Optimization Problems with Stochastic Dominance Constraints  Institute … · 20110912 ·...
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Optimization Problems with StochasticDominance Constraints
Darinka Dentcheva
Stevens Institute of Technology, Hoboken, New Jersey, USA
Research supported by NSF awards DMS0604060 and CMII0965702
Borovec, Bulgaria, September 1216, 2011
Motivation
RiskAverse Optimization Models
Choose a decision z ∈ Z , which results in a random outcomeG(z) ∈ Lp(Ω,F ,P) with “good" characteristics; special attention tolow probabilityhigh impact events.
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))]Probabilistic / chance constraints impose prescribed probabilityon some events: P[G(z) ≥ η]Mean–risk models optimize a composite objective of theexpected performance and a scalar measure of undesirablerealizationsE[G(z)]− %[G(z)] (risk/ deviation measures)Stochastic ordering constraints compare random outcomes usingstochastic orders and random benchmarks
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Outline
1 Stochastic orders
2 Stochastic dominance constraints
3 Optimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelation to coherent measures of risk
4 Stability of dominance constrained problems
5 Numerical methods
6 Applications
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
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.
Stochastic Dominance Generators
F1 ={
nondecreasing functions u : R→ R}
generates the usualstochastic order or first order stochastic dominance (X �
(1) Y )Introduced in statistics in 1947 by Mann and WhitneyBlackwell (1953), Lehmann (1955)F2 =
{nondecreasing concave u : R→ R
}generates the second
order stochastic dominance relation (X �(2) Y )
Riskconsistency facilitates applications in economics pioneered by Quirkand Saposnik (1962), Fishburn (1964), Hadar and Russell (1969)
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Stochastic Dominance Relation via Distribution Functions
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, . . .
k th order Stochastic Dominance
X �(k) Y ⇔ Fk (X , η) ≤ Fk (Y , η) for all η ∈ R
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
First Order Stochastic DominanceUsual Stochastic Order


0
1 6
ηµXµY
F1(X ; η)F1(Y ; η)
X �(1) Y ⇔ F1(X ; η) ≤ F1(Y ; η) for all η ∈ R
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
SecondOrder Stochastic Dominance
F2(X ; η) =∫ η−∞
F1(X ; t) dt =∫Ω
max(0, η − X (ω)) P(dω) 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
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Characterization via Inverse Distribution Functions
First order dominance ≡ Continuum of probability inequalities
X �(1) Y ⇔ F(−1)(X ; p) ≥ F(−1)(Y ; p) for all 0 < p < 1F(−1)(X ; p) = inf{η : F1(X ; η) ≥ p}
Absolute Lorenz function (Max Otto Lorenz, 1905)
F(−2)(X ; p) =∫ p
0F(−1)(X ; t) dt for 0 < p ≤ 1,
F(−2)(X ; 0) = 0 and F(−2)(X ; p) = +∞ for p 6∈ [0, 1].
Lorenz function and Expected shortfall are Fenchel conjugates
F(−2)(X ; ·) = [F2(X ; ·)]∗ and F2(X ; ·) = [F(−2)(X ; ·)]∗
Second order dominance ≡ Relation between Lorenz functionX �(2) Y ⇔ F(−2)(X ; p) ≥ F(−2)(Y ; p) for all 0 ≤ p ≤ 1.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
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 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
X �(2) Y ⇔ F(−2)(X ; p) ≥ F(−2)(Y ; p) for all 0 ≤ p ≤ 1.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Characterization by Rank Dependent Utility Functions
Rank dependent utility are introduced by Quiggin (1982), Schmeidler(1986–89), Yaari (1987).
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 any two random variables X ,Y ∈ L1(Ω,F ,P) the relationX �(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 true if and only if (1) is satisfied for all w ∈ W2.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Dominance Relation in Optimization
Introduced by Dentcheva and Ruszczyński (2003)
min f (z)subject to Gi (z) �(ki ) Yi , i = 1..m
z ∈ Z
Yi  benchmark random outcome
The dominance constraints reflect risk aversion
Gi (z) is preferred over Yi by all riskaverse decision makers withutility functions in the generator Fki .
