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  • Optimization Problems with StochasticDominance Constraints

    Darinka Dentcheva

    Stevens Institute of Technology, Hoboken, New Jersey, USA

    Research supported by NSF awards DMS-0604060 and CMII-0965702

    Borovec, Bulgaria, September 12-16, 2011

  • Motivation

    Risk-Averse Optimization Models

    Choose a decision z ∈ Z , which results in a random outcomeG(z) ∈ Lp(Ω,F ,P) with “good" characteristics; special attention tolow probability-high impact events.

    Utility models apply a nonlinear transformation to the realizationsof G(z) (expected utility) or to the probability of events (rankdependent utility/distortion). Expected utility models optimizeE[u(G(z))]Probabilistic / chance constraints impose prescribed probabilityon some events: P[G(z) ≥ η]Mean–risk models optimize a composite objective of theexpected performance and a scalar measure of undesirablerealizationsE[G(z)]− %[G(z)] (risk/ deviation measures)Stochastic ordering constraints compare random outcomes usingstochastic orders and random benchmarks

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Outline

    1 Stochastic orders

    2 Stochastic dominance constraints

    3 Optimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelation to coherent measures of risk

    4 Stability of dominance constrained problems

    5 Numerical methods

    6 Applications

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Stochastic Orders

    Integral Univariate Stochastic Orders

    For X ,Y ∈ L1(Ω,F ,P)

    X �F

    Y ⇔∫Ω

    u(X (ω)) P(dω) ≥∫Ω

    u(Y (ω)) P(dω) ∀ u(·) ∈ F

    Collection of functions F is the generator of the order.

    Stochastic Dominance Generators

    F1 ={

    nondecreasing functions u : R→ R}

    generates the usualstochastic order or first order stochastic dominance (X �

    (1) Y )Introduced in statistics in 1947 by Mann and WhitneyBlackwell (1953), Lehmann (1955)F2 =

    {nondecreasing concave u : R→ R

    }generates the second

    order stochastic dominance relation (X �(2) Y )

    Risk-consistency facilitates applications in economics pioneered by Quirkand Saposnik (1962), Fishburn (1964), Hadar and Russell (1969)

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Stochastic Dominance Relation via Distribution Functions

    Distribution Functions

    F1(X ; η) =∫ η−∞

    PX (dt) = P{X ≤ η} for all η ∈ R

    Fk (X ; η) =∫ η−∞

    Fk−1(X ; t) dt for all η ∈ R, k = 2,3, . . .

    k th order Stochastic Dominance

    X �(k) Y ⇔ Fk (X , η) ≤ Fk (Y , η) for all η ∈ R

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • First Order Stochastic DominanceUsual Stochastic Order

    -

    -

    0

    1 6

    ηµXµY

    F1(X ; η)F1(Y ; η)

    X �(1) Y ⇔ F1(X ; η) ≤ F1(Y ; η) for all η ∈ R

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Second-Order Stochastic Dominance

    F2(X ; η) =∫ η−∞

    F1(X ; t) dt =∫Ω

    max(0, η − X (ω)) P(dω) for all η ∈ R

    -

    6

    η

    F2(Y ; η)F2(X ; η)

    X �(2) Y ⇔ F2(X ; η) ≤ F2(Y ; η) for all η ∈ R

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Characterization via Inverse Distribution Functions

    First order dominance ≡ Continuum of probability inequalities

    X �(1) Y ⇔ F(−1)(X ; p) ≥ F(−1)(Y ; p) for all 0 < p < 1F(−1)(X ; p) = inf{η : F1(X ; η) ≥ p}

    Absolute Lorenz function (Max Otto Lorenz, 1905)

    F(−2)(X ; p) =∫ p

    0F(−1)(X ; t) dt for 0 < p ≤ 1,

    F(−2)(X ; 0) = 0 and F(−2)(X ; p) = +∞ for p 6∈ [0, 1].

