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Page 1: Stochastic Orders in Risk-averse Optimizationcermics.enpc.fr/~delara/SESO/SESO2016/SESO2016-Wednesday_De… · 1 Stochastic orders Stochastic dominance and the increasing convex order

Stochastic Orders in Risk-averse Optimization

Darinka Dentcheva

Stevens Institute of Technology Hoboken, New Jersey, USA

Research supported by NSF award DMS-1311978

June 1, 2016

Paris, France

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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.

I 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))]

I Probabilistic / chance constraints impose prescribed probabilityon some events: P[G(z) ≥ η]

I Mean–risk models optimize a composite objective of theexpected performance and a scalar measure of undesirablerealizations E[G(z)]− %[G(z)] (risk/ deviation measures)

I Stochastic-order constraints compare random outcomes usingstochastic orders and random benchmarks

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Outline

1 Stochastic ordersStochastic dominance and the increasing convex orderMultivariate and Dynamic Orders

2 Optimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelations to coherent measures of risk

3 Stochastic dominance constraints in nonconvex problems

4 Numerical methodsPrimal outer approximation methodQuantile Cutting Plane Methods for General Probability SpacesDecomposition methods for two-stage problemsDual Approach and Regularization

5 Applications

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References

1 D. Dentcheva, A. Ruszczynski: Stochastic optimization with dominanceconstraints, SIAM J. Optimization, 14 (2003) 548–566.

2 — " — Inverse stochastic dominance constraints and rank dependentutility theory, Mathematical Programming 108 (2006), 297–311.

3 — " — Optimization with multivariate stochastic dominance constraints,Mathematical Programming, 117 (2009) 111–127.

4 — " — Duality between coherent risk measures and stochasticdominance constraints in risk-averse optimization, Pacific Journal onOptimization, 4 (2008), No. 3, 433–446.

5 — " — Inverse cutting plane methods for optimization problems withsecond-order stochastic-dominance constraints, Optimization, 59 (2010)323–338.

6 D. Dentcheva, G. Martinez, Two-stage stochastic optimization problemswith stochastic-order constraints on the recourse, European Journal ofOperations Research 1 (2012), 1, 1–8.

7 D. Dentcheva, E. Wolfhagen: Optimization with multivariate stochasticdominance constraints, SIAM J. Optimization, 25 (2015) 1, 564–588.

8 D. Dentcheva, W. Römisch: Stability and sensitivity of stochasticdominance constrained optimization models, SIAM J. Optimization, 23(2014) 3, 1672–1688.

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Integral Univariate Stochastic Orders

For X ,Y ∈ L1(Ω,F ,P)

X U Y ⇔∫Ω

u(X (ω)) P(dω) ≥∫Ω

u(Y (ω)) P(dω) ∀ u(·) ∈ U

Collection of functions U is the generator of the order.

Generators:I U1 =

nondecreasing functions u : R→ R

generates the usual

stochastic order or first order stochastic dominance (X (1) Y )Mann and Whitney (1947), Blackwell (1953), Lehmann (1955)

I U2 =

nondecreasing concave u : R→ R

generates the secondorder stochastic dominance relation (X (2) Y )Quirk and Saposnik (1962), Fishburn (1964), Hadar and Russell (1969)

I U2 =

nondecreasing convex u : R→ R

generates theincreasing convex order (X ic Y )

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First Order Stochastic Dominance

-

-

0

16

ηµXµY

F (1)(X ; η)F (1)(Y ; η)

Distribution function F (X ; η) =

∫ η

−∞PX (dt) = PX ≤ η, η ∈ R

Quantile function F (−1)(X ; p) = infη : F (X ; η) ≥ p, p ∈ (0,1)

Survival function F (X ; η) = 1− F (X ; η) = PX > η, η ∈ R

The usual stochastic order

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

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

⇔ F (X ; η) ≥ F (Y ; η) for all η ∈ R.

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Second-Order Stochastic Dominance

Shortfall function F (2)(X , η) =∫ η−∞ F (X , t) dt = E[(η − X )+] η ∈ R.

Lorenz function: F (−2)(X ; p) =∫ p

0 F (−1)(X ; t) dt p ∈ (0,1].

-

6

ηµx

η − µxF (2)(X ; η)

6

-

1

F (−2)(X ; p)µx

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 p

0

Fenchel conjugate function F ∗(p) = supupu − F (u).

Theorem (Ogryczak and Ruszczynski, 2002)

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

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Second-Order Stochastic Dominance

X (2) Y ⇔ E[(η − X )+] ≤ E[(η − Y )+]

⇔ F (−2)(X ; p) ≥ F (−2)(Y ; p) ∀p ∈ [0,1].

-

6

η

F (2)(Y , η)F (2)(X , η)

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

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Increasing convex order

Characterization by the integrated survival function

For X ,Y ∈ L1, the relation X ic Y holds if and only if∫ ∞η

P(X > t) dt ≤∫ ∞η

P(Y > t) dt for all η ∈ R.

