, 466

isometric, 410, 470isotonic, 469–472, 474, 476list, 466unique, 389, 411, 412, 428, 429

recursion, 372, 529, 581, 599reflection, 426, 427, 652

invariant, 426vector, 638

reflexivity, 100, 596regular

lattice, 30, 311simplex, 482tetrahedron, 479, 493

regularizationcardinality, 333, 361norm, 355, 690rank, 324, 346, 356, 379, 380, 565, 577

relaxation, 69, 83, 222, 286, 289, 409, 451,473

reweighting, 327, 564Riemann, 270robotics, 25, 31root, see function, fractionalrotation, 426, 638, 639, 651

invariant, 426of matrix, 640of range, 638vector, 125, 426, 637

round, 769

saddle value, 153, 154scaling, 466

multidimensional, 24, 466ordinal, 471

unidimensional, 566Schoenberg, 385, 396, 448, 455, 457, 473,

488, 544, 546, 708auxiliary matrix, 394, 634criterion, 30, 397, 461, 531, 717

Schur, 585-form, 601

anomaly, 240constraint, 70convex set, 114norm, Frobenius, 239norm, spectral, 258, 642rank, 604semidefinite program, 227, 239, 240,

258, 475, 558, 570, 571, 642, 730sparse, 603

complement, 316, 409, 601, 602, 604self-distance, 387semidefinite

domain of test, 586program, see programsymmetry versus, 586

sensor, 307, 308network localization, 307, 308, 407, 411

sequence, 741set

Cartesian product, 47closed, 36, 39–41, 72–74, 175

cone, 111, 116, 137, 140, 141, 156,162, 175, 176, 276–278, 489

exposed, 508polyhedron, 137, 140

connected, 36convex, 36

invariance, 46projection on, 166, 721–723

difference, 47, 156empty, 38, 39, 46, 641

affine, 38closed, 41cone, 97face, 90hull, 62, 65, 70

Page 827: Convex Optimization & Euclidean Distance Geometry

INDEX 827

open, 41feasible, 45, 69, 166, 195, 214, 266, 766

dual, 276primal, 270, 276

intersection, see intersectioninvariant, 457level, 196, 229, 232, 241, 250, 262, 266

affine, 231minimal, 64, 77, 137, 763

affine, 708extreme, 107, 178halfspace, 138hyperplane, 138orthant, 213positive semidefinite cone, 112, 173,

254nonconvex, 40, 94–98open, 36, 38, 40, 41, 161projection of, 50Schur-form, 114smooth, 449solution, 214, 767sublevel, 196, 233, 242, 251, 256, 261,

602convex, 234, 250quasiconvex, 261

sum, 46superlevel, 234, 261union, 46, 95, 97

Shannon, 363shape, 23Sherman-Morrison, 626, 675shift, 424shroud, 427, 428, 512

cone, 174, 533SIAM, 266signal

dropout, 332processing

digital, 332, 363, 376, 466simplex, 144, 145

area, 481content, 483method, 324, 404nonnegative, 337, 338

regular, 482unit, 144, 145volume, 483

singular value, 610, 642σ , 582, 761decomposition, 610–615, 643

compact, 610diagonal, 656full, 611geometrical, 613pseudoinverse, 614subcompact, 611symmetric, 614, 655

eigenvalue, 610, 614, 615inequality, 70inverse, 614largest, 70, 258, 642, 732, 771normal matrix, 53, 610sum of, 53, 67, 642, 643symmetric matrix, 614, 615triangle inequality, 67

skinny, 183, 755slab, 38, 316, 709slack

complementary, 282variable, 267, 276, 299

Slater, 154, 208, 277, 281slice, 123, 173, 415, 668, 669slope, 242solid, 140, 144, 479solution

analytical, 239, 265, 641feasible, 166, 277, 738–742, 767

dual, 281strictly, 277, 281

numerical, 322optimal, 83, 88, 89, 214, 767problem statement as, 7, 265set, 214, 767trivial, 37unique, 45, 213, 214, 409, 410, 555, 557,

