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 outcomeG(z) ∈ Lp(Ω,F ,P) with “good" characteristics; special attention tolow probability-high impact events.
I Utility models apply a nonlinear transformation to the realizationsof G(z) (expected utility) or to the probability of events (rankdependent utility/distortion). Expected utility models optimizeE[u(G(z))]
I Probabilistic / chance constraints impose prescribed probabilityon some events: P[G(z) ≥ η]
I Mean–risk models optimize a composite objective of theexpected performance and a scalar measure of undesirablerealizations E[G(z)]− %[G(z)] (risk/ deviation measures)
I Stochastic-order constraints compare random outcomes usingstochastic orders and random benchmarks
Darinka Dentcheva Stochastc-orders in Optimization
Outline
1 Stochastic ordersStochastic dominance and the increasing convex orderMultivariate and Dynamic Orders
2 Optimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelations to coherent measures of risk
3 Stochastic dominance constraints in nonconvex problems
4 Numerical methodsPrimal outer approximation methodQuantile Cutting Plane Methods for General Probability SpacesDecomposition methods for two-stage problemsDual Approach and Regularization
5 Applications
Darinka Dentcheva Stochastc-orders in Optimization
References
1 D. Dentcheva, A. Ruszczynski: Stochastic optimization with dominanceconstraints, SIAM J. Optimization, 14 (2003) 548–566.
2 — " — Inverse stochastic dominance constraints and rank dependentutility 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 stochasticdominance constraints in risk-averse optimization, Pacific Journal onOptimization, 4 (2008), No. 3, 433–446.
5 — " — Inverse cutting plane methods for optimization problems withsecond-order stochastic-dominance constraints, Optimization, 59 (2010)323–338.
6 D. Dentcheva, G. Martinez, Two-stage stochastic optimization problemswith stochastic-order constraints on the recourse, European Journal ofOperations Research 1 (2012), 1, 1–8.
7 D. Dentcheva, E. Wolfhagen: Optimization with multivariate stochasticdominance constraints, SIAM J. Optimization, 25 (2015) 1, 564–588.
8 D. Dentcheva, W. Römisch: Stability and sensitivity of stochasticdominance 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 secondorder stochastic dominance relation (X (2) Y )Quirk and Saposnik (1962), Fishburn (1964), Hadar and Russell (1969)
I U2 =
nondecreasing convex u : R→ R
generates theincreasing 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) = PX ≤ η, η ∈ R
Quantile function F (−1)(X ; p) = infη : F (X ; η) ≥ p, p ∈ (0,1)
Survival function F (X ; η) = 1− F (X ; η) = PX > 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.
⇔ F (X ; η) ≥ F (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) = supupu − F (u).
Theorem (Ogryczak and Ruszczynski, 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 pp p p p p p p p p p p p p p p 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+qF (−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 )
) 1k and
F (k+1)(X ; η) =
∫ η
−∞F (k)(X ; t)dt =
1k !
∫ η
−∞(η − t)k PX (dt) =
1k !‖max
(0, η − X
)‖k
k ∀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−1
k−1 ≤ ‖max(0, η − Y
)‖k−1
k−1
The generator
Uk contains all functions u : R→ R such that a non-increasing,left-continuous, bounded function ϕ : R→ R+ exists such thatu(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 ) withconcave 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 functionsu : 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
DefinitionGiven a closed convex set C ∈ Rm
+ and a mapping M : c 7→ Uc , c ∈ C,where Uc are univariate generators, a random vector X ∈ Lm
1 isstochastically larger than a random vector Y ∈ Lm
1 with respect to Mand 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 ∈ Lm
1 ,
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 orderdominance.
Other definitions: A. Müller, D. Stoyan, Homem-de-Mello and Mehrotra: lineardominance 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 withfinite 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 = ϕφ,µ : φ ∈ Ψ(M), µ ∈M+(S).
Theorem
For each X ,Y ∈ Lm1 the relation X M
(2) Y is equivalent to
E[ϕ(X )] ≥ E[ϕ(Y )] for all ϕ ∈ UmM.
Moreover, UmM is the maximal generator of the order M
(2) .
Darinka Dentcheva Stochastc-orders in Optimization
Discounted Dynamic Stochastic Order
Definition
A sequence X ∈ LT1 is stochastically larger than a sequence Y ∈ LT
1X dis
(k) Y if
〈λ,X 〉 (k) 〈λ,Y 〉 ∀λ ∈ D ⊆λ ∈ RT : 1 ≥ λ1 ≥ λ2 ≥ · · · ≥ λT ≥ 0
.
Example Finite set D consisting of T elements λ`, ` = 1, . . . ,T , with
λ`t =
1 if t ≤ `,0 if t > `.
