Chance constrained optimization - applications,properties and numerical issues
Dr. Abebe Geletu
Ilmenau University of TechnologyDepartment of Simulation and Optimal Processes (SOP)
May 31, 2012
Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
This Chancy, Chancy, Chancy WorldLeonard Rastrigin
Uncertainty is the only certainty ...John Allen Paulos, Temple University, Philadelphia
Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
Topics
Chance constrained optimization problems
History and major contributions
Some application areas
Properties and difficulties of chance constraints
Approximation strategies for chance constraints
A new analytic approximation
Conclusion
Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
Chance constrained optimization problems
(CCOPT ) minx
E [f (x , ξ)] (1)
s.t.
either Prgi (x , ξ) ≤ 0 ≥ αi , i = 1, . . . ,m. (2)
or Prgi (x , ξ) ≤ 0, i = 1, . . . ,m ≥ α. (3)
f , g : Rn × Rp → R are at least differentiable w.r.t. x ∈ X ⊂ Rn;
x- a vector of deterministic variables;
ξ ⊂ Ω ⊂ Rp- a vector of random variables with joint probabilitydensity function φ(ξ).
Chance constrained optimization - applications, properties and numerical issues
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Chance constrained optimization problems...
Pr·,E [·] are probability, expectation operators;
Prg(x , ξ) ≤ 0 ≥ α - chance or probability constraint.12 ≤ α ≤ 1 probability (reliability) level.
• Single chance constraints
Prgi (x , ξ) ≤ 0 ≥ αi , i = 1, . . . ,m.
• Joint chance constraints
Prgi (x , ξ) ≤ 0, i = 1, . . . ,m ≥ α.
• The random vector ξ:can be either a Gaussian or non-Gaussian.(See Geletu et al. 2012 for recent review article.)
Chance constrained optimization - applications, properties and numerical issues
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History and major contributionsThe beginning:
Charnes, Cooper & Symonds 1958, 1959.
Major contribution:
Prekopa 1972,1973, 1995, 2001, 2011.
Major references:
Ben-Tal, El Ghaoui & Nemirovski: Robust optimization, 2009.
Birge & Lauveaux: Introduction to stochastic programming, 1997.
Kall & Wallace : Stochastic Programming, 1995.
Kibzun & Kan : Stochastic programming problems. withprobability and quantile functions, 1996.
Marti: Stochastic Optimization Methods, 2005.
Prekopa: Stochastic programming, 1995.
Ruszczynski & Shapiro: Stochastic programming, 2003.
Chance constrained optimization - applications, properties and numerical issues
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Some application areasClassical application areas
Water reservoir management
Optimal power flow (OPF)
Financial risk management (using risk metrics like VaR andcVaR)
Reliability based (engineering) design optimization (RBDO)
Modern applications
Control and optimization based on prediction (eg. using weatherforecast data, etc.)
Reliable navigation and obstacle avoidance in unmannedautonomous ground/areal vehicles
Optimal and reliable wind/soalr power generation
Reliability and fault-tolerance in mechatronic systems
etc.Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
Some application areas ...
Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
Some application areas ....
Chance constrained optimization - applications, properties and numerical issues
TU Ilmenau
Some application areas ....Some related terminologies
RobustnessReliability /ZuverlassigkeitRisk metricsRisk managementFault tolerance
Chance Constraints
♣
Worst case scenario
Chance Constraints
F
min PrFailure
max PrReliability
Chance Constraints
Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraintsMeaning of chance constraints:• Z := g(x , ξ) is a random variable .• Hence,
Prg(x , ξ) ≤ 0+ Prg(x , ξ) > 0 = 1.
irrespective of the distribution of Z = g(x , ξ).
reliability constraint
For α near 1, Prg(x , ξ) ≤ 0 ≥ α; eg. α = 0.95, α = 0.99.
risk constraint
Prg(x , ξ) > 0 ≤ 1− α; eg. 1− α = 0.05, α = 0.01.