    Lorenz function and Expected shortfall are Fenchel conjugates

    F(−2)(X ; ·) = [F2(X ; ·)]∗ and F2(X ; ·) = [F(−2)(X ; ·)]∗

    Second order dominance ≡ Relation between Lorenz functionX �(2) Y ⇔ F(−2)(X ; p) ≥ F(−2)(Y ; p) for all 0 ≤ p ≤ 1.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Characterization of Stochastic Dominance by Lorenz Functions

    ���������

    6

    -���������

    1

    F(−2)(X ; p)

    F(−2)(Y ; p)

    µx

    µy

    p

    p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

    0

    X �(2) Y ⇔ F(−2)(X ; p) ≥ F(−2)(Y ; p) for all 0 ≤ p ≤ 1.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Characterization by Rank Dependent Utility Functions

    Rank dependent utility are introduced by Quiggin (1982), Schmeidler(1986–89), Yaari (1987).

    W1 contains all continuous nondecreasing functions w : [0,1]→ R.W2 ⊂ W1 contains all concave subdifferentiable at 0 functions.

    Theorem [DD, A. Ruszczyński, 2006]

    (i) For any two random variables X ,Y ∈ L1(Ω,F ,P) the relationX �(1) Y holds if and only if for all w ∈ W1

    1∫0

    F(−1)(X ; p) dw(p) ≥1∫

    0

    F(−1)(Y ; p) dw(p) (1)

    (ii) X �(2) Y holds true if and only if (1) is satisfied for all w ∈ W2.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Dominance Relation in Optimization

    Introduced by Dentcheva and Ruszczyński (2003)

    min f (z)subject to Gi (z) �(ki ) Yi , i = 1..m

    z ∈ Z

    Yi - benchmark random outcome

    The dominance constraints reflect risk aversion

    Gi (z) is preferred over Yi by all risk-averse decision makers withutility functions in the generator Fki .

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Semi-infinite composite optimization

    X, Y, Z are Banach spaces, Y is separable, T is a compact Hausdorff space

    min ϕ(f (z))

    s.t. G (G(z), t) ≤ 0 for all t ∈ Tz ∈ Z

    f : X→ Z, G : X→ Y are continuously Fréchet differentiable; ϕ : Z→ R isconvex, continuous; G : Y× T → R is continuous, G (·, t) is convex ∀t ∈ TProblem (P2) with second order dominance constraint is obtained whenY = L1(Ω,F ,P), X = G(z), and G is defined as follows:

    G (X , t) =∫Ω

    max(0, t − X (ω)) P(dω)− v(t), t ∈ [a, b] ⊂ R,

    where v(t) =∫Ω

    max(0, t − Y (ω)) P(dω) for some Y ∈ L1(Ω,F ,P).

    D. Dentcheva, A Ruszczyński: Composite semi-infinite optimization, Controland Cybernetics, 36 (2007) 3, 633-646.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Sets Defined by Dominance Relation

    For all k ≥ 1, Y - benchmark outcome in Lk−1(Ω,F ,P), [a,b] ⊆ R.

    Acceptance sets Ak (Y ; [a,b]) = {X ∈ Lk−1 : X �(k) Y in [a,b]}

    Theorem

    The set Ak (Y ; [a,b]) is convex and closed for all [a,b], all Y , andk ≥ 2 . Its recession cone has the form

    A∞k (Y ; [a,b]) ={

    H ∈ Lk−1(Ω,F ,P) : H ≥ 0 a.s. on [a,b]}

    A1(Y ; [a,b]) is closed and Ak (Y ; [a,b]) ⊆ Ak+1(Y ; [a,b]) ∀k ≥ 1.Ak (Y ; [a, b]) is a cone pointed at Y if and only if Y is a constant in [a, b].