Then X dis(k) Y is equivalent to the dominance of partial sums
∑t=1
Xt (k)
∑t=1
Yt , ` = 1, . . . ,T .
Darinka Dentcheva Stochastc-orders in Optimization
Rank Dependent Utility Functions/Distortions
W1 = w : [0, 1]→ R : w is continuous nondecreasing.W2 = w ∈ W1 : w concave subdifferentiable at 0.
Theorem
For all random variables X ,Y ∈ L1, the relation X (i) Y , i = 1, 2 holds if andonly if for all w ∈ Wi∫ 1
0F (−1)(X ; p) dw(p) ≥
∫ 1
0F (−1)(Y ; p) dw(p). (1)
Corollary: X ic Y ⇔ (1) for all convex functions w .
Let M− : S ⇒ (0, 1) have closed convex images.
Multivariate distortions
For X ,Y ∈ Lm1 , the relation X (i) Y , i = 1, 2 holds if and only if for all
measurable ϑ : c ∈ S →Wi (M−(c)) and all µ ∈M+(S)
∫S
1∫0
F (−1)(c>X ; p) dϑc(p) dµ(c) ≥∫S
1∫0
F (−1)(c>Y ; p) dϑc(p) dµ(c) (2)
Quiggin (1982), Schmeidler (1986–89), Yaari (1987)Darinka Dentcheva Stochastc-orders in Optimization
Acceptance Sets
For all k ≥ 1, Y - benchmark outcome in Lk−1(Ω,F ,P).
Acceptance sets Ak (Y ) = X ∈ Lk−1 : X (k) Y
Theorem
The set Ak (Y ) is convex and closed for all Y , and k ≥ 2 . Itsrecession cone has the form
A∞k (Y ) =
H ∈ Lk−1(Ω,F ,P) : H ≥ 0 a.s.
A1(Y ) is closed and Ak (Y ) ⊆ Ak+1(Y ) ∀k ≥ 1. Ak (Y ) is a conepointed at Y if and only if Y is a constant.
Theorem
If (Ω,F ,P) is atomless, then A2(Y ) = conv A1(Y ) If Ω = 1..N, andP[k ] = 1/N, then A2(Y ) = conv A1(Y )
The result is not true for general probability spaces
Darinka Dentcheva Stochastc-orders in Optimization
Dominance Relation as Constraints in Optimization
min f (z)
subject to Gi (z) Ui Yi , i = 1..mz ∈ Z
Yi - benchmark random outcome
The stochastic order constraints reflect risk aversionI Gi (z) is preferred over Yi by all risk-averse decision makers with
utility functions in the generator Ui ;I Easier consensus on a benchmark rather than a utility function;I Data of a benchmark is readily available.
G(z) (1) Y ≡ continuum of chance constraints;G(z) (2) Y ≡ continuum of CV@R constraints.
Introduced by Dentcheva and Ruszczynski (2003)
Darinka Dentcheva Stochastc-orders in Optimization
Second Order Multivariate Dominance Constraints
Given Y ∈ Lm1 - benchmark random vector
Direct Stochastic Order Constraints
(P) min f (z)
s. t. E[(η − c>G(z))+] ≤ E[(η − c>Y )+]
∀ (c, η) ∈ graphM,
z ∈ Z .
Let M− : S ⇒ (0, 1) have closed convex images.
Inverse Stochastic Order Constraints
(Q) min f (z)
s. t. F (−2)(c>G(z); p) ≥ F (−2)(c>Y ; p)
∀ (c, p) ∈ graphM−,
z ∈ Z .
Z is a closed subset of a Banach space X ; M(c), resp. M−(c) are compact,G : X → Lm
1 is continuous and for P-almost all ω ∈ Ω the functions [Gi (·)](ω)
are concave and continuous. f : X → R is convex and continuous.Darinka Dentcheva Stochastc-orders in Optimization
Optimality Conditions Using von Neumann Utility Functions
The Lagrangian-like functional L : X × UmM → R
L(z,u) = f (z)− E[u(G(z))− u(Y )
]Uniform Dominance Condition (UDC) for problem (P)
∃z ∈ Z : inf(η,c)∈graph M
F (2)(c>Y ; η)− F (2)(c>G(z); η)
> 0.
Theorem
Assume UDC. If z is an optimal solution of (P) then u ∈ UmM exists:
L(z, u) = minz∈Z
L(z, u) (3)
E[u(G(z)
]= E
[u(Y )
](4)
If for some u ∈ UmM an optimal solution z of (3) satisfies the
dominance constraints and (4), then z solves (P).