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Semiinfinite composite optimization
X, Y, Z are Banach spaces, Y is separable, T is a compact Hausdorff space
min ϕ(f (z))
s.t. G (G(z), t) ≤ 0 for all t ∈ Tz ∈ Z
f : X→ Z, G : X→ Y are continuously Fréchet differentiable; ϕ : Z→ R isconvex, continuous; G : Y× T → R is continuous, G (·, t) is convex ∀t ∈ TProblem (P2) with second order dominance constraint is obtained whenY = L1(Ω,F ,P), X = G(z), and G is defined as follows:
G (X , t) =∫Ω
max(0, t − X (ω)) P(dω)− v(t), t ∈ [a, b] ⊂ R,
where v(t) =∫Ω
max(0, t − Y (ω)) P(dω) for some Y ∈ L1(Ω,F ,P).
D. Dentcheva, A Ruszczyński: Composite semiinfinite optimization, Controland Cybernetics, 36 (2007) 3, 633646.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Sets Defined by Dominance Relation
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 , andk ≥ 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 [DD, A. Ruszczyński, 2004]
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 under Stochastic Ordering Constraints
Second Order Dominance Constraints
Given Y ∈ L1(Ω,F ,P)  benchmark random outcome
Primal Stochastic Dominance Constraints
max f (z)(P2) subject to F2(G(z); η) ≤ F2(Y ; η), ∀ η ∈ [a,b],
z ∈ Z
Inverse Stochastic Dominance Constraints
max f (z)(P−2) subject to F(−2)(G(z); p) ≥ F(−2)(Y ; p), ∀ p ∈ [α, β],
z ∈ Z
Z is a closed subset of a Banach space X, [α, β] ⊂ (0,1), [a,b] ⊂ RG : X→ L1(Ω,F ,P) is continuous and for Palmost all ω ∈ Ω thefunctions [G(·)](ω) are concave and continuousf : X→ R is concave and continuous
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Optimality Conditions Using von Neumann Utility Functions
The Lagrangianlike functional L : Z × F2([a,b])→ R
L(z,u) := f (z) +∫Ω
[u(G(z))− u(Y )
]P(dω)
F2([a,b]) modified generator.
Uniform Dominance Condition (UDC) for problem (P2)A point z̃ ∈ Z exists such that inf
η∈[a,b]
{F2(Y ; η)− F2(G(z̃); η)
}> 0.
Theorem Assume UDC. If ẑ is an optimal solution of (P2) thenû ∈ F2([a,b]) exists:
L(ẑ, û) = maxz∈Z
L(z, û) (2)∫Ω
û (G(ẑ)) P(dω) =∫Ω
û(Y ) P(dω) (3)
If for some û ∈ F2([a,b]) an optimal solution ẑ of (2) satisfies thedominance constraints and (3), then ẑ solves (P2).
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Optimality Conditions Using Rank Dependent Utility Function
Lagrangianlike functional Φ : Z ×W ([α, β])→ R
Φ(z,w) = f (z) +∫ 1
0F(−1)(G(z); p) dw(p)−
∫ 10
F(−1)(Y ; p) dw(p)
W ([α, β]) is the modified generator of the relaxed order
Uniform inverse dominance condition (UIDC) for (P−2)∃z̃ ∈ Z such that inf
p∈[α,β]
{F(−2)(G(z̃); p)− F(−2)(Y ; p)
}> 0.
Theorem
Assume UIDC. If ẑ is a solution of (P−2), then ŵ ∈ W ([α, β]) exists:Φ(ẑ, ŵ) = max
z∈ZΦ(z, ŵ) (4)∫ 1
0F(−1)(G(ẑ); p) dŵ(p) =
∫ 10
F(−1)(Y ; p) dŵ(p) (5)
If for some ŵ ∈ W ([α, β]) and a solution ẑ of (4) the dominanceconstraint and (5) are satisfied, then ẑ is a solution of (P−2).
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Duality Relations to Utility Theories
The Dual Functionals
D(u) = supz∈Z
L(z,u) Ψ(w) = supz∈Z
Φ(z,w)
The Dual Problems
(D2) minu∈F2([a,b])
D(u) (D−2) minw∈W ([α,β])
Ψ(w).
