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Variational Analysis of Convexly GeneratedSpectral Max Functions

James V BurkeMathematics, University of Washington

Joint work withJulie Eation (UW), Adrian Lewis (Cornell),

Michael Overton (NYU)

Universitat BonnHaursdorff Center for Mathematics

May 22, 2017

Convexly Generated Spectral Max Functions

f : Cnn R {+} =: R

f(X) := max{f() | C and det(I X) = 0},

where f : C R is closed, proper, convex.

Spectral abscissa: f() = Re() and we write f = .

Spectral radius: f() = || and we write f = .

Convexly Generated Spectral Max Functions

f : Cnn R {+} =: R

f(X) := max{f() | C and det(I X) = 0},

where f : C R is closed, proper, convex.

Spectral abscissa: f() = Re() and we write f = .

Spectral radius: f() = || and we write f = .

Convexly Generated Spectral Max Functions

f : Cnn R {+} =: R

f(X) := max{f() | C and det(I X) = 0},

where f : C R is closed, proper, convex.

Spectral abscissa: f() = Re() and we write f = .

Spectral radius: f() = || and we write f = .

Example: Damped Oscillator: w + w + w = 0

u =

[0 11

]u, u =

(ww

)A() =

[0 11 0

]+

[0 00 1

]

(A()) =

{

2 4

2, (A()) =

+24

2 , || > 2 ,

/2 , || 2 .

4 3 2 1 0 1 2 3 41

0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Fundamentally Non-Lipschitzian

N() =

0 1 0 0 0 00 0 1 0 0 0...

. . .

0 0 1 0 0

nn

det[IN()] = n = k := ()1/ne2ki/n k = 0, . . . , n1.

Active eigenvalue hypothesis(H) For all active eigenvalues one of the following holds:

(A) f is quadratic, or f is C2 and positive definite(B) rspan (f()) = C

For : C R, define : R2 R by

= ,

where : R2 C is the R-linear transformation

(x) = x1 + ix2.

Note 1 = =

[ReIm

].

is R-differentiable if is differentiable, and by the chain rule

() = ().

Similarly,2() = 2().

Active eigenvalue hypothesis(H) For all active eigenvalues one of the following holds:

(A) f is quadratic, or f is C2 and positive definite(B) rspan (f()) = C

For : C R, define : R2 R by

= ,

where : R2 C is the R-linear transformation

(x) = x1 + ix2.

Note 1 = =

[ReIm

].

is R-differentiable if is differentiable, and by the chain rule

() = ().

Similarly,2() = 2().

Key Result: B-Eaton (2016)

Suppose that X Cnn is such that hypothesis (H) holds at allactive eigenvalues of X with f() 6= {0}.

Then f is subdifferentially regular at X if and only if the activeeigenvalues of X are nonderogatory.

Nonderogatory Matrices:

An eigenvalue if nonderogatory if it has only one eigenvector.

The set of nonderogatory matrices is an open dense set in Cnn.

If matrices are stratified by Jordan structure, then within everystrata the submanifold of nonderogatory matrices has the sameco-dimension as the strata.

Spectral abscissa case, B-Overton (2001).

Key Result: B-Eaton (2016)

Suppose that X Cnn is such that hypothesis (H) holds at allactive eigenvalues of X with f() 6= {0}.

Then f is subdifferentially regular at X if and only if the activeeigenvalues of X are nonderogatory.

Nonderogatory Matrices:

An eigenvalue if nonderogatory if it has only one eigenvector.

The set of nonderogatory matrices is an open dense set in Cnn.

If matrices are stratified by Jordan structure, then within everystrata the submanifold of nonderogatory matrices has the sameco-dimension as the strata.

Spectral abscissa case, B-Overton (2001).

Key Result: B-Eaton (2016)

Suppose that X Cnn is such that hypothesis (H) holds at allactive eigenvalues of X with f() 6= {0}.

Then f is subdifferentially regular at X if and only if the activeeigenvalues of X are nonderogatory.

Nonderogatory Matrices:

An eigenvalue if nonderogatory if it has only one eigenvector.

The set of nonderogatory matrices is an open dense set in Cnn.

If matrices are stratified by Jordan structure, then within everystrata the submanifold of nonderogatory matrices has the sameco-dimension as the strata.

Spectral abscissa case, B-Overton (2001).

Key Result: B-Eaton (2016)

Suppose that X Cnn is such that hypothesis (H) holds at allactive eigenvalues of X with f() 6= {0}.

Then f is subdifferentially regular at X if and only if the activeeigenvalues of X are nonderogatory.

Nonderogatory Matrices:

An eigenvalue if nonderogatory if it has only one eigenvector.

The set of nonderogatory matrices is an open dense set in Cnn.

