Optimization Problems with Stochastic Order ... Optimization Problems with Stochastic Order...
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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