    Theorem [DD, A. Ruszczyński, 2004]

    If (Ω,F ,P) is atomless, then A2(Y ;R) = co A1(Y ;R) If Ω = {1..N},and P[k ] = 1/N, then A2(Y ;R) = co A1(Y ;R)

    The result is not true for general probability spaces

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Second Order Dominance Constraints

    Given Y ∈ L1(Ω,F ,P) - benchmark random outcome

    Primal Stochastic Dominance Constraints

    max f (z)(P2) subject to F2(G(z); η) ≤ F2(Y ; η), ∀ η ∈ [a,b],

    z ∈ Z

    Inverse Stochastic Dominance Constraints

    max f (z)(P−2) subject to F(−2)(G(z); p) ≥ F(−2)(Y ; p), ∀ p ∈ [α, β],

    z ∈ Z

    Z is a closed subset of a Banach space X, [α, β] ⊂ (0,1), [a,b] ⊂ RG : X→ L1(Ω,F ,P) is continuous and for P-almost all ω ∈ Ω thefunctions [G(·)](ω) are concave and continuousf : X→ R is concave and continuous

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Optimality Conditions Using von Neumann Utility Functions

    The Lagrangian-like functional L : Z × F2([a,b])→ R

    L(z,u) := f (z) +∫Ω

    [u(G(z))− u(Y )

    ]P(dω)

    F2([a,b]) modified generator.

    Uniform Dominance Condition (UDC) for problem (P2)A point z̃ ∈ Z exists such that inf

    η∈[a,b]

    {F2(Y ; η)− F2(G(z̃); η)

    }> 0.

    Theorem Assume UDC. If ẑ is an optimal solution of (P2) thenû ∈ F2([a,b]) exists:

    L(ẑ, û) = maxz∈Z

    L(z, û) (2)∫Ω

    û (G(ẑ)) P(dω) =∫Ω

    û(Y ) P(dω) (3)

    If for some û ∈ F2([a,b]) an optimal solution ẑ of (2) satisfies thedominance constraints and (3), then ẑ solves (P2).

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Optimality Conditions Using Rank Dependent Utility Function

    Lagrangian-like functional Φ : Z ×W ([α, β])→ R

    Φ(z,w) = f (z) +∫ 1

    0F(−1)(G(z); p) dw(p)−

    ∫ 10

    F(−1)(Y ; p) dw(p)

    W ([α, β]) is the modified generator of the relaxed order

    Uniform inverse dominance condition (UIDC) for (P−2)∃z̃ ∈ Z such that inf

    p∈[α,β]

    {F(−2)(G(z̃); p)− F(−2)(Y ; p)

    }> 0.

    Theorem

    Assume UIDC. If ẑ is a solution of (P−2), then ŵ ∈ W ([α, β]) exists:Φ(ẑ, ŵ) = max

    z∈ZΦ(z, ŵ) (4)∫ 1

    0F(−1)(G(ẑ); p) dŵ(p) =

    ∫ 10

    F(−1)(Y ; p) dŵ(p) (5)

    If for some ŵ ∈ W ([α, β]) and a solution ẑ of (4) the dominanceconstraint and (5) are satisfied, then ẑ is a solution of (P−2).

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Duality Relations to Utility Theories

    The Dual Functionals

    D(u) = supz∈Z

    L(z,u) Ψ(w) = supz∈Z

    Φ(z,w)

    The Dual Problems

    (D2) minu∈F2([a,b])

    D(u) (D−2) minw∈W ([α,β])

    Ψ(w).

    TheoremUnder UDC/UIDC, if problem (P−2) resp. (P−2) has an optimalsolution, then the corresponding dual problem has an optimal solutionand the same optimal value. The optimal solutions of the dualproblem (D2) are the utility functions û ∈ F2([a,b]) satisfying (2)–(3)for an optimal solution ẑ of problem (P2). The optimal solutions of(D−2) are the rank dependent utility functions ŵ ∈ W ([α, β])satisfying (4)–(5) for an optimal solution ẑ of problem (P−2).

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Mean-Risk Models as a Lagrangian Relaxation

    A measure of risk % assigns to an uncertain outcomeX ∈ Lp(Ω,F ,P) a real value %(X ) on the extended real lineR = R ∪ {+∞} ∪ {−∞}.A coherent measure of risk is a convex, positive homogeneous, p.w.monotone functional % : L1(Ω,F ,P)→ R satisfying the%(X + a) = %(X )− a for all a ∈ R.