563, 573vertex, 83, 222, 350, 404

sortfunction, 223, 325, 549, 553, 575, 761

Page 828: Convex Optimization & Euclidean Distance Geometry

828 INDEX

index matrix, 471largest entries, 223monotone nonnegative, 551smallest entries, 223

span, 63, 763sparsity, 219–222, 225, 284, 324, 325, 328,

330, 331, 337–340, 342, 343, 360,363

gradient, 364nonnegative, 221, 330, 337, 338, 340theorem, 221, 330, 363

spectralcone, 460, 462, 549

dual, 465orthant, 552

inequality, 70, 460norm, 70, 258, 542, 555, 612, 624, 631,

642, 732, 771ball, 70, 732dual, 643, 771inequality, 70, 771Schur-form, 258, 642

projection, see projectionsphere packing, 403steepest descent, 242, 668strict

complementarity, 283feasibility, 277, 281positivity, 441triangle inequality, 444

Sturm, 620subject to, 768submatrix

principal, 407, 477, 482, 506, 599leading, 442, 444theorem, 599

subspace, 36algebra, 36, 85antihollow, 59

antisymmetric, 59, 60, 539Cartesian, 327, 732complementary, 56fundamental, 49, 84, 85, 688, 698geometric center, 429, 430, 488, 498,

522–524, 560, 619, 719

orthogonal complement, 118, 425,500, 719

hollow, 58, 430, 523intersection, 89

hyperplanes, 86nullspace basis

span, 87proper, 36representation, 84symmetric hollow, 58tangent, 118translation invariant, 425, 431, 500, 719

successiveapproximation, 737projection, 737

sum, 46eigenvalues of, 598of eigenvalues, 53, 592of functions, 214, 247, 263of matrices

eigenvalues, 598rank, 128, 129, 593

of singular values, 53, 67, 642, 643of squares, 602vector, 46

orthogonal, 759unique, 56, 148, 627, 758, 759

supremum, 64, 263, 641–644, 768of affine functions, 232, 643of convex functions, 258, 263

surjective, 54, 429, 431, 433, 550, 763linear, 433, 434, 451, 453

svec, 57, 267, 770Swiss roll, 26symmetry, 387

tangentline, 43, 409subspace, 118

tangential, 43, 82, 410Taylor series, 667, 673–675tensor, 660–663tesseract, 52tetrahedron, 144, 145, 337, 481

angle inequality, 479regular, 479, 493

Page 829: Convex Optimization & Euclidean Distance Geometry

INDEX 829

Theobald, 593theorem

0 eigenvalues, 615alternating projection

distance, 740alternative, 162–165, 172

EDM, 534semidefinite, 279weak, 165

Barvinok, 132Bunt-Motzkin, 722Caratheodory, 161, 714compressed sensing, 330, 363cone

faces, 103intersection, 99

conic coordinates, 205convexity condition, 248decomposition

dyad, 620directional derivative, 669discretized membership, 168dual cone intersection, 201duality, weak, 280EDM, 451eigenvalue

of difference, 598order, 597sum, 598zero, 615

elliptope vertices, 451exposed, 108extreme existence, 90extremes, 107Farkas’ lemma, 159, 164

not positive definite, 279positive definite, 278positive semidefinite, 276

fundamentalalgebra, 607algebra, linear, 85convex optimization, 266

generalized inequality and membership,159

Gersgorin discs, 125

gradient monotonicity, 246Hahn-Banach, 74, 79, 84halfspaces, 74Hardy-Littlewood-Polya, 550hypersphere, 451inequalities, 550intersection, 46inverse image, 47, 51

closedness, 175Klee, 107line, 257, 262linearly independent dyads, 627majorization, 585mean value, 674Minkowski, 141monotone nonnegative sort, 551Moreau decomposition, 198, 728Motzkin, 722

transposition, 165nonexpansivity, 730pointed cones, 100positive semidefinite, 590