TheoremUnder UDC/UIDC, if problem (P−2) resp. (P−2) has an optimalsolution, then the corresponding dual problem has an optimal solutionand the same optimal value. The optimal solutions of the dualproblem (D2) are the utility functions û ∈ F2([a,b]) satisfying (2)–(3)for an optimal solution ẑ of problem (P2). The optimal solutions of(D−2) are the rank dependent utility functions ŵ ∈ W ([α, β])satisfying (4)–(5) for an optimal solution ẑ of problem (P−2).
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
MeanRisk Models as a Lagrangian Relaxation
A measure of risk % assigns to an uncertain outcomeX ∈ Lp(Ω,F ,P) a real value %(X ) on the extended real lineR = R ∪ {+∞} ∪ {−∞}.A coherent measure of risk is a convex, positive homogeneous, p.w.monotone functional % : L1(Ω,F ,P)→ R satisfying the%(X + a) = %(X )− a for all a ∈ R.
Theorem (Necessary Optimality Conditions):
Under the UIDC, if ẑ is an optimal solution of (P−2), then a riskmeasure %̂ and a constant κ ≥ 0 exist such that G(ẑ) is also anoptimal solution of the problem
maxz∈Z
{f (z)− κ%̂(G(z))
}and (6)
κ%̂(G(ẑ)) = κ%̂(Y ). (7)
If the dominance constraint is active, then condition (7) takes on theform %̂(G(ẑ)) = %̂(Y ).
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Extensions and Further Research Directions
Nonconvex problems Optimality conditions for problems withFSD constraints and problems with higher order dominanceconstraints with nonconvex functions (DD, A Ruszczyński 2004,2007)Multivariate orders based on scalarization of random vectors(DD, A Ruszczyński, 2007, T. HomemdeMello, S. Mehrotra, 2009)Dynamic orders Scalarizations as discounts for stochasticsequences; maximum prnciple; (DD, A Ruszczyński, 2008);Twostage problems with dominance constraints on the recoursefunction (R. Schultz, F. Neise, R. Gollmer, U. Gotzes, D. Drapkin, 2007,2009, 2010)Stochastic dominance efficiency in multiobjective optimizationand its relations to dominance constraints (G. Mitra, C. Fabian,K. DarbyDowman, D. Roman, 2006, 2009)Stability and sensitivity analysis (DD, R. Henrion, A Ruszczyński,2007; Y. Liu, H. Xu, 2010; DD W. Römisch 2011)Robust Dominance Relation (DD, A Ruszczyński, 2010)
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Problem and Assumptions
k th Order Stochastic Dominance Constraints
max f (z)(Pk ) subject to Fk (G(z, ξ); η) ≤ Fk (Y ; η), ∀ η ∈ [a,b],
z ∈ Z .
k ≥ 2, Z ⊂ Rn is nonempty closed convex, Y ∈ Lk−1(Ω,F ,P)f : Rn → R is convex;ξ is a random vector with closed convex support Ξ ⊂ Rs;G : Rm × Rs → R is continuous, concave w.r.to the first argument,satisfies the linear growth condition
G(z, ξ) ≤ C(B) max{1, ‖ξ‖} (z ∈ B, ξ ∈ Ξ,∀B bounded)
Theorem
The feasible set Z (ξ,Y ) is closed and convex.
Z (ξ,Y ) = {z ∈ Z : Fk (G(z, ξ), η) ≤ Fk (Y , η), ∀η ∈ R}
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Distances between random variables
Rachev metrics on Lk (Ω,F ,P)
Dk,p(X ,Y ) =
(∫
R
∣∣Fk (X , η)− Fk (Y , η)∣∣pdη) 1p for 1 ≤ p
Quantitative Stability
Constraints qualification: k UDC
A point z̄ ∈ Z exists such that minη∈[a,b]
(Fk (Y , η)− Fk (G(z̄, ξ), η)) > 0.
v(ξ,Y ) is the optimal value of (Pk ) and S(ξ,Y ) is its solution set.Consider nondecreasing growth function ψ:
ψ(τ) = inf{f (z)− v(ξ,Y ) : d(z,S(ξ,Y )) ≥ τ, z ∈ Z (ξ,Y )}
Θ(t) = t + ψ−1(2t) (t ∈ R+).
