Optimization Problems with Stochastic Order Constraints

Darinka Dentcheva

Stevens Institute of Technology, Hoboken, New Jersey, USA

Research supported by NSF awards CMII-0965702

Ann Arbor, February 28th 2013

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 paying special attention to low-probability-high-impact events.

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))] Probabilistic / chance constraints impose prescribed probability on some events: P[G(z) ≥ η] 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) Stochastic ordering constraints compare random outcomes using stochastic orders and random benchmarks

Darinka Dentcheva Optimization and Stochastic Orders

Outline

1 Stochastic orders

2 Stochastic orders as constraints

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

4 Multivariate and Dynamic Orders

5 Numerical methods

6 Applications Portfolio optimization Beyond portfolio optimization

Darinka Dentcheva Optimization and 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.

Generators

F1 = {

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)

F2 = {

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)

F̄2 = {

nondecreasing convex u : R→ R }

generates the increasing convex order (X �ic Y ) counterpart of stochastic dominance of second order when small values are preferred

Darinka Dentcheva Optimization and Stochastic Orders

Stochastic Orders and Distribution Functions

For any X ∈ Lk (Ω,F ,P)(Ω,F ,P), we define

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, . . .

The function F (k)X is nondecreasing for k ≥ 1 and convex for k ≥ 2.

Quantile function

F(−1)(X ; p) = inf{η : F1(X ; η) ≥ p}, p ∈ (0,1)

Survival function

F 1(X ; η) = 1− F1(X ; η) = P{X > η}, η ∈ R

Darinka Dentcheva Optimization and Stochastic Orders

First Order Stochastic Dominance

-

-

0

1 6

ηµXµY

F1(X ; η) F1(Y ; η)

The usual stochastic order

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

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

Darinka Dentcheva Optimization and Stochastic Orders

Second-Order Stochastic Dominance

F2(X ; η) = ∫ η −∞

F1(X ; t) dt = E [

max(0, η − X ) ]

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

⇔ E [

max(0, η − X ) ] ≤ E

[ max(0, η − Y )

] for all η ∈ R

Darinka Dentcheva Optimization and Stochastic Orders

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

‖max ( 0, η − X

) ‖k−1k−1 ≤ ‖max

( 0, η − Y

) ‖k−1k−1

The generator

Fk 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 Optimization and Stochastic Orders

Second Order Dominance and Inverse Distribution Functions

Absolute Lorenz function (Max Otto Lorenz, 1905)

F(−2)(X ; p) = ∫ p

0 F(−1)(X ; t) dt for 0 < p ≤ 1,

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

Fenchel conjugate function of F : F ∗(p) = sup u {pu − F (u)}

Lorenz function and Expected shortfall are Fenchel conjugates

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

Ogryczak - Ruszczyński (2002)

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

Darinka Dentcheva Optimization and Stochastic Orders

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 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

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

Darinka Dentcheva Optimization and Stochastic Orders

Preference to small values: Increasing convex order

Characterization via integrated survival function

For X ,Y ∈ Lp(Ω,F ,P), X is smaller than Y (X �ic Y ) 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 Optimization and Stochastic Orders

Characterization via Rank Dependent Utility Functions

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 all random variables X ,Y ∈ L1(Ω,F ,P) the relation X �(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 if and only if (1) is satisfied for all w ∈ W2.

Corollary

X �ic Y holds if and only if (1) is satisfied for all convex functions w which are subdifferentiable at zero.

Quiggin (1982), Schmeidler (1986–89), Yaari (1987)

Darinka Dentcheva Optimization and Stochastic Orders

Acceptance Sets

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 , and k ≥ 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

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 and Stochastic Orders

Dominance Relation in Optimization

Introduced by Dentcheva and Ruszczyński in 2003

min f (z) (P) s.t. Gi (z) �(ki ) Yi i = 1, . . . ,m,

z ∈ Z .

Yi - benchmark random outcome

Z - convex subset of a separable Banach space Z , Gi – continuous operators from Z to the space Lki−1(Ω,F ,P;R), ki ≥ 1, f – continuous function defined on Z .

The stochastic order constraints reflect risk aversion

Gi (z) is preferred over Yi by all risk-averse decision makers with utility functions in the generator Fki ; Easier consensus on a benchmark rather than a utility function; Data of a benchmark is readily available.

Darinka Dentcheva Optimization and Stochastic Orders

Portfolio Optimization

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

∑n j=1 zjRj

max f (z)

s.t. n∑

j=1

zjRj � Y

z ∈ Z

f (x) = E [ R(x)

] or f (x) = −%

[ R(x)

] : measure of risk.

Darinka De