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  • 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