If matrices are stratified by Jordan structure, then within everystrata the submanifold of nonderogatory matrices has the sameco-dimension as the strata.

Spectral abscissa case, B-Overton (2001).

Key Result: B-Eaton (2016)

Suppose that X Cnn is such that hypothesis (H) holds at allactive eigenvalues of X with f() 6= {0}.

Then f is subdifferentially regular at X if and only if the activeeigenvalues of X are nonderogatory.

Nonderogatory Matrices:

An eigenvalue if nonderogatory if it has only one eigenvector.

The set of nonderogatory matrices is an open dense set in Cnn.

If matrices are stratified by Jordan structure, then within everystrata the submanifold of nonderogatory matrices has the sameco-dimension as the strata.

Spectral abscissa case, B-Overton (2001).

Convexly Generated Polynomial Root Max Functions

The characteristic polynomial mapping n : Cnn Pn:

n(X)() := det(I X).

Polynomial root max function generated by f is the mappingf : Pn R defined by

f(p) := max{f() | C and p() = 0}.

Key Result:Suppose that p Pn is such that (H) holds at all active roots of p with f() 6= {0}.Then f is subdifferentially regular at p.

The abscissa case, B-Overton (2001).

Convexly Generated Polynomial Root Max Functions

The characteristic polynomial mapping n : Cnn Pn:

n(X)() := det(I X).

Polynomial root max function generated by f is the mappingf : Pn R defined by

f(p) := max{f() | C and p() = 0}.

Key Result:Suppose that p Pn is such that (H) holds at all active roots of p with f() 6= {0}.Then f is subdifferentially regular at p.

The abscissa case, B-Overton (2001).

The Plan

Cnn

G(0,)

f // R

SpFp

// Pnf

OO

p() :=

mj=1

( j)nj

e(nj ,j)() := ( j)nj

G(0, X) := (0, e(n1,1), . . . , e(nm,m)) are the active factors

of the characteristic polynomial n(X).

The mapping G : C Cnn Sp takes a matrix X Cnn tothe active factors (of degree n n) associated with itscharacteristic polynomial n(X).

p has a local factorization based at its roots giving rise to afactorization space Sp and an associated diffeomorphismFp : Sp Pn.

Apply a nonsmooth chain rule.

The Plan

Cnn

G(0,)

f // R

SpFp

// Pnf

OOp() :=

mj=1

( j)nj

e(nj ,j)() := ( j)nj

G(0, X) := (0, e(n1,1), . . . , e(nm,m)) are the active factors

of the characteristic polynomial n(X).

The mapping G : C Cnn Sp takes a matrix X Cnn tothe active factors (of degree n n) associated with itscharacteristic polynomial n(X).

p has a local factorization based at its roots giving rise to afactorization space Sp and an associated diffeomorphismFp : Sp Pn.

Apply a nonsmooth chain rule.

Inner Products

Lemma (Inner product construction)

Let L1 and L2 be finite dimensional vector spaces over F = C orR, and suppose

L : L1 L2 is an F-linear isomorphism.

If L2 has inner product , 2, then the bilinear formB : L1 L1 F given by

B(x, y) := Lx,Ly2 x, y L1

is an inner product on L1.The adjoint mappings L? : L2 L1 and (L1)? : L1 L2 withrespect to the inner products , 1 := B(x, y) and , 2 satisfy

L? = L1 and (L1)? = L.

Real and complex inner products, , and , c, resp.ly.

Inner Products

Lemma (Inner product construction)

Let L1 and L2 be finite dimensional vector spaces over F = C orR, and suppose

L : L1 L2 is an F-linear isomorphism.

If L2 has inner product , 2, then the bilinear formB : L1 L1 F given by

B(x, y) := Lx,Ly2 x, y L1

is an inner product on L1.The adjoint mappings L? : L2 L1 and (L1)? : L1 L2 withrespect to the inner products , 1 := B(x, y) and , 2 satisfy

L? = L1 and (L1)? = L.

Real and complex inner products, , and , c, resp.ly.

Factorization Spaces

Given p Mn (monic polynomials of degree at most n), write

p :=mj=1e(nj ,j),

where 1, . . . , m are the distinct roots of p, orderedlexicographically with multiplicities n1, . . . , nm, and themonomials e(`,j) are defined by

e(`,0)() := ( 0)`.

The factorization space Sp for p is given by

Sp := C Pn11 Pn21 Pnm1,

where the component indexing for elements of Sp starts at zeroso that the jth component is an element of Pnj1.

Taylor Mappings Tp : Sp Cn+1

For each 0 C, the scalar Taylor maps (k,0) : Pn C are

(k,0)(q) := q(k)(0)/k! , for k = 0, 1, 2, . . . , n,

where q(`) is the `th derivative of q.

Define the C-linear isomorphism Tp : Sp Cn+1 by