Darinka Dentcheva Stochastc-orders in Optimization
Optimality Conditions Using Rank Dependent Utility Function
Lagrangian-like functional
Λ(z, ϑ, µ) = f (z)−∫S
1∫0
F (−1)(c>G(z); p)− F (−1)(c>Y ; p) dϑc(p) dµ(c)
Uniform inverse dominance condition (UIDC) for (Q)∃z ∈ Z inf
(p,c)∈graph M−
F (−2)(c>G(z); p)− F (−2)(c>Y ; p)
> 0.
Theorem
Assume UIDC. If z is a solution of (Q), then ϑ : S →W2(M−) andµ ∈M+(S) exist:
Λ(z, ϑ, µ) = minz∈Z
Λ(z, ϑ, µ) (5)∫S
∫ 1
0F (−1)(G(z); p) d ϑc(p) d µ(c) =
∫S
∫ 1
0F (−1)(Y ; p) d ϑc(p) d µ(c) (6)
If for some ϑ : S →W(M−), µ ∈M+(S) and a solution z of (5) the orderconstraint and (6) are satisfied, then z is a solution of (Q).
Darinka Dentcheva Stochastc-orders in Optimization
Duality Relations to Utility Theories
The Dual Functionals
D(u) = infz∈Z
L(z,u) ∆(ϑ, µ) = infz∈Z
Λ(z, ϑ, µ)
The Dual Problems
(D2) maxu∈Um(M)
D(u) (D−2) maxϑ,µ
Φ(ϑ, µ).
TheoremUnder UDC/UIDC, if problem (P) resp. (Q) has an optimal solution,then the corresponding dual problem has an optimal solution and thesame optimal value. The optimal solutions of the dual problem are theutility functions u ∈ Um(M) satisfying (3)–(4) for an optimal solution zof problem (P). The optimal solutions of (D−2) provide rankdependent utility functions ϑc ∈ W(M−) and non-negative measureson S satisfying (5)–(6) for an optimal solution z of problem (Q).
Darinka Dentcheva Stochastc-orders in Optimization
Measures of Risk
A coherent measure of risk for random variables expressing cost orlosses is a functional % : L1(Ω,F ,P)→ R satisfying the axioms:
I Convexity: %(αX + (1− α)Y ) ≤ α%(X ) + (1− α)%(Y ) for allX ,Y ∈ L1, ∀α ∈ [0,1].
I Monotonicity: If X (ω) ≤ Y (ω) ∀ω ∈ Ω, then %(X ) ≤ %(Y ).I Translation Equivariance : %(X + a) = %(X ) + a ∀a ∈ R.I Positive homogeneity: %(tX ) = t%(X ) ∀t > 0.
A coherent measure of risk for random variables expressing gains isa functional % : L1(Ω,F ,P)→ R satisfying the convexity, positivehomogeneity, and the following modified axioms:
I Monotonicity: If X (ω) ≤ Y (ω) ∀ω ∈ Ω, then %(X ) ≥ %(Y ).I Translation Equivariance : %(X + a) = %(X )− a ∀a ∈ R.
Definition
A risk measure % : Z → R is law invariant, if it assigns the same valueto random variables which have the same distribution.
Darinka Dentcheva Stochastc-orders in Optimization
Relations to Measures of Risk
Consistency of law-invariant risk measure with orders
Suppose the probability space is nonatomic. A risk measure isconsistent with the usual stochastic order iff it satisfies themonotonicity axiom. A proper lower semicontinuous convex riskmeasure % : Z → R for variables expressing losses (gains) isconsistent with the increasing convex order (second-order stochasticdominance), i.e.
X ic Y (X (2) Y ) ⇒ %(X ) ≤ %(Y ).
maxz,σf (z)− λσ : z ∈ Z , G(z) + σ (2) Y
λ > 0 is a tradeoff between f (·) and the error in dominating.
Proposition
The optimal value of σ is a coherent measure of risk expressinglosses.
Darinka Dentcheva Stochastc-orders in Optimization
Mean-risk models as Lagrangian Relaxation
Kusuoka representation
If Ω is atomless, then for every law invariant measure of risk on% : L∞(Ω,F ,P)→ R a convex setM% of probability measures on (0, 1]exists such that
%(X ) = supµ∈M%
−∫ 1
0
1p
F (−2)(X ; p)µ(dp) ∀X ∈ L∞.
(Q′) max
f (X ) : F (−2)(X ; p) ≥ F (−2)(Y ; p), p ∈ [α, β],X ∈ C.
Theorem
Under the UIDC, if X is an optimal solution of (Q’), then a law-invariantcoherent risk measure % (withM% being a singleton) and κ ≥ 0 exist suchthat X is a solution of the mean-risk problem
maxX∈C
f (X )− κ%(X )
and κ%(X ) = κ%(Y ).