The excess function and its Fenchel conjugate

H(Z , η) =

∫ ∞η

F (Z , t) dt = E(Z − η)+

L(Z ,q) = −∫ 1

1+qF (−1)(Z , t) dt for − 1 ≤ q < 0,

L(Z ,0) = 0, L(Z ,q) =∞ for q 6∈ [−1,0]

Increasing convex order vs. Second order dominance

X ic Y ⇔ −X (2) −Y .

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Higher order relation

For any X ∈ Lk (Ω,F ,P)(Ω,F ,P), ‖X‖k =(E(|X |k )

) 1k and

F (k+1)(X ; η) =

∫ η

−∞F (k)(X ; t)dt =

1k !

∫ η

−∞(η − t)k PX (dt) =

1k !‖max

(0, η − X

)‖k

k ∀η ∈ R,

k th degree Stochastic Dominance (kSD), k ≥ 2

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

‖max(0, η − X

)‖k−1

k−1 ≤ ‖max(0, η − Y

)‖k−1

k−1

The generator

Uk contains all functions u : R→ R such that a non-increasing,left-continuous, bounded function ϕ : R→ R+ exists such thatu(k−1)(η) = (−1)kϕ(η) for a.a. η ∈ R.

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Multivariate and Dynamic Orders

Consider X = (X1, . . . ,Xm) and Y = (Y1, . . . ,Ym) in Lm1 (Ω,F ,P).

Coordinate Order

X sep(2) Y ⇔ Xt (2) Yt , t = 1, . . . ,m

The generator consists of all functions u(X ) =∑m

t=1 ui (Xi ) withconcave nondecreasing ui : R→ R.

Ignores temporal structure and dependency

Increasing Convex Order

X Y ⇔ E[u(X )

]≥ E

[u(Y )

]∀ u ∈ F

The generator consists of all concave nondecreasing functionsu : Rm → R

Hard to treat analytically, the generator is too rich

Adopted approach

Define the mutlivariate order via a family of univariate orders

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Multivariate Orders: Definition

DefinitionGiven a closed convex set C ∈ Rm

+ and a mapping M : c 7→ Uc , c ∈ C,where Uc are univariate generators, a random vector X ∈ Lm

1 isstochastically larger than a random vector Y ∈ Lm

1 with respect to Mand C if

c>X Uc c>Y for all c ∈ C.

Example

I Set M(c) ⊂ R and S = c ∈ Rm+ : ‖c‖1 = 1. For X ,Y ∈ Lm

1 ,

X M(2) Y ⇔ E[(η − c>X )+] ≤ E[(η − c>Y )+] ∀(c, η) ∈ graphM.

I If M(c) = [a,b], the order is known as linear second orderdominance.

Other definitions: A. Müller, D. Stoyan, Homem-de-Mello and Mehrotra: lineardominance with S a polyhedron, or a compact convex set.

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Multivariate Stochastic Dominance: Generator of the order

The set Ψ(M) contains all mappings φ : c ∈ S 7→ U2(M(c)

)such that

(c, x)→ [φ(c)](c>x) is Lebesgue measurable on S × Rm.

M(S) is the space of regular countably additive measures on S withfinite variation;M+(S) is its subset of nonnegative measures.

With every mapping φ ∈ Ψ(M) and every finite measure µ ∈M+(S)we associate a function ϕφ,µ : Rm → R as follows:

ϕφ,µ(x) =

∫S

[φ(c)](c>x)µ(dc).

Define UmM = ϕφ,µ : φ ∈ Ψ(M), µ ∈M+(S).

Theorem

For each X ,Y ∈ Lm1 the relation X M

(2) Y is equivalent to

E[ϕ(X )] ≥ E[ϕ(Y )] for all ϕ ∈ UmM.

Moreover, UmM is the maximal generator of the order M

(2) .

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Discounted Dynamic Stochastic Order

Definition

A sequence X ∈ LT1 is stochastically larger than a sequence Y ∈ LT

1X dis

(k) Y if

〈λ,X 〉 (k) 〈λ,Y 〉 ∀λ ∈ D ⊆λ ∈ RT : 1 ≥ λ1 ≥ λ2 ≥ · · · ≥ λT ≥ 0

.

Example Finite set D consisting of T elements λ`, ` = 1, . . . ,T , with

λ`t =

1 if t ≤ `,0 if t > `.

Then X dis(k) Y is equivalent to the dominance of partial sums

∑t=1

Xt (k)

∑t=1

Yt , ` = 1, . . . ,T .

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Rank Dependent Utility Functions/Distortions

W1 = w : [0, 1]→ R : w is continuous nondecreasing.W2 = w ∈ W1 : w concave subdifferentiable at 0.

Theorem

For all random variables X ,Y ∈ L1, the relation X (i) Y , i = 1, 2 holds if andonly if for all w ∈ Wi∫ 1

0F (−1)(X ; p) dw(p) ≥

∫ 1

0F (−1)(Y ; p) dw(p). (1)

Corollary: X ic Y ⇔ (1) for all convex functions w .

