Optimization Problems with Stochastic Dominance Constraints - Institute … · 2011-09-12 ·...

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


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)


Stochastic Dominance Relation via Distribution Functions

Distribution Functions

F1(X ; η) =

∫ η

−∞PX (dt) = PX ≤ 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


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


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


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) = infη : 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 function

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


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.


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. Ruszczynski, 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.


Dominance Relation in Optimization

Introduced by Dentcheva and Ruszczynski (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 .


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 ∈ T

z ∈ 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 Ruszczynski: Composite semi-infinite optimization, Controland Cybernetics, 36 (2007) 3, 633-646.


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. Ruszczynski, 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


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


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 z ∈ Z exists such that inf

η∈[a,b]

F2(Y ; η)− F2(G(z); η)

> 0.

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

L(z, u) = maxz∈Z

L(z, u) (2)∫Ω

u (G(z)) P(dω) =

∫Ω

u(Y ) P(dω) (3)

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


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)−

∫ 1

0F(−1)(Y ; p) dw(p)

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

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

p∈[α,β]

F(−2)(G(z); p)− F(−2)(Y ; p)

> 0.

Theorem

Assume UIDC. If z is a solution of (P−2), then w ∈ W ([α, β]) exists:

Φ(z, w) = maxz∈Z

Φ(z, w) (4)∫ 1

0F(−1)(G(z); p) dw(p) =

∫ 1

0F(−1)(Y ; p) dw(p) (5)

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


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 u ∈ F2([a,b]) satisfying (2)–(3)for an optimal solution z of problem (P2). The optimal solutions of(D−2) are the rank dependent utility functions w ∈ W ([α, β])satisfying (4)–(5) for an optimal solution z of problem (P−2).


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

maxz∈Z

f (z)− κ%(G(z))

and (6)

κ%(G(z)) = κ%(Y ). (7)

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


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 Ruszczynski 2004,2007)Multivariate orders based on scalarization of random vectors(DD, A Ruszczynski, 2007, T. Homem-de-Mello, S. Mehrotra, 2009)Dynamic orders Scalarizations as discounts for stochasticsequences; maximum prnciple; (DD, A Ruszczynski, 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 Ruszczynski,2007; Y. Liu, H. Xu, 2010; DD W. Römisch 2011)Robust Dominance Relation (DD, A Ruszczynski, 2010)


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) max1, ‖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


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 <∞

supη∈R

∣∣Fk (X , η)− Fk (Y , η)∣∣ for p =∞

(8)

Lk−1(Ω,F ,P)-minimal distance (k ≥ 2)

`k−1(ξ, ξ) = inf‖ζ − ζ‖k−1 : ζ

d= ξ, ζ

d= ξ

and d= means equality in distribution.

Distance on Lk−1(Ω,F ,P)×Lk−1(Ω,F ,P)

dk ((ξ,Y ), (ξ, Y )) = `k−1(ξ, ξ) + Dk,∞(Y , Y ),


Quantitative Stability

Constraints qualification: k UDC

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

(Fk (Y , η)− Fk (G(z, 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) = inff (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 (ξ, Y )) ≤ Ldk ((ξ,Y ), (ξ, Y )),

|v(ξ,Y )− v(ξ, Y )| ≤ L dk ((ξ,Y ), (ξ, Y ))

supz∈S(ξ,Y )

d(z,S(ξ,Y )) ≤ Θ(L dk ((ξ,Y ), (ξ, Y ))

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


Robust Dominance Relation

Uncertainty of the probability measure P0

Assume 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 P0

with 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∫

Ω

u(X )P(dω) ≥∫Ω

u(Y )P(dω) ∀ u ∈ U , ∀ P ∈ Q.


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 u : R→ R and aset of measures Q such that

X is preferred over Y if and only if

infP∈Q

∫Ω

u(X ) P(dω) ≥ infP∈Q

∫Ω

u(Y ) P(dω).

In the case, U =

u

, our definition implies

infP∈Q

∫Ω

[u(X )− u(Y )]P(dω)≥ 0.

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


Numerical Methods

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

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

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

Combinatorial methods for FSD constraints (Rudolf, Noyan,Ruszczynski 2006) based on second order stochastic dominancerelaxation (X : X (2) Y = coX : X (1) Y)Subgradient methods based on quantile functions and conditionalexpectations (DD, Ruszczynski, 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)


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


Optimization Problems with Stochastic Dominance Constraints - Institute … · 2011-09-12 ·...

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