If the dominance constraint is active, then %(X ) = %(Y ).
Darinka Dentcheva Stochastc-orders in Optimization
The Implied Dominance Constraint
Given the problem
(R) minX∈C
f (X )− κ%(X )
%(·) a coherent law invariant measure of risk and κ > 0
Theorem IfM% is compact∗ inM([0,1]) and X is a solution ofproblem (R), then ∃ µ ∈M such that
%(X ) =
∫ 1
0
1p
F (−2)(X ; p) µ(dt),
and for every Y ∈ L1(Ω,F ,P) satisfying the conditions
F (−2)(Y ; t) ≤ F (−2)(X ; t), for all t ∈ [0,1],
F (−2)(Y ; t) = F (−2)(X ; t), for all t ∈ supp(µ),
the point X is also a solution of problem (Q′) with [α, β] = [0,1].
Darinka Dentcheva Stochastc-orders in Optimization
Stochastic Dominance Constraints: The Nonconvex Case
min f (H(z))
(P ′) s.t. F (2)(G(z); η) ≤ F (2)(Y ; η) η ∈ [a,b]
z ∈ Z
Assumption
H(·) and G(·) are continuously Fréchet differentiable.
We define I0(z) =η ∈ [a,b] : F (2)(G(z); η) = F (2)(Y ; η)
.
The differential constraint qualification condition at z0 ∈ Z
A point zS ∈ Z and a constant δ > 0 exist such that for all η ∈ I0(z0)∫G(z0)<η
[G′(z0)(zS − z0)](ω) P(dω)
≥∫
G(z0)=η
max(0,−[G′(z0)(zS − z0)](ω)
)P(dω) + δ.
Darinka Dentcheva Stochastc-orders in Optimization
The non-convex case: continued
The normal cone of Z at z is denoted by NZ (z)
Theorem
If the point z is a local minimum of problem P ′ and the differentialconstraint qualification condition is satisfied at z, then a functionu ∈ U2 exists such that
0 ∈ [H ′(z)]∗∂f (H(z))− [G′(z)]∗∫Ω
∂u(G(z)) P(dω) + NZ (z),
E[u(G(z))
]= E
[u(Y0)
].
The integral∫Ω∂u(Y ) P(dω) is understood as a collection of integrals
of all measurable selections of the multifunction ω → ∂u(Y (ω)).
Key idea of the proof Local linearization of H and G around z andconvex approximation of the objective function and the feasible setsuch that the tangent cones of both feasible sets at z are the same.(DD, A. Ruszczynski, Composite semi-infinite optimization, Control andCybernetics, 36 (2007) 3, 633–646).
Darinka Dentcheva Stochastc-orders in Optimization
Control problem with order constraint and its risk-neutralequivalent
(C)
maxT∑
t=1
E[Gt (st , vt )
]+ E
[GT+1(sT+1)
]s.t. st+1 = Atst + Btvt + et , t = 1, . . . ,T ,
(G1(s1, v1), . . . ,G1(sT , vT ),GT+1(sT+1)) dis(2) (Y1, . . . ,YT ,YT+1)
vt ∈ Vt a.s., t = 1, . . . ,T .
Theorem
If (s, v) is an optimal solution of problem (C), then a random discountsequence ξt ∈ L∞(Ω,Ft ,P), t = 1, . . . ,T + 1, exists such that (s, v) is anoptimal solution of the problem
maxT∑
t=1
E[(1 + ξt )Gt (st , vt )
]+ E
[(1 + ξT+1)GT+1(sT+1)
]s.t. st+1 = Atst + Btvt + et , t = 1, . . . ,T ,
vt ∈ Vt a.s., t = 1, . . . ,T .
Darinka Dentcheva Stochastc-orders in Optimization
Maximum Principle
If (s, v) is an optimal solution of the control problem, then there exist:discount factors ξt ∈ L∞(Ω,Ft ,P), t = 1, . . . ,T + 1,
ξ1 ≥ ξ2 ≥ · · · ≥ ξT ≥ ξT+1 ≥ 0 a.s.
subgradients (σst , σ
vt ) ∈ Lns
q (Ω,Ft ,P)× Lnvq (Ω,Ft ,P) satisfying
(σst (ω), σv
t (ω)) ∈ ∂gt (st (ω), vt (ω))
σsT+1(ω) ∈ ∂gT+1(sT+1(ω))
and dual variables yt ∈ Lnsq (Ω,Ft+1,P), t = 1, . . . ,T , satisfying
the adjoint equations
yT = (1 + ξT+1)σsT+1
yt−1 = A′tE[yt |Ft ] + (1 + ξt )σst , t = T , . . . ,2
such that for almost all ω ∈ Ω the control vt (ω) is a solution of
maxc∈Vt
(1 + ξt (ω))gt (st (ω), c) +⟨E[yt |Ft ](ω),Bt (ω)c
⟩Challenge: The decisions are not time-consistent.