• x is feasible iff Prg(x , ξ) ≤ 0 ≥ α holds with reliability α.• Let p(x) := Pr g(x , ξ) ≤ 0. Then the feasible set of CCOPT is
P := x ∈ X | p(x) ≥ α .Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraintsImportant issues
(A) Continuity (see Kibzun & Kan 1996, Birge & Lauveaux 1997,Prekopa 1995).
(B) Differentiability of p(·)Uryasev 1994, Marti 1996.
(C) Convexity of the feasible set PPrekopa 1995, Henrion & Strugarek .
(D) Stability and regularityHenrion and Romisch 1998.
(E) How to determine if a given x is feasible for CCOPT or not?
(F) For a given x , how to determine the valuep(x) = Pr g(x , ξ) ≤ 0?
In general, (E) & (F) are not trivial issues.Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraintsMajor difficulty:the value
p(x) = Prg(x , ξ) ≤ 0 =
∫ξ∈Ω | g(x ,ξ)≤0
φ(ξ)dξ
is difficult to compute. Hence, CCOPT is known to be ahard-problem. But there are somespecial cases:
Example 1:
g(x , ξ) = a>x + b − ξ, ξ ∈ R and ξ ∼ N(µ, σ2). Then
Prg(x , ξ) ≤ 0 = Pra>x + b ≤ ξ = 1− Pra>x + b ≥ ξ
= 1− Pr
ξ − µσ≤(a>x + b
)− µ
σ
= 1− Φ
(a>x + b
).
Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraints
Example 1: contd...
Hence, if g(x , ξ) = a>x + b − ξ, ξ ∈ R and ξ ∼ N(µ, σ2), then
Prg(x , ξ) ≤ 0 ≥ α⇔ Φ−1 (1− α)−(a>x + b
)≥ 0.
The expression on the left is an exact analytic representation of thechance constraint.
Example 2:(separable form)
g(x , ξ) = g(x) + c>ξ + b with ξ ∼ N(µ, σ2).
Example 3:(a linear transformation of ξ)
g(x , ξ) = (Ax + b) ξ + b with ξ ∼ N(µ, σ2).
Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraints...
An analytic representation can lead to a nonlinear optimizationproblem.
Example 4 (Theorem 6.2., P. 83, B. Liu 2002)
Let ξ = (ξ1, . . . , ξn, ξn+1) and g(x , ξ) = x1ξ1 + . . .+ xnξn − ξn+1. Ifξ1, . . . , ξn, ξn+1 are independently normally distributed randomvariables, then
Prg(x , ξ) ≤ 0 ≥ α
⇔n∑
i=1
E [ξi ]xi+Φ−1(α)
√√√√ n∑i=1
V [ξi ]x2i + V [ξn+1] ≤ E [ξn+1)],
where Φ is the standard normal distribution.
Chance constrained optimization - applications, properties and numerical issues
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Properties and difficulties of chance constraints...• The above result can be extended to
Pr
w0(x) +
p∑k=1
ξkwk(x) ≤ 0
≥ α. (4)
Garnier et al. 2008 (Linearization)
If ξ = (ξ1, . . . , ξp) is a normal multivariate random vector with mean0 and small variance, then
g(x , ξ) ≈ g(x , 0) +
p∑i=1
∂g
∂ξi(x , 0)ξ,
which leads to (??)
• In general, there is no closed-form analytic representation.• Nemirovski 2012 studies tractability of (??) for non-Gaussian ξ.
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Approximation strategies for chance constraintoptimization
It remains to use approximation methods.Approximation strategies
(I) Back-mapping
(II) robust optimization
(III) sample average approximation
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I. Back-mapping (projection) method
Idea of back-projection (Wendt, Li and Wozny 2002):Find a monotonic relation between Z = g(x , ξ) and some randomvariable ξj ; i.e. verify theoretically (see Geletu et al. 2010) orexperimentally that there is a real-valued function ϕ such that, forany x ∈ X ,
Z = ϕx(ξj);
ϕx(·) is strictly increasing (ξj ↑ Z ) or decreasing (ξj ↓ Z ).