Let M− : S ⇒ (0, 1) have closed convex images.

Multivariate distortions

For X ,Y ∈ Lm1 , the relation X (i) Y , i = 1, 2 holds if and only if for all

measurable ϑ : c ∈ S →Wi (M−(c)) and all µ ∈M+(S)

∫S

1∫0

F (−1)(c>X ; p) dϑc(p) dµ(c) ≥∫S

1∫0

F (−1)(c>Y ; p) dϑc(p) dµ(c) (2)

Quiggin (1982), Schmeidler (1986–89), Yaari (1987)Darinka Dentcheva Stochastc-orders in Optimization

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Acceptance Sets

For all k ≥ 1, Y - benchmark outcome in Lk−1(Ω,F ,P).

Acceptance sets Ak (Y ) = X ∈ Lk−1 : X (k) Y

Theorem

The set Ak (Y ) is convex and closed for all Y , and k ≥ 2 . Itsrecession cone has the form

A∞k (Y ) =

H ∈ Lk−1(Ω,F ,P) : H ≥ 0 a.s.

A1(Y ) is closed and Ak (Y ) ⊆ Ak+1(Y ) ∀k ≥ 1. Ak (Y ) is a conepointed at Y if and only if Y is a constant.

Theorem

If (Ω,F ,P) is atomless, then A2(Y ) = conv A1(Y ) If Ω = 1..N, andP[k ] = 1/N, then A2(Y ) = conv A1(Y )

The result is not true for general probability spaces

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Dominance Relation as Constraints in Optimization

min f (z)

subject to Gi (z) Ui Yi , i = 1..mz ∈ Z

Yi - benchmark random outcome

The stochastic order constraints reflect risk aversionI Gi (z) is preferred over Yi by all risk-averse decision makers with

utility functions in the generator Ui ;I Easier consensus on a benchmark rather than a utility function;I Data of a benchmark is readily available.

G(z) (1) Y ≡ continuum of chance constraints;G(z) (2) Y ≡ continuum of CV@R constraints.

Introduced by Dentcheva and Ruszczynski (2003)

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Second Order Multivariate Dominance Constraints

Given Y ∈ Lm1 - benchmark random vector

Direct Stochastic Order Constraints

(P) min f (z)

s. t. E[(η − c>G(z))+] ≤ E[(η − c>Y )+]

∀ (c, η) ∈ graphM,

z ∈ Z .

Let M− : S ⇒ (0, 1) have closed convex images.

Inverse Stochastic Order Constraints

(Q) min f (z)

s. t. F (−2)(c>G(z); p) ≥ F (−2)(c>Y ; p)

∀ (c, p) ∈ graphM−,

z ∈ Z .

Z is a closed subset of a Banach space X ; M(c), resp. M−(c) are compact,G : X → Lm

1 is continuous and for P-almost all ω ∈ Ω the functions [Gi (·)](ω)

are concave and continuous. f : X → R is convex and continuous.Darinka Dentcheva Stochastc-orders in Optimization

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Optimality Conditions Using von Neumann Utility Functions

The Lagrangian-like functional L : X × UmM → R

L(z,u) = f (z)− E[u(G(z))− u(Y )

]Uniform Dominance Condition (UDC) for problem (P)

∃z ∈ Z : inf(η,c)∈graph M

F (2)(c>Y ; η)− F (2)(c>G(z); η)

> 0.

Theorem

Assume UDC. If z is an optimal solution of (P) then u ∈ UmM exists:

L(z, u) = minz∈Z

L(z, u) (3)

E[u(G(z)

]= E

[u(Y )

](4)

If for some u ∈ UmM an optimal solution z of (3) satisfies the

dominance constraints and (4), then z solves (P).

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Optimality Conditions Using Rank Dependent Utility Function

Lagrangian-like functional

Λ(z, ϑ, µ) = f (z)−∫S

1∫0

F (−1)(c>G(z); p)− F (−1)(c>Y ; p) dϑc(p) dµ(c)

Uniform inverse dominance condition (UIDC) for (Q)∃z ∈ Z inf

(p,c)∈graph M−

F (−2)(c>G(z); p)− F (−2)(c>Y ; p)

> 0.

Theorem

Assume UIDC. If z is a solution of (Q), then ϑ : S →W2(M−) andµ ∈M+(S) exist:

Λ(z, ϑ, µ) = minz∈Z

Λ(z, ϑ, µ) (5)∫S

∫ 1

0F (−1)(G(z); p) d ϑc(p) d µ(c) =

∫S

∫ 1

0F (−1)(Y ; p) d ϑc(p) d µ(c) (6)

If for some ϑ : S →W(M−), µ ∈M+(S) and a solution z of (5) the orderconstraint and (6) are satisfied, then z is a solution of (Q).

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Duality Relations to Utility Theories

The Dual Functionals

D(u) = infz∈Z

L(z,u) ∆(ϑ, µ) = infz∈Z

Λ(z, ϑ, µ)

The Dual Problems

(D2) maxu∈Um(M)

D(u) (D−2) maxϑ,µ

Φ(ϑ, µ).