Darinka Dentcheva Stochastc-orders in Optimization
Two-stage stochastic optimization problems with order constrainton the recourse
First Stage Problem:
minx
f (x) + E[Q(x , ξ)
]s.t. Q(x , ξ) ic Z ,
x ∈ D.
where Q(x , ξ) is the optimal value of the second stage problem
Second Stage Problem:
Q(x , ξ) = minyq>y : Tx + Wy = h, y ∈ Y.
D ⊂ Rn and Y ⊆ Rm are closed convex sets,f : Rn → R is a convex function,ξ = (q,W ,T ,h); q,T ,h are random.
R. Schultz, R. Gollmer, and U. Gotzes (2008)
Darinka Dentcheva Stochastc-orders in Optimization
Numerical Methods: Finite localizations
If z ∈ Rn, then at most n + 2 target values ηk , scalarizations ck andshortfall levels E
[(ηk − Y>ck )+
], k = 1, . . . ,n + 2, exists such that
problem (P) is equivalent to
min f (z)
s.t. E[(ηk − 〈ck ,G(z)〉)+
]≤ E
[(ηk − 〈ck ,Y 〉)+
],
k = 1, . . . ,n + 2,z ∈ Z .
Corollary
If x is an optimal solution of problem (P), then a piecewise linearfunction ϕ ∈ Um(M) exists with no more than n + 2 pieces such thatconditions (5)-(6) are satisfied.
Similar result is established for (Q).
Homem de Mello– Merothra (2009), Noyan–Rudolf (2013) haveshown that for a finite probability space (c j , ηj ) are vertices of aparticular polyhedron.
Darinka Dentcheva Stochastc-orders in Optimization
Numerical Approaches
Primal Dominance Constraints
min f (z)
(P) s.t. E([η −G(z)]+) ≤ E([η − Y ]+) ∀ η ∈ [a,b],
z ∈ Z
Nonsmooth problem even for linear f and G.Denote v(η) = E([η − Y ]+).
Equivalent formulation
min f (z)
s.t.∫A
(η −G(z)
)dP ≤ v(η) ∀η ∈ [a,b], ∀A ∈ F
z ∈ Z
If Y has finitely many realizations, then the dominance constraintreduces to finitely many inequalities.
Darinka Dentcheva Stochastc-orders in Optimization
Primal Outer Approximation Method
Ω = ω1, . . . , ωN with probabilities p1, . . . ,pN .The realizations of Y are yj with shortfalls vj = E([yj − Y ]+) andGj (z) = [G(z)](ωj ).
Step 1: At iteration k , solve the relaxation to obtain zk :
min f (z)
s.t.∑i∈Aj
pi(yj −Gi (z)
)≤ vj j = 1, . . . , k − 1
z ∈ Z .
Step 2: Check the order: calculate
δk = max1≤j≤N
E[(yj −G(zk ))+ − (yj − Y )+
].
If δk ≤ 0, stop; otherwise, continue.Step 3: Determine j∗, for which E([yj∗ −G(zk )]+) > vj∗
and define a new event Ak = 1 ≤ i ≤ N : yj∗ > Gi (zk )Increase k by one, and go to Step 1.
Darinka Dentcheva Stochastc-orders in Optimization
Multivariate version
Step 1: At iteration k , solve the master problem:
min f (z)
s.t.∑i∈Aj
pi(ηj − 〈c j ,Gi (z)〉
)≤ E
[(ηj − 〈c j ,Y 〉
)+
], j ∈ Jk .
z ∈ Z .
Let zk denote its solution and let X k = G(zk ).
Step 2: Calculate the quantity
δk = supη,c
E[(η−c>G(zk ))+−(η−c>Y )+
]: (η, c) ∈ [a, b]×S
.
If δk ≤ 0, stop; otherwise, continue.
Step 3: Determine (ηk , ck ) such that
E[(η − c>G(zk ))+ − (η − c>Y )+
]≥ δk
2.
Set Jk+1 = Jk ∪ (ηk , ck ); k ← k + 1, and go to Step 1.
Darinka Dentcheva Stochastc-orders in Optimization
Order-Verification Problem
minE[η − c>Y
]+− E
[η − c>X k]
+: η ∈ [a, b], c ∈ S
≥ 0?
I DC-optimization Linearization of the function E[
max(0, η − c>X k )],
X k = G(zk ), at each point (c i , ηi ) ∈ Jk by subdifferentiation. Event
Aik = ω ∈ Ω : 〈c i ,X k (ω)〉 ≤ ηi,
Subgradient of hk at (c i , ηi ) by Strassen’s Theorem:(P(Aik )
−E[X k 1Aik ]
)∈ ∂hk (ηi , c i ).
