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Transcript of Stochastic Orders in Risk-averse delara/SESO/SESO2016/SESO2016-Wednesday_De · PDF file 1...

  • 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 = {