TheoremUnder UDC/UIDC, if problem (P) resp. (Q) has an optimal solution,then the corresponding dual problem has an optimal solution and thesame optimal value. The optimal solutions of the dual problem are theutility functions u ∈ Um(M) satisfying (3)–(4) for an optimal solution zof problem (P). The optimal solutions of (D−2) provide rankdependent utility functions ϑc ∈ W(M−) and non-negative measureson S satisfying (5)–(6) for an optimal solution z of problem (Q).

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Measures of Risk

A coherent measure of risk for random variables expressing cost orlosses is a functional % : L1(Ω,F ,P)→ R satisfying the axioms:

I Convexity: %(αX + (1− α)Y ) ≤ α%(X ) + (1− α)%(Y ) for allX ,Y ∈ L1, ∀α ∈ [0,1].

I Monotonicity: If X (ω) ≤ Y (ω) ∀ω ∈ Ω, then %(X ) ≤ %(Y ).I Translation Equivariance : %(X + a) = %(X ) + a ∀a ∈ R.I Positive homogeneity: %(tX ) = t%(X ) ∀t > 0.

A coherent measure of risk for random variables expressing gains isa functional % : L1(Ω,F ,P)→ R satisfying the convexity, positivehomogeneity, and the following modified axioms:

I Monotonicity: If X (ω) ≤ Y (ω) ∀ω ∈ Ω, then %(X ) ≥ %(Y ).I Translation Equivariance : %(X + a) = %(X )− a ∀a ∈ R.

Definition

A risk measure % : Z → R is law invariant, if it assigns the same valueto random variables which have the same distribution.

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Relations to Measures of Risk

Consistency of law-invariant risk measure with orders

Suppose the probability space is nonatomic. A risk measure isconsistent with the usual stochastic order iff it satisfies themonotonicity axiom. A proper lower semicontinuous convex riskmeasure % : Z → R for variables expressing losses (gains) isconsistent with the increasing convex order (second-order stochasticdominance), i.e.

X ic Y (X (2) Y ) ⇒ %(X ) ≤ %(Y ).

maxz,σf (z)− λσ : z ∈ Z , G(z) + σ (2) Y

λ > 0 is a tradeoff between f (·) and the error in dominating.

Proposition

The optimal value of σ is a coherent measure of risk expressinglosses.

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Mean-risk models as Lagrangian Relaxation

Kusuoka representation

If Ω is atomless, then for every law invariant measure of risk on% : L∞(Ω,F ,P)→ R a convex setM% of probability measures on (0, 1]exists such that

%(X ) = supµ∈M%

−∫ 1

0

1p

F (−2)(X ; p)µ(dp) ∀X ∈ L∞.

(Q′) max

f (X ) : F (−2)(X ; p) ≥ F (−2)(Y ; p), p ∈ [α, β],X ∈ C.

Theorem

Under the UIDC, if X is an optimal solution of (Q’), then a law-invariantcoherent risk measure % (withM% being a singleton) and κ ≥ 0 exist suchthat X is a solution of the mean-risk problem

maxX∈C

f (X )− κ%(X )

and κ%(X ) = κ%(Y ).

If the dominance constraint is active, then %(X ) = %(Y ).

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The Implied Dominance Constraint

Given the problem

(R) minX∈C

f (X )− κ%(X )

%(·) a coherent law invariant measure of risk and κ > 0

Theorem IfM% is compact∗ inM([0,1]) and X is a solution ofproblem (R), then ∃ µ ∈M such that

%(X ) =

∫ 1

0

1p

F (−2)(X ; p) µ(dt),

and for every Y ∈ L1(Ω,F ,P) satisfying the conditions

F (−2)(Y ; t) ≤ F (−2)(X ; t), for all t ∈ [0,1],

F (−2)(Y ; t) = F (−2)(X ; t), for all t ∈ supp(µ),

the point X is also a solution of problem (Q′) with [α, β] = [0,1].

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Stochastic Dominance Constraints: The Nonconvex Case

min f (H(z))

(P ′) s.t. F (2)(G(z); η) ≤ F (2)(Y ; η) η ∈ [a,b]

z ∈ Z

Assumption

H(·) and G(·) are continuously Fréchet differentiable.

We define I0(z) =η ∈ [a,b] : F (2)(G(z); η) = F (2)(Y ; η)

.

The differential constraint qualification condition at z0 ∈ Z

A point zS ∈ Z and a constant δ > 0 exist such that for all η ∈ I0(z0)∫G(z0)<η

[G′(z0)(zS − z0)](ω) P(dω)

≥∫

G(z0)=η

max(0,−[G′(z0)(zS − z0)](ω)

)P(dω) + δ.

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The non-convex case: continued

The normal cone of Z at z is denoted by NZ (z)

Theorem

If the point z is a local minimum of problem P ′ and the differentialconstraint qualification condition is satisfied at z, then a functionu ∈ U2 exists such that

0 ∈ [H ′(z)]∗∂f (H(z))− [G′(z)]∗∫Ω

∂u(G(z)) P(dω) + NZ (z),

E[u(G(z))

]= E

[u(Y0)

].