I Combinatorial methods for finite probability space.
Every event A is represented by αi =
1, if i ∈ A;0, if i 6∈ A
for i = 1, . . . ,N.
min E[(η − c>Y )+
]−
N∑i=1
piαi (η − c>X i )
s. t. (c, η) ∈ S × [a, b], α ∈ 0, 1N .
We alternate between minimizing in α for a fixed (c, η) and minimizing in(c, η) keeping α fixed.
Darinka Dentcheva Stochastc-orders in Optimization
Inverse Stochastic Dominance Constraints
max f (z)
(Q) s.t. F (−2)(G(z); p) ≥ F (−2)(Y ; p), ∀ p ∈ [α, β],
z ∈ Z
Z is a closed subset of a vector space Z, [α, β] ⊂ (0,1)G : Z → L1(Ω,F ,P) and f : Z → R are continuous.
Orders and conditional expectationI For X ,Y ∈ L1(Ω,F ,P), X (2) Y if and only if
E[X |A] ≥ 1P(A)
F (−2)(Y ; P(A)) ∀ A ∈ F : P(A) > 0.
I For X ,Y ∈ L1(Ω,F ,P), X ic Y if and only if
E[X |A] +1
P(A)L(Y ;−P(A)) ≤ 0 ∀ A ∈ F : P(A) > 0.
Darinka Dentcheva Stochastc-orders in Optimization
Quantile Cutting Plane Methods: The Scaled Method
Step 1: Solve the problem to obtain zk :
max f (z)
s.t. E[G(z)|Bj ] ≥ 1P(Bj )
F (−2)(Y ; P(Bj )) j = 1, . . . , k
z ∈ Z .
Step 2: Consider the sets Akt = G(zk ) ≤ t and let
δk = supt
1P(Ak
t )F (−2)(Y ; P(Ak
t ))−E[G(zk )|Akt ] : P(Ak
t ) > 0.
If δk ≤ 0, stop; otherwise, continue.Step 3: Find tk such that P(G(zk ) ≤ tk ) > 0 and
E[G(zk )|Aktk ]− 1
P(Aktk )
F (−2)(Y ; P(Aktk )) ≤ −δk
2.
Set Bk+1 = Aktk , increase k by one, and go to Step 1.
Darinka Dentcheva Stochastc-orders in Optimization
Two-stage problems: Multicut decomposition method
For the finite probability space Ω = ω1, . . . , ωN with probabilitiesP(ωi ) = pi , i = 1, . . . ,N, denote T i = T (ωi ), hi = h(ωi ), y i = y(ωi ),and q i = q(ωi ).
min f (x) +N∑
i=1
piQ i (x)
s.t. Q(x , ξ) ic Z ,x ∈ D.
Q i (x) := infy(q i )>y : W iy = hi − T ix , y ≥ 0, i = 1, . . . ,N.
Idea: Approximate the functions Q i (·) and their domains by cuttingplanes and use the approximate models to construct anapproximation of the distribution of the recourse function Q(·).
Darinka Dentcheva Stochastc-orders in Optimization
Two-stage problems: Objective cuts
The functions Q i (·) are convex.The subdifferential of Q i (·) is ∂Q i (xk ) = −(T i )>ψi (xk ), where
ψi (xk ) = argmaxπ>(hi − T ixk )|(W i )Tπ ≤ q i
is the set of optimal solutions of the dual to the second stage problemat x = xk . For πk,i ∈ ψi (xk ), it holds
Q i (x) ≥ Q i (xk )− 〈(T i )Tπk,i , x − xk 〉.
Objective cuts
Qs(x)≥ αk,i + 〈gk,i , x〉,where gk,i = −(T i )Tπk,i ,
αk,i = Q i (xk ) + 〈πk,i ,T ixk 〉.
Darinka Dentcheva Stochastc-orders in Optimization
Two-stage problems: Feasibility Cuts
We derive an inequality that must be satisfied x ∈ dom Q i . Denote byU i (x) the optimal value of
minx,y‖u‖
s.t. Wy + u = hi − T ix ,y ≥ 0.
U(·) is convex and U i (x) = 0 if and only if Q i (x) <∞. Letr k,i ∈ ∂U i (xk ). Then,
0 ≥ U i (xk ) + 〈r k,i , x − xk 〉 = U i (xk )− 〈r k,i , xk 〉+ 〈r k,i , x〉.
Feasibility cuts
βk,i + (r k,i )>x ≤ 0,
where r k,i = (−T i )>∆i (xk )
βk,i = U i (xk )− 〈r k,i , xk 〉.
