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

Motivation

Risk-Averse Optimization Models

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

I Utility models apply a nonlinear transformation to the realizations of G(z) (expected utility) or to the probability of events (rank dependent utility/distortion). Expected utility models optimize E[u(G(z))]

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

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

I Stochastic-order constraints compare random outcomes using stochastic orders and random benchmarks

Darinka Dentcheva Stochastc-orders in Optimization

Outline

1 Stochastic orders Stochastic dominance and the increasing convex order Multivariate and Dynamic Orders

2 Optimality conditions and duality Relation to von Neumann utility theory Relation to rank dependent utility Relations to coherent measures of risk

3 Stochastic dominance constraints in nonconvex problems

4 Numerical methods Primal outer approximation method Quantile Cutting Plane Methods for General Probability Spaces Decomposition methods for two-stage problems Dual Approach and Regularization

5 Applications

Darinka Dentcheva Stochastc-orders in Optimization

References

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

2 — " — Inverse stochastic dominance constraints and rank dependent utility 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 stochastic dominance constraints in risk-averse optimization, Pacific Journal on Optimization, 4 (2008), No. 3, 433–446.

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

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

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

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

Darinka Dentcheva Stochastc-orders in Optimization

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 second order stochastic dominance relation (X �(2) Y ) Quirk and Saposnik (1962), Fishburn (1964), Hadar and Russell (1969)

I Ū2 = {

nondecreasing convex u : R→ R }

generates the increasing convex order (X �ic Y )

Darinka Dentcheva Stochastc-orders in Optimization

First Order Stochastic Dominance

-

-

0

16

ηµXµY

F (1)(X ; η) F (1)(Y ; η)

Distribution function F (X ; η) = ∫ η −∞

PX (dt) = P{X ≤ η}, η ∈ R

Quantile function F (−1)(X ; p) = inf{η : F (X ; η) ≥ p}, p ∈ (0,1) Survival function F̄ (X ; η) = 1− F (X ; η) = P{X > 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. ⇔ F̄ (X ; η) ≥ F̄ (Y ; η) for all η ∈ R.

Darinka Dentcheva Stochastc-orders in Optimization

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) = sup u {pu − F (u)}.

Theorem (Ogryczak and Ruszczyński, 2002)

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

Darinka Dentcheva Stochastc-orders in Optimization

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 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

Darinka Dentcheva Stochastc-orders in Optimization

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+q F (−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 .

Darinka Dentcheva Stochastc-orders in Optimization

Higher order relation

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

) 1 k and

F (k+1)(X ; η) = ∫ η −∞

F (k)(X ; t)dt = 1 k !

∫ η −∞

(η − t)k PX (dt) =

1 k ! ‖max

( 0, η − X

) ‖kk ∀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−1k−1 ≤ ‖max

( 0, η − Y

) ‖k−1k−1

The generator

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

Darinka Dentcheva Stochastc-orders in Optimization

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 ) with concave 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 functions u : Rm → R

Hard to treat analytically, the generator is too rich

Adopted approach

Define the mutlivariate order via a family of univariate orders

Darinka Dentcheva Stochastc-orders in Optimization

Multivariate Orders: Definition

Definition Given a closed convex set C ∈ Rm+ and a mapping M : c 7→ Uc , c ∈ C, where Uc are univariate generators, a random vector X ∈ Lm1 is stochastically larger than a random vector Y ∈ Lm1 with respect to M and 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 ∈ Lm1 ,

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 order dominance.

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

Darinka Dentcheva Stochastc-orders in Optimization

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 with finite 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 = {