The integral∫Ω∂u(Y ) P(dω) is understood as a collection of integrals

of all measurable selections of the multifunction ω → ∂u(Y (ω)).

Key idea of the proof Local linearization of H and G around z andconvex approximation of the objective function and the feasible setsuch that the tangent cones of both feasible sets at z are the same.(DD, A. Ruszczynski, Composite semi-infinite optimization, Control andCybernetics, 36 (2007) 3, 633–646).

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Control problem with order constraint and its risk-neutralequivalent

(C)

maxT∑

t=1

E[Gt (st , vt )

]+ E

[GT+1(sT+1)

]s.t. st+1 = Atst + Btvt + et , t = 1, . . . ,T ,

(G1(s1, v1), . . . ,G1(sT , vT ),GT+1(sT+1)) dis(2) (Y1, . . . ,YT ,YT+1)

vt ∈ Vt a.s., t = 1, . . . ,T .

Theorem

If (s, v) is an optimal solution of problem (C), then a random discountsequence ξt ∈ L∞(Ω,Ft ,P), t = 1, . . . ,T + 1, exists such that (s, v) is anoptimal solution of the problem

maxT∑

t=1

E[(1 + ξt )Gt (st , vt )

]+ E

[(1 + ξT+1)GT+1(sT+1)

]s.t. st+1 = Atst + Btvt + et , t = 1, . . . ,T ,

vt ∈ Vt a.s., t = 1, . . . ,T .

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Maximum Principle

If (s, v) is an optimal solution of the control problem, then there exist:discount factors ξt ∈ L∞(Ω,Ft ,P), t = 1, . . . ,T + 1,

ξ1 ≥ ξ2 ≥ · · · ≥ ξT ≥ ξT+1 ≥ 0 a.s.

subgradients (σst , σ

vt ) ∈ Lns

q (Ω,Ft ,P)× Lnvq (Ω,Ft ,P) satisfying

(σst (ω), σv

t (ω)) ∈ ∂gt (st (ω), vt (ω))

σsT+1(ω) ∈ ∂gT+1(sT+1(ω))

and dual variables yt ∈ Lnsq (Ω,Ft+1,P), t = 1, . . . ,T , satisfying

the adjoint equations

yT = (1 + ξT+1)σsT+1

yt−1 = A′tE[yt |Ft ] + (1 + ξt )σst , t = T , . . . ,2

such that for almost all ω ∈ Ω the control vt (ω) is a solution of

maxc∈Vt

(1 + ξt (ω))gt (st (ω), c) +⟨E[yt |Ft ](ω),Bt (ω)c

⟩Challenge: The decisions are not time-consistent.

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Two-stage stochastic optimization problems with order constrainton the recourse

First Stage Problem:

minx

f (x) + E[Q(x , ξ)

]s.t. Q(x , ξ) ic Z ,

x ∈ D.

where Q(x , ξ) is the optimal value of the second stage problem

Second Stage Problem:

Q(x , ξ) = minyq>y : Tx + Wy = h, y ∈ Y.

D ⊂ Rn and Y ⊆ Rm are closed convex sets,f : Rn → R is a convex function,ξ = (q,W ,T ,h); q,T ,h are random.

R. Schultz, R. Gollmer, and U. Gotzes (2008)

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Numerical Methods: Finite localizations

If z ∈ Rn, then at most n + 2 target values ηk , scalarizations ck andshortfall levels E

[(ηk − Y>ck )+

], k = 1, . . . ,n + 2, exists such that

problem (P) is equivalent to

min f (z)

s.t. E[(ηk − 〈ck ,G(z)〉)+

]≤ E

[(ηk − 〈ck ,Y 〉)+

],

k = 1, . . . ,n + 2,z ∈ Z .

Corollary

If x is an optimal solution of problem (P), then a piecewise linearfunction ϕ ∈ Um(M) exists with no more than n + 2 pieces such thatconditions (5)-(6) are satisfied.

Similar result is established for (Q).

Homem de Mello– Merothra (2009), Noyan–Rudolf (2013) haveshown that for a finite probability space (c j , ηj ) are vertices of aparticular polyhedron.

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Numerical Approaches

Primal Dominance Constraints

min f (z)

(P) s.t. E([η −G(z)]+) ≤ E([η − Y ]+) ∀ η ∈ [a,b],

z ∈ Z

Nonsmooth problem even for linear f and G.Denote v(η) = E([η − Y ]+).

Equivalent formulation

min f (z)

s.t.∫A

(η −G(z)

)dP ≤ v(η) ∀η ∈ [a,b], ∀A ∈ F

z ∈ Z

If Y has finitely many realizations, then the dominance constraintreduces to finitely many inequalities.

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Primal Outer Approximation Method

Ω = ω1, . . . , ωN with probabilities p1, . . . ,pN .The realizations of Y are yj with shortfalls vj = E([yj − Y ]+) andGj (z) = [G(z)](ωj ).