cone subsets, 129matrix sum, 128principal submatrix, 599symmetric, 600

projectionalgebraic complement, 728minimum-distance, 722, 723on affine, 703on cone, 726on convex set, 722, 723on PSD geometric intersection, 524on subspace, 50via dual cone, 729via normal cone, 723

projectorrank/trace, 699semidefinite, 600

proper-cone boundary, 103Pythagorean, 351, 421, 734range of dyad sum, 628rank

affine dimension, 439partitioned matrix, 604

Page 830: Convex Optimization & Euclidean Distance Geometry

830 INDEX

Schur-form, 604trace, 696

real eigenvector, 606sparse sampling, 330, 363sparsity, 221Tour, 455Weyl, 141

eigenvalue, 598tight, 389, 767trace, 229, 581, 582, 642, 644, 681

/rank gap, 561eigenvalues, 592heuristic, 302–304, 375, 560–563, 643inequality, 596, 597Kronecker, 584maximization, 498minimization, 302–304, 375, 409, 560,

562, 563, 603, 643of product, 51, 593, 596product, 584, 596projection matrix, 696zero, 618

trajectory, 457, 518sensor, 413

transformationaffine, 47, 49, 79, 175, 247, 262

positive semidefinite cone, 114, 115bijective, see bijectiveFourier, discrete, 54, 366injective, see injectiveinvertible, 49, 53, 55linear, 37, 49, 78, 135, 688

cone, 99, 103, 137, 140, 162, 175–177cone, dual, 157, 162, 530inverse, 49, 489, 688polyhedron, 137, 140, 162

rigid, 23, 410similarity, 714surjective, see surjective

transitivity, 93, 100, 596trefoil, 496, 497, 499triangle, 388

inequality, 387, 388, 421, 442–444, 446,454, 477, 478, 480–482, 491, 493

norm, 217

strict, 444unique, 477

triangulation, 23trilateration, 23, 45, 408

tandem, 411Tucker, 164, 198, 547

unbounded below, 83, 88, 89, 164underdetermined, 378, 756unfolding, 496, 500unfurling, 494unimodal, 260unique, 8

eigenvalues, 122, 607extreme direction, 106localization, 23, 308, 310, 408, 411minimum element, 102, 214, 215solution, see solutionsum, 758

orthogonal, 759unraveling, 496, 497, 499untieing, 497, 499

valueabsolute, 216, 252minimum

unique, 214objective, 280, 296

unique, 267variable

matrix, 77slack, 267, 276, 299

variationcalculus, 166total, 370

vec, 51, 367, 582, 583, 770vector, 36, 766

binary, 62, 289, 353, 449direction, 328

interpretation, 326, 327, 562dual, 749normal, 722optimization, 101, 102, 214–216, 324,

371parallel, 701Perron, 459

Page 831: Convex Optimization & Euclidean Distance Geometry

INDEX 831

primal, 749product

inner, 50, 261, 268, 363, 421, 591,710, 760

Kronecker, 664quadrature, 426reflection, 638rotation, 125, 426, 637space, 36, 763standard basis, 58, 145, 396, 768

vectorization, 51projection, 514symmetric, 57

hollow, 60Venn, 142, 269, 540, 541vertex, 41, 45, 69, 91, 92, 220

cone, 100description, 71, 72, 107

affine, 87dual cone, 186, 200halfspace, 138hyperplane, 77polyhedral cone, 70, 143, 162, 177polyhedron, 64, 142projection, 708

elliptope, 287, 450, 451polyhedron, 69, 142, 220solution, 83, 222, 350, 404

Vitulli, 623vortex, 505

Wıκımization, 33, 134, 166, 199, 200, 292,336, 353, 354, 360, 374, 384, 390,397, 400, 470, 472, 473, 620, 732

wavelet, 363, 365wedge, 104, 105, 159Weyl, 141, 598Wiener, 737wireless location, 24, 307, 415Wolkowicz, 653womb, 658Wuthrich, 24

Young, 546, 548, 579

zero, 615

definite, 619entry, 615gradient, 166, 242, 243, 718matrix product, 618norm-, 615trace, 618

Page 832: Convex Optimization & Euclidean Distance Geometry

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