    Theorem (Necessary Optimality Conditions):

    Under the UIDC, if ẑ is an optimal solution of (P−2), then a riskmeasure %̂ and a constant κ ≥ 0 exist such that G(ẑ) is also anoptimal solution of the problem

    maxz∈Z

    {f (z)− κ%̂(G(z))

    }and (6)

    κ%̂(G(ẑ)) = κ%̂(Y ). (7)

    If the dominance constraint is active, then condition (7) takes on theform %̂(G(ẑ)) = %̂(Y ).

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Extensions and Further Research Directions

    Non-convex problems Optimality conditions for problems withFSD constraints and problems with higher order dominanceconstraints with non-convex functions (DD, A Ruszczyński 2004,2007)Multivariate orders based on scalarization of random vectors(DD, A Ruszczyński, 2007, T. Homem-de-Mello, S. Mehrotra, 2009)Dynamic orders Scalarizations as discounts for stochasticsequences; maximum prnciple; (DD, A Ruszczyński, 2008);Two-stage problems with dominance constraints on the recoursefunction (R. Schultz, F. Neise, R. Gollmer, U. Gotzes, D. Drapkin, 2007,2009, 2010)Stochastic dominance efficiency in multi-objective optimizationand its relations to dominance constraints (G. Mitra, C. Fabian,K. Darby-Dowman, D. Roman, 2006, 2009)Stability and sensitivity analysis (DD, R. Henrion, A Ruszczyński,2007; Y. Liu, H. Xu, 2010; DD W. Römisch 2011)Robust Dominance Relation (DD, A Ruszczyński, 2010)

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Problem and Assumptions

    k -th Order Stochastic Dominance Constraints

    max f (z)(Pk ) subject to Fk (G(z, ξ); η) ≤ Fk (Y ; η), ∀ η ∈ [a,b],

    z ∈ Z .

    k ≥ 2, Z ⊂ Rn is nonempty closed convex, Y ∈ Lk−1(Ω,F ,P)f : Rn → R is convex;ξ is a random vector with closed convex support Ξ ⊂ Rs;G : Rm × Rs → R is continuous, concave w.r.to the first argument,satisfies the linear growth condition

    |G(z, ξ)| ≤ C(B) max{1, ‖ξ‖} (z ∈ B, ξ ∈ Ξ,∀B bounded)

    Theorem

    The feasible set Z (ξ,Y ) is closed and convex.

    Z (ξ,Y ) = {z ∈ Z : Fk (G(z, ξ), η) ≤ Fk (Y , η), ∀η ∈ R}

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Distances between random variables

    Rachev metrics on Lk (Ω,F ,P)

    Dk,p(X ,Y ) =

    (∫

    R

    ∣∣Fk (X , η)− Fk (Y , η)∣∣pdη) 1p for 1 ≤ p

  • Quantitative Stability

    Constraints qualification: k UDC

    A point z̄ ∈ Z exists such that minη∈[a,b]

    (Fk (Y , η)− Fk (G(z̄, ξ), η)) > 0.

    v(ξ,Y ) is the optimal value of (Pk ) and S(ξ,Y ) is its solution set.Consider nondecreasing growth function ψ:

    ψ(τ) = inf{f (z)− v(ξ,Y ) : d(z,S(ξ,Y )) ≥ τ, z ∈ Z (ξ,Y )}

    Θ(t) = t + ψ−1(2t) (t ∈ R+).