Step 1: At iteration k , solve the relaxation to obtain zk :

min f (z)

s.t.∑i∈Aj

pi(yj −Gi (z)

)≤ vj j = 1, . . . , k − 1

z ∈ Z .

Step 2: Check the order: calculate

δk = max1≤j≤N

E[(yj −G(zk ))+ − (yj − Y )+

].

If δk ≤ 0, stop; otherwise, continue.Step 3: Determine j∗, for which E([yj∗ −G(zk )]+) > vj∗

and define a new event Ak = 1 ≤ i ≤ N : yj∗ > Gi (zk )Increase k by one, and go to Step 1.

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Multivariate version

Step 1: At iteration k , solve the master problem:

min f (z)

s.t.∑i∈Aj

pi(ηj − 〈c j ,Gi (z)〉

)≤ E

[(ηj − 〈c j ,Y 〉

)+

], j ∈ Jk .

z ∈ Z .

Let zk denote its solution and let X k = G(zk ).

Step 2: Calculate the quantity

δk = supη,c

E[(η−c>G(zk ))+−(η−c>Y )+

]: (η, c) ∈ [a, b]×S

.

If δk ≤ 0, stop; otherwise, continue.

Step 3: Determine (ηk , ck ) such that

E[(η − c>G(zk ))+ − (η − c>Y )+

]≥ δk

2.

Set Jk+1 = Jk ∪ (ηk , ck ); k ← k + 1, and go to Step 1.

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Order-Verification Problem

minE[η − c>Y

]+− E

[η − c>X k]

+: η ∈ [a, b], c ∈ S

≥ 0?

I DC-optimization Linearization of the function E[

max(0, η − c>X k )],

X k = G(zk ), at each point (c i , ηi ) ∈ Jk by subdifferentiation. Event

Aik = ω ∈ Ω : 〈c i ,X k (ω)〉 ≤ ηi,

Subgradient of hk at (c i , ηi ) by Strassen’s Theorem:(P(Aik )

−E[X k 1Aik ]

)∈ ∂hk (ηi , c i ).

I Combinatorial methods for finite probability space.

Every event A is represented by αi =

1, if i ∈ A;0, if i 6∈ A

for i = 1, . . . ,N.

min E[(η − c>Y )+

]−

N∑i=1

piαi (η − c>X i )

s. t. (c, η) ∈ S × [a, b], α ∈ 0, 1N .

We alternate between minimizing in α for a fixed (c, η) and minimizing in(c, η) keeping α fixed.

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Inverse Stochastic Dominance Constraints

max f (z)

(Q) s.t. F (−2)(G(z); p) ≥ F (−2)(Y ; p), ∀ p ∈ [α, β],

z ∈ Z

Z is a closed subset of a vector space Z, [α, β] ⊂ (0,1)G : Z → L1(Ω,F ,P) and f : Z → R are continuous.

Orders and conditional expectationI For X ,Y ∈ L1(Ω,F ,P), X (2) Y if and only if

E[X |A] ≥ 1P(A)

F (−2)(Y ; P(A)) ∀ A ∈ F : P(A) > 0.

I For X ,Y ∈ L1(Ω,F ,P), X ic Y if and only if

E[X |A] +1

P(A)L(Y ;−P(A)) ≤ 0 ∀ A ∈ F : P(A) > 0.

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Quantile Cutting Plane Methods: The Scaled Method

Step 1: Solve the problem to obtain zk :

max f (z)

s.t. E[G(z)|Bj ] ≥ 1P(Bj )

F (−2)(Y ; P(Bj )) j = 1, . . . , k

z ∈ Z .

Step 2: Consider the sets Akt = G(zk ) ≤ t and let

δk = supt

1P(Ak

t )F (−2)(Y ; P(Ak

t ))−E[G(zk )|Akt ] : P(Ak

t ) > 0.

If δk ≤ 0, stop; otherwise, continue.Step 3: Find tk such that P(G(zk ) ≤ tk ) > 0 and

E[G(zk )|Aktk ]− 1

P(Aktk )

F (−2)(Y ; P(Aktk )) ≤ −δk

2.

Set Bk+1 = Aktk , increase k by one, and go to Step 1.

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Two-stage problems: Multicut decomposition method

For the finite probability space Ω = ω1, . . . , ωN with probabilitiesP(ωi ) = pi , i = 1, . . . ,N, denote T i = T (ωi ), hi = h(ωi ), y i = y(ωi ),and q i = q(ωi ).

min f (x) +N∑

i=1

piQ i (x)

s.t. Q(x , ξ) ic Z ,x ∈ D.

Q i (x) := infy(q i )>y : W iy = hi − T ix , y ≥ 0, i = 1, . . . ,N.

Idea: Approximate the functions Q i (·) and their domains by cuttingplanes and use the approximate models to construct anapproximation of the distribution of the recourse function Q(·).