Darinka Dentcheva Stochastc-orders in Optimization
Quantile function-based decomposition method for two-stageproblems
The cuts constructed until iteration k are used in the following
Approximate problem
minx,υ
f (x) +N∑
i=1
piv i
s.t. αj,i + (g j,i )>x ≤ v i j ∈ Jko (i), i = 1, . . . ,N,
β j,i + (r j,i )>x ≤ 0 j ∈ Jkf (i), i = 1, . . . ,N,
1P(Aj )
[∑i∈Aj
piv i]
+1
P(Aj )L(Z ; P(Aj )− 1) ≤ 0 Aj ∈ Jk
e , (7)
x ∈ D.
Here υ = (v1, . . . , vN)> and V is a random variable with realizationsv1, . . . , vN . Inequalities (7) define the event cuts which approximatethe ordering constraint.
Darinka Dentcheva Stochastc-orders in Optimization
Quantile function-based decomposition method: Algorithm
Step 1. Solve the second stage problems with x = xk .(a) If Q i (xk ) <∞, then set Jk
f (i) = Jk−1f (i). If
Q i (xk ) > vk,i , then construct an objective cut andset Jk
o (i) = Jk−1o (i) ∪ k; otherwise set
Jko (i) = Jk−1
o (i).(b) If Q i (xk ) =∞, construct a feasibility cut and
update Jkf (i) = Jk−1
f (i) ∪ k.Step 2. If Qs(xk ) <∞ for all i = 1, . . . ,N, then construct an
event cut. Set Ak to be the new event andJk
e = Jk−1e ∪ Ak.
Step 3. If Q(xk ) =∑N
i=1 pivk,i and δ ≤ 0, then stop; otherwisecontinue.
Step 4. Solve the approximate problem. If it is infeasible, thenstop. Otherwise, denote by (xk+1,V k+1) its solution;increase k by one, and go to Step 1.
Darinka Dentcheva Stochastc-orders in Optimization
Dual approach: Reduced Problems and Their Duals
Given a collection of point (ηj , c j ), j ∈ Jk for some index set Jk , k ∈ N, thereduced problem is
min f (z)
s.t. E[(ηj − (c j )>G(z)
)+
]≤ E
[(ηj − (c j )>Y
)+
], j ∈ Jk .
z ∈ Z
The Lagrangian of the reduced problem is defined on Z × R|Jk |+ is
Lk (z, µk ) = f (z) + E[∑
j∈Jk
µkj(ηj − (c j )>G(z)
)+−∑j∈Jk
µkj(ηj − (c j )>Y
)+
].
Extension µk of µk to the entire set [a, b]× S by setting
µk (A) =∑j∈Jk
µkj 1A∩(ηj ,cj ).
The reduced dual function Dk : R|Jk | → R at the point µk
Dk (µk ) = minz∈Z
Lk (z, µk ) = D(µk ).
Darinka Dentcheva Stochastc-orders in Optimization
Subgradients of the Reduced Dual Function
The subgradients of Dk (µ)
Γkj (µ) = E
[(ηj − (c j )>G(zk
µ))+−(ηj − (c j )>Y
)+
], j = 1, . . . , |Jk |,
where zkµ is such that Dk (µ) = Lk (zk
µ, µ).
For atomic measure µ` with atoms on the set (ηj , c j ), j ∈ J`, we define anextension µ`,k ∈ R|Jk |
+ to the set (ηj , c j ), j ∈ Jk ⊃ J` by setting
µ`,kj =
µ`j if j ∈ J`0 if j 6∈ J`.
Note: the subgradients Γ`,k (µ`,k ) are not subgradients of the dual function.
Piecewise linear model Dk : R|Jk | → R of the reduced dual function
Dk (µ) = min1≤`≤k
D(µ`) + (Γ`,k )>(µ− µ`,k ).
Darinka Dentcheva Stochastc-orders in Optimization
Dual Approximative Bundle Method
Step 1: Solve minx∈X Lk (x , µk ) and calculate new subgradients Γ`,k at µk .Step 2: If k = 1 or if D(µk ) ≥ (1− γ)D(wk−1) + γϑk−1(µk ),
then set wk := µk ; else set wk := wk−1,k .Step 3: Calculate a solution (θk+1, µ
k+1) of the master problem:
maxµ≥0θ−%
2‖µ−wk‖2
2 : D(µ`)+(Γ`,k )>(µ−µ`) ≥ θ ` = 1 . . . k.