⇒ ξj = ϕ−1x (Z ).
• ξ ↑ Z ⇒ Pr g(x , ξ) ≤ 0 = Prξj ≤ ϕ−1
x (0).
• ξ ↓ Z ⇒ Pr g(x , ξ) ≤ 0 = Prξj ≥ ϕ−1
x (0).
Chance constrained optimization - applications, properties and numerical issues
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Back projection ....
Abbildung: Back Projection of chance constraints
• Requires a global implicit function theorem (see Geletu et al. 2011).Chance constrained optimization - applications, properties and numerical issues
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I. Back-projection ...
Now, for ξj ↑ Z , (CCOPT) is equivalent to
(CCOPT ) minx
E [f (x , ξ)]
s.t. p(x) = Prξj ≤ ϕ−1
x (0)≥ α,
u ∈ U .
p(x) =
∫ +∞
−∞. . .
∫ +∞
−∞
∫ ϕ−1x (0)
−∞φ(ξ)dξ (5)
∇p(x) =
∫ +∞
−∞. . .
∫ +∞
−∞∇xϕ
−1x (0)φ(ξ)dξ (6)
Chance constrained optimization - applications, properties and numerical issues
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I. Back-projection ...
Advantages:
usable if monotonic relations are easy to find;
provides direct representation of chance constraints.
Disadvantages:
monotonic relations may not exist;
monotonic relations can be difficult to verify.
Chance constrained optimization - applications, properties and numerical issues
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II. Sample average approximation (SAA)
Shapiro 2003, Pagnoncelli et al. 2009.• Define
I(0,+∞] (g(x , ξ)) =
0, if g(x , ξ) > 01, if g(x , ξ) ≤ 0.
• Determine samples ξ1, . . . , ξN ⊂ Ω (e.g. low discrepancysequences like Fourer, Sobol or Niederreiter sequences).• Replace the chance constrains with
pN(x) =1
N
N∑k=1
I(−∞,0]
(g(x , ξk)
)≥ α.
(pN(x) =Relative-frequency count for the satisfaction of g(x , ξ) ≤ 0. )
Chance constrained optimization - applications, properties and numerical issues
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II. Sample average approximation...
Advantages:
(SAA) avoids computation of multidimensional integrals;
Convexity structures are preserved.
Disadvantages:
SAA leads to a non-smooth optimization problem.
Feasibility of the obtained solution to the (CCOPT) isguaranteed only when N →∞.
Chance constrained optimization - applications, properties and numerical issues
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III. Robust optimization technique
Robust optimization considers the (worst-case) problem
(RO) minx
E [f (x , ξ)]
s.t. g(x , ξ) ≤ 0, ξ ∈ Ω,
x ∈ X ,
where g(x , ξ) ≤ 0 is required to be satisfied for as many realizationsof ξ from Ω as possible.
Randomized solution based on scenario generation:• Generate independent identically distributed random samplesξ1, . . . , ξN from Ω (Monte-Carlo method).
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III. Robust optimization technique ...• Solve the optimization problem
(NLP)RO minx
1
N
N∑k=1
f (x , ξk)
s.t. g(x , ξk) ≤ 0, k = 1, . . . ,N;
x ∈ X .
Theorem (Califore & Campi 2005)
Suppose α ∈ (0, 1) and f (·, ξ) is convex w.r.t. x ∈ Rn. If the numberof random samples ξ1, . . . , ξN
N ≥ 2n
(1− α)ln
(1
1− α
)+
(2
1− α
)ln
(1
α
)+ 2n,
then the optimal solution obtained from (NLP)RO is an optimalsolution of (RO) with reliability α.
Chance constrained optimization - applications, properties and numerical issues
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III. Robust optimization technique ...
Advantages:
there is no need to compute integrals;
the problem (NLP)RO simple to implement and solve;
it also preserves convexity structures.