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Two-stage problems: Objective cuts

The functions Q i (·) are convex.The subdifferential of Q i (·) is ∂Q i (xk ) = −(T i )>ψi (xk ), where

ψi (xk ) = argmaxπ>(hi − T ixk )|(W i )Tπ ≤ q i

is the set of optimal solutions of the dual to the second stage problemat x = xk . For πk,i ∈ ψi (xk ), it holds

Q i (x) ≥ Q i (xk )− 〈(T i )Tπk,i , x − xk 〉.

Objective cuts

Qs(x)≥ αk,i + 〈gk,i , x〉,where gk,i = −(T i )Tπk,i ,

αk,i = Q i (xk ) + 〈πk,i ,T ixk 〉.

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Two-stage problems: Feasibility Cuts

We derive an inequality that must be satisfied x ∈ dom Q i . Denote byU i (x) the optimal value of

minx,y‖u‖

s.t. Wy + u = hi − T ix ,y ≥ 0.

U(·) is convex and U i (x) = 0 if and only if Q i (x) <∞. Letr k,i ∈ ∂U i (xk ). Then,

0 ≥ U i (xk ) + 〈r k,i , x − xk 〉 = U i (xk )− 〈r k,i , xk 〉+ 〈r k,i , x〉.

Feasibility cuts

βk,i + (r k,i )>x ≤ 0,

where r k,i = (−T i )>∆i (xk )

βk,i = U i (xk )− 〈r k,i , xk 〉.

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Quantile function-based decomposition method for two-stageproblems

The cuts constructed until iteration k are used in the following

Approximate problem

minx,υ

f (x) +N∑

i=1

piv i

s.t. αj,i + (g j,i )>x ≤ v i j ∈ Jko (i), i = 1, . . . ,N,

β j,i + (r j,i )>x ≤ 0 j ∈ Jkf (i), i = 1, . . . ,N,

1P(Aj )

[∑i∈Aj

piv i]

+1

P(Aj )L(Z ; P(Aj )− 1) ≤ 0 Aj ∈ Jk

e , (7)

x ∈ D.

Here υ = (v1, . . . , vN)> and V is a random variable with realizationsv1, . . . , vN . Inequalities (7) define the event cuts which approximatethe ordering constraint.

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Quantile function-based decomposition method: Algorithm

Step 1. Solve the second stage problems with x = xk .(a) If Q i (xk ) <∞, then set Jk

f (i) = Jk−1f (i). If

Q i (xk ) > vk,i , then construct an objective cut andset Jk

o (i) = Jk−1o (i) ∪ k; otherwise set

Jko (i) = Jk−1

o (i).(b) If Q i (xk ) =∞, construct a feasibility cut and

update Jkf (i) = Jk−1

f (i) ∪ k.Step 2. If Qs(xk ) <∞ for all i = 1, . . . ,N, then construct an

event cut. Set Ak to be the new event andJk

e = Jk−1e ∪ Ak.

Step 3. If Q(xk ) =∑N

i=1 pivk,i and δ ≤ 0, then stop; otherwisecontinue.

Step 4. Solve the approximate problem. If it is infeasible, thenstop. Otherwise, denote by (xk+1,V k+1) its solution;increase k by one, and go to Step 1.

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Dual approach: Reduced Problems and Their Duals

Given a collection of point (ηj , c j ), j ∈ Jk for some index set Jk , k ∈ N, thereduced problem is

min f (z)

s.t. E[(ηj − (c j )>G(z)

)+

]≤ E

[(ηj − (c j )>Y

)+

], j ∈ Jk .

z ∈ Z

The Lagrangian of the reduced problem is defined on Z × R|Jk |+ is

Lk (z, µk ) = f (z) + E[∑

j∈Jk

µkj(ηj − (c j )>G(z)

)+−∑j∈Jk

µkj(ηj − (c j )>Y

)+

].

Extension µk of µk to the entire set [a, b]× S by setting

µk (A) =∑j∈Jk

µkj 1A∩(ηj ,cj ).

The reduced dual function Dk : R|Jk | → R at the point µk

Dk (µk ) = minz∈Z

Lk (z, µk ) = D(µk ).

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Subgradients of the Reduced Dual Function

The subgradients of Dk (µ)

Γkj (µ) = E

[(ηj − (c j )>G(zk

µ))+−(ηj − (c j )>Y

)+

], j = 1, . . . , |Jk |,

where zkµ is such that Dk (µ) = Lk (zk

µ, µ).

For atomic measure µ` with atoms on the set (ηj , c j ), j ∈ J`, we define anextension µ`,k ∈ R|Jk |

+ to the set (ηj , c j ), j ∈ Jk ⊃ J` by setting

µ`,kj =

µ`j if j ∈ J`0 if j 6∈ J`.

Note: the subgradients Γ`,k (µ`,k ) are not subgradients of the dual function.

Piecewise linear model Dk : R|Jk | → R of the reduced dual function

Dk (µ) = min1≤`≤k

D(µ`) + (Γ`,k )>(µ− µ`,k ).