    Theorem [DD and W. Römisch 2011]

    Assume k UDC and let Z be compact and G(z, ·) be Lipschitz for allz ∈ Z . Then constants L > 0 and δ > 0 exist such that

    dH(Z (ξ,Y ),Z (ξ̃, Ỹ )) ≤ Ldk ((ξ,Y ), (ξ̃, Ỹ )),|v(ξ,Y )− v(ξ̃, Ỹ )| ≤ L dk ((ξ,Y ), (ξ̃, Ỹ ))sup

    z∈S(ξ̃,Ỹ )d(z,S(ξ,Y )) ≤ Θ

    (L dk ((ξ,Y ), (ξ̃, Ỹ ))

    )for all (ξ̃, Ỹ ) such that dk ((ξ,Y ), (ξ̃, Ỹ )) < δ.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Robust Dominance Relation

    Uncertainty of the probability measure P0Assume that P0 may vary in a set of probability measures Q .

    Consider Lp(Ω,F ,P0), p ∈ [1,∞) and

    Lq(Ω,F ,P0) ≡{

    measures Q : absolutely continuous w.r.to P0with densities dQ/dP0 ∈ Lq(Ω,F ,P0)

    For any X ∈ Lp(Ω,F ,P0) and Q ∈ Lq(Ω,F ,P0)

    〈Q,X 〉 =∫Ω

    X (ω) Q(dω) =∫Ω

    X (ω)dQdP0

    (ω) P0(dω).

    Let Q ⊂ Lq(Ω,F ,P0) be a convex, closed, and bounded set ofprobability measures.

    Definition

    X ∈ Lp(Ω,F ,P0) dominates robustly Y ∈ Lp(Ω,F ,P0) over a set ofprobability measures Q ⊂ Lq(Ω,F ,P0) if∫

    u(X )P(dω) ≥∫Ω

    u(Y )P(dω) ∀ u ∈ U , ∀ P ∈ Q.

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Robust Preferences

    Savage 1954, Gilboa and Schmeidler 1989,F. Maccheroni, A. Rustichini and M. Marinacci, Econometrica 74 (2006)

    Set of axioms imply the existence of a utility function ū : R→ R and aset of measures Q such that

    X is preferred over Y if and only if

    infP∈Q

    ∫Ω

    ū(X ) P(dω) ≥ infP∈Q

    ∫Ω

    ū(Y ) P(dω).

    In the case, U ={

    ū}

    , our definition implies

    infP∈Q

    {∫Ω

    [ū(X )− ū(Y )]P(dω)}≥ 0.

    D. Dentcheva, A Ruszczynski: Robust stochastic dominance constraints,Mathematical Programming, 2010

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Numerical Methods

    Large scale convex optimization methods for second order dominanceconstraints: applicable only to small problems

    Dual methods for SSD constraints (DD, Ruszczyński, 2005); ( Rudolf,Ruszczyński 2006, Luedtke 2008).

    Subgradient Based Approximation Methods for SSD constraints withlinear G(·) (Rudolf, Ruszczyński, 2006; Fabian, Mitra, and Roman,2008)

    Combinatorial methods for FSD constraints (Rudolf, Noyan,Ruszczyński 2006) based on second order stochastic dominancerelaxation ({X : X �(2) Y} = co{X : X �(1) Y})Subgradient methods based on quantile functions and conditionalexpectations (DD, Ruszczyński, 2010)

    Methods for two-stage problems with dominance constraints on therecourse (Schultz, Neise, Gollmer, Drapkin; DD and G. Martinez, 2011)

    Methods for multivariate linear dominance constraints(Homem-de-Mello, Mehrotra, 2009; Hu, Homem-de-Mello, and Mehrotra2010, Armbruster and Luedtke 2010)

    Sample average approximation methods (Sun, Xu and Wang, 2011)

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

  • Applications

    Finance: portfolio optimizationElectricity markets: portfolio of contracts and/or acceptability ofcontractsInverse models and forecasting: Compare the forecast errors viastochastic dominance and design data collection for modelcalibrationNetwork design: assign capacity to optimize network throughputMedicine: radiation therapy designs

    Darinka Dentcheva Optimization under Stochastic Ordering Constraints

    Stochastic ordersStochastic dominance constraintsOptimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelation to coherent measures of risk

    Stability of dominance constrained problemsNumerical methodsApplications