Theorem [DD and W. Römisch 2011]
Assume k UDC and let Z be compact and G(z, ·) be Lipschitz for allz ∈ Z . Then constants L > 0 and δ > 0 exist such that
dH(Z (ξ,Y ),Z (ξ̃, Ỹ )) ≤ Ldk ((ξ,Y ), (ξ̃, Ỹ )),v(ξ,Y )− v(ξ̃, Ỹ ) ≤ L dk ((ξ,Y ), (ξ̃, Ỹ ))sup
z∈S(ξ̃,Ỹ )d(z,S(ξ,Y )) ≤ Θ
(L dk ((ξ,Y ), (ξ̃, Ỹ ))
)for all (ξ̃, Ỹ ) such that dk ((ξ,Y ), (ξ̃, Ỹ )) < δ.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Robust Dominance Relation
Uncertainty of the probability measure P0Assume that P0 may vary in a set of probability measures Q .
Consider Lp(Ω,F ,P0), p ∈ [1,∞) and
Lq(Ω,F ,P0) ≡{
measures Q : absolutely continuous w.r.to P0with densities dQ/dP0 ∈ Lq(Ω,F ,P0)
For any X ∈ Lp(Ω,F ,P0) and Q ∈ Lq(Ω,F ,P0)
〈Q,X 〉 =∫Ω
X (ω) Q(dω) =∫Ω
X (ω)dQdP0
(ω) P0(dω).
Let Q ⊂ Lq(Ω,F ,P0) be a convex, closed, and bounded set ofprobability measures.
Definition
X ∈ Lp(Ω,F ,P0) dominates robustly Y ∈ Lp(Ω,F ,P0) over a set ofprobability measures Q ⊂ Lq(Ω,F ,P0) if∫
Ω
u(X )P(dω) ≥∫Ω
u(Y )P(dω) ∀ u ∈ U , ∀ P ∈ Q.
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Robust Preferences
Savage 1954, Gilboa and Schmeidler 1989,F. Maccheroni, A. Rustichini and M. Marinacci, Econometrica 74 (2006)
Set of axioms imply the existence of a utility function ū : R→ R and aset of measures Q such that
X is preferred over Y if and only if
infP∈Q
∫Ω
ū(X ) P(dω) ≥ infP∈Q
∫Ω
ū(Y ) P(dω).
In the case, U ={
ū}
, our definition implies
infP∈Q
{∫Ω
[ū(X )− ū(Y )]P(dω)}≥ 0.
D. Dentcheva, A Ruszczynski: Robust stochastic dominance constraints,Mathematical Programming, 2010
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Numerical Methods
Large scale convex optimization methods for second order dominanceconstraints: applicable only to small problems
Dual methods for SSD constraints (DD, Ruszczyński, 2005); ( Rudolf,Ruszczyński 2006, Luedtke 2008).
Subgradient Based Approximation Methods for SSD constraints withlinear G(·) (Rudolf, Ruszczyński, 2006; Fabian, Mitra, and Roman,2008)
Combinatorial methods for FSD constraints (Rudolf, Noyan,Ruszczyński 2006) based on second order stochastic dominancerelaxation ({X : X �(2) Y} = co{X : X �(1) Y})Subgradient methods based on quantile functions and conditionalexpectations (DD, Ruszczyński, 2010)
Methods for twostage problems with dominance constraints on therecourse (Schultz, Neise, Gollmer, Drapkin; DD and G. Martinez, 2011)
Methods for multivariate linear dominance constraints(HomemdeMello, Mehrotra, 2009; Hu, HomemdeMello, and Mehrotra2010, Armbruster and Luedtke 2010)
Sample average approximation methods (Sun, Xu and Wang, 2011)
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Applications
Finance: portfolio optimizationElectricity markets: portfolio of contracts and/or acceptability ofcontractsInverse models and forecasting: Compare the forecast errors viastochastic dominance and design data collection for modelcalibrationNetwork design: assign capacity to optimize network throughputMedicine: radiation therapy designs
Darinka Dentcheva Optimization under Stochastic Ordering Constraints
Stochastic ordersStochastic dominance constraintsOptimality conditions and dualityRelation to von Neumann utility theoryRelation to rank dependent utilityRelation to coherent measures of risk
Stability of dominance constrained problemsNumerical methodsApplications