Set ϑk (µk+1) = θk+1. Let π` be the optimal Lagrange multiplier ofthe `-th constraint. Define xk =
∑k`=1 π`x
`.Step 4: Calculate the quantities
δk = sup(η,c)∈[a,b]×S
E[(η − 〈c,G(xk )〉
)+−(η − 〈c,Y 〉
)+
],
δ′k = maxj∈Jk
E[(ηj − 〈c j ,G(xk )〉
)+−(ηj − 〈c j ,Y 〉
)+
].
Step 5: If D(wk ) ≥ θk+1 − ε and δk ≤ ε, then stop; otherwise continue.Step 6: If δk > ε and δ′k ≤
δk4 , then determine (η∗, c∗) such that
E[(η∗ − 〈c∗,G(xk )〉)+ − (η∗ − 〈c∗,Y 〉)+
]≥ 1
2δk
and set Jk+1 = Jk ∪ (η∗, c∗); else set Jk+1 = Jk .Set k ← k + 1 and go to Step 1.
Darinka Dentcheva Stochastc-orders in Optimization
Further Numerical Methods
I Subgradient-Based Approximation Methods for second-orderdominance constraints with linear G(·) in primal formC. Fabian, G. Mitra, and D. Roman, 2008;
I Combinatorial methods for first-order dominance constraintsG. Rudolf, N. Noyan, A. Ruszczynski 2006 based on second-orderstochastic dominance relaxation(X : X (2) Y = convX : X (1) Y);
I Cutting Surface Method and Sample Average ApproximationHomem de Mello-Mehrotra 2009, 2012;
I Exact Penalty Method and Sample Average ApproximationMeskarian-Fliege-Xu 2014;
I Strassen theorem representationLuedtke 2012, Armbruster-Luedtke 2014;
I Augmented Lagrangian methodDentcheva-Martinez-Wolfhagen 2016.
Darinka Dentcheva Stochastc-orders in Optimization
Further research
I Non-convex problems Optimality conditions for problems withfirst- and higher-order dominance constraints with non-convexfunctions (DD–A.Ruszczynski 2004.)
I Stochastic dominance efficiency in multi-objective optimizationand its relations to dominance constraints (G. Mitra, C. Fabian,K. Darby-Dowman, D. Roman, 2006, 2009)
I Two-stage problems with order constraints on the recoursefunction (R. Schultz, F. Neise, R. Gollmer, U. Gotzes, D. Drapkin, 2007,2009, 2010, DD–E.Wolfhagen 2016)
I Stability and sensitivity analysis (DD-R. Henrion-A Ruszczynski,2007; Y. Liu-H.Xu, 2010; DD-Römisch 2013; Klaus-Schultz 2015)
I Robust Dominance Relation and relations to robust preferences(DD-A. Ruszczynski, 2010)
I Shape optimization with stochastic-order constraintsGotzes-Schultz 2014
I Asymptotic distributions in Markov decision processesHaskell-Jane 2012, Haskell-Shanthikumar-Shen 2015
Darinka Dentcheva Stochastc-orders in Optimization
Portfolio Optimization
Assets j = 1, . . . ,n with random return rates RjReference return rate Y (e.g. index, existing portfolio, etc.)Decision variables zj , j = 1, . . . ,n, Z -polyhedral setPortfolio return rate R(z) =
∑nj=1 zjRj
max f (z)− λσ
s.t.n∑
j=1
zjRj (2) 〈c,Y 〉 − σ for all c ∈ S
z ∈ Z
f (x) = E[R(x)
]or f (x) = −%
[R(x)
]: measure of risk.
S is the M-dimensional simplex, λ > 0.
Darinka Dentcheva Stochastc-orders in Optimization
All Statements are Equivalent
n∑j=1
zjRj (2) c>Y
F (−2)( n∑
j=1
zjRj ; p)≥ F (−2)(c>; p) for all p ∈ [0,1]
continuum of CVaR constraints for every risk level p ∈ [0,1]
E[u( n∑
j=1
zjRj
)]≥ E
[u(c>Y )
]for all concave nondecreasing functions u (von Neuman-Morgenstern utility)
1∫0
F (−1)( n∑
j=1
zjRj ; p)
dw(p) ≥1∫
0
F (−1)(c>Y ; p) dw(p)
for all concave nondecreasing functions w (rank dependent utility)
Darinka Dentcheva Stochastc-orders in Optimization
Other Applications
I Electricity markets: portfolio of contracts and/or acceptability ofcontracts; risk-averse power generation planning;
I Inverse models, forecasting, learning: Compare the forecasterrors via stochastic dominance and design data collection formodel calibration;
I Network design: assign capacity to optimize network throughput;I Robotics: control of position and communication of robots;I Budget allocation in disaster prevention and relief planning;I Supply-chain models: cost vs. market share;I Medicine: radiation therapy design.
Darinka Dentcheva Stochastc-orders in Optimization
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