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Dual Approximative Bundle Method

Step 1: Solve minx∈X Lk (x , µk ) and calculate new subgradients Γ`,k at µk .Step 2: If k = 1 or if D(µk ) ≥ (1− γ)D(wk−1) + γϑk−1(µk ),

then set wk := µk ; else set wk := wk−1,k .Step 3: Calculate a solution (θk+1, µ

k+1) of the master problem:

maxµ≥0θ−%

2‖µ−wk‖2

2 : D(µ`)+(Γ`,k )>(µ−µ`) ≥ θ ` = 1 . . . k.

Set ϑk (µk+1) = θk+1. Let π` be the optimal Lagrange multiplier ofthe `-th constraint. Define xk =

∑k`=1 π`x

`.Step 4: Calculate the quantities

δk = sup(η,c)∈[a,b]×S

E[(η − 〈c,G(xk )〉

)+−(η − 〈c,Y 〉

)+

],

δ′k = maxj∈Jk

E[(ηj − 〈c j ,G(xk )〉

)+−(ηj − 〈c j ,Y 〉

)+

].

Step 5: If D(wk ) ≥ θk+1 − ε and δk ≤ ε, then stop; otherwise continue.Step 6: If δk > ε and δ′k ≤

δk4 , then determine (η∗, c∗) such that

E[(η∗ − 〈c∗,G(xk )〉)+ − (η∗ − 〈c∗,Y 〉)+

]≥ 1

2δk

and set Jk+1 = Jk ∪ (η∗, c∗); else set Jk+1 = Jk .Set k ← k + 1 and go to Step 1.

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Further Numerical Methods

I Subgradient-Based Approximation Methods for second-orderdominance constraints with linear G(·) in primal formC. Fabian, G. Mitra, and D. Roman, 2008;

I Combinatorial methods for first-order dominance constraintsG. Rudolf, N. Noyan, A. Ruszczynski 2006 based on second-orderstochastic dominance relaxation(X : X (2) Y = convX : X (1) Y);

I Cutting Surface Method and Sample Average ApproximationHomem de Mello-Mehrotra 2009, 2012;

I Exact Penalty Method and Sample Average ApproximationMeskarian-Fliege-Xu 2014;

I Strassen theorem representationLuedtke 2012, Armbruster-Luedtke 2014;

I Augmented Lagrangian methodDentcheva-Martinez-Wolfhagen 2016.

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Further research

I Non-convex problems Optimality conditions for problems withfirst- and higher-order dominance constraints with non-convexfunctions (DD–A.Ruszczynski 2004.)

I Stochastic dominance efficiency in multi-objective optimizationand its relations to dominance constraints (G. Mitra, C. Fabian,K. Darby-Dowman, D. Roman, 2006, 2009)

I Two-stage problems with order constraints on the recoursefunction (R. Schultz, F. Neise, R. Gollmer, U. Gotzes, D. Drapkin, 2007,2009, 2010, DD–E.Wolfhagen 2016)

I Stability and sensitivity analysis (DD-R. Henrion-A Ruszczynski,2007; Y. Liu-H.Xu, 2010; DD-Römisch 2013; Klaus-Schultz 2015)

I Robust Dominance Relation and relations to robust preferences(DD-A. Ruszczynski, 2010)

I Shape optimization with stochastic-order constraintsGotzes-Schultz 2014

I Asymptotic distributions in Markov decision processesHaskell-Jane 2012, Haskell-Shanthikumar-Shen 2015

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Portfolio Optimization

Assets j = 1, . . . ,n with random return rates RjReference return rate Y (e.g. index, existing portfolio, etc.)Decision variables zj , j = 1, . . . ,n, Z -polyhedral setPortfolio return rate R(z) =

∑nj=1 zjRj

max f (z)− λσ

s.t.n∑

j=1

zjRj (2) 〈c,Y 〉 − σ for all c ∈ S

z ∈ Z

f (x) = E[R(x)

]or f (x) = −%

[R(x)

]: measure of risk.

S is the M-dimensional simplex, λ > 0.

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All Statements are Equivalent

n∑j=1

zjRj (2) c>Y

F (−2)( n∑

j=1

zjRj ; p)≥ F (−2)(c>; p) for all p ∈ [0,1]

continuum of CVaR constraints for every risk level p ∈ [0,1]

E[u( n∑

j=1

zjRj

)]≥ E

[u(c>Y )

]for all concave nondecreasing functions u (von Neuman-Morgenstern utility)

1∫0

F (−1)( n∑

j=1

zjRj ; p)

dw(p) ≥1∫

0

F (−1)(c>Y ; p) dw(p)

for all concave nondecreasing functions w (rank dependent utility)

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Other Applications

I Electricity markets: portfolio of contracts and/or acceptability ofcontracts; risk-averse power generation planning;

I Inverse models, forecasting, learning: Compare the forecasterrors via stochastic dominance and design data collection formodel calibration;

I Network design: assign capacity to optimize network throughput;I Robotics: control of position and communication of robots;I Budget allocation in disaster prevention and relief planning;I Supply-chain models: cost vs. market share;I Medicine: radiation therapy design.

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