Disadvantages:
solution of (NLP)RO may not be feasible to the (CCOPT);
for a higher reliability level α, very large number of scenariosξ1, . . . , ξN are required.
Recent suggestion: scenarios reduction methods:Henrion, Kuchler & Romisch 2009; Campi & Garetti 2011.
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A new analytic approximation strategyLet
h(x , ξ) =
0, if g(x , ξ) ≤ 01, if g(x , ξ) > 0.
This impliesPrg(x , ξ) > 0 = E [h(x , ξ)]
⇒
Prg(x , ξ) ≤ 0 ≥ α⇔ E [h(x , ξ)] ≤ 1− α.
A general idea of analytic approximation (Geletu et al. 2012):Find a continuous (possibly smooth) parametric function ψ(τ, ·) sothat
E [h(x , ξ)] ≤ ψ(τ, x),
where τ > 0.Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...Hence
M(τ) = x | ψ(τ, x) ≤ 1− α ⊂ P := x | E [h(x , ξ)] ≤ 1− α.
Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...
• Requirements on ψ(τ, x):
(a) for each fixed τ , ψ(τ, x) should be easy to compute.
(b) For each fixed x
infτ>0
ψ(τ, x) = E [h(x , ξ)]
• (b) ⇒ supτ M(τ) = P.Question:
How to construct such a function ψ(τ, x)?
Basically, ψ(τ, x) = E [Ψ(τ, g(x , ξ))], for Ψ(τ, s) : R+ × R→ R.
Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...
Some suggestions:
Ψ1(τ, s) = exp(τs), τ > 0 (Pinter 1984)
Ψ2(τ, s) = exp(τ−1s), τ > 0 (Nemirovski and Shapiro 2006,Nemirovski 2011).
Ψ3(τ, s) = τ + 11−α [s − τ, 0]+, τ > 0 (Rockafellar and Uryasev
2000), where [s − τ, 0]+ = maxs − τ, 0.A new analytic approximation function (Geletu et al. 2012)
Ψ4(τ, s) =1 + m1τ
1 + m2τe− 1
τs
with 0 < m2 ≤ m1 and τ > 0.
Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...
Abbildung: Comparison of Ψ2 and Ψ4
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A new analytic approximation strategy...
Theorem (A. Geletu 2012)
1 Let τ > 0. Then
Ψ4(τ, s) > 0 for any value of s;Ψ4(τ, s) ≥ 1 for s ≥ 0.
2 For 0 < τ ≤ 12 , then
the function Ψ4(·, s) non-decreasing w.r.t. τ ;
limτ0+
Ψ4(τ, s) =
1, if s ≥ 00, if s < 0
Hence, for any s ∈ R : limτ0+ |Ψ4(τ, s)− I[0,+∞)(s)| = 0.
Hence,
limτ0+
ψ(τ, x) = limτ0+
E [Ψ4(τ, g(x , ξ))] = limτ0+
E [h(x , ξ)]
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A new analytic approximation strategy...
Instead of CCOPT solve the problem
(NLP)τ minx
E [f (x , ξ)] (7)
s.t. (8)
x ∈ M(τ) = x ∈ X | ψ(τ, x) ≤ 1− α,
for each τ with the same objective function E [f (x , ξ)] as in (CCOPT).
Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...
Theorem (Geletu et al. 2012)
If τk any sequence, such that τk 0+ and X is a compact set,then
for each τk , the feasible set M(τk) is compact.
limk→+∞M(τk) = P. (in fact, limk→+∞H(M(τk),P) = 0 )
If x∗ ∈ X is a limit point of a sequence xk of optimal solutionsof (NLP)τk , then x∗ is a solution of CCOPT.
Computational advantage:Solution of NLPτ for a sufficiently small τ provides a goodapproximation for CCOPT.
Chance constrained optimization - applications, properties and numerical issues
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A new analytic approximation strategy...
Advantages:
CCOPT becomes more tractable
NLPτ can be solved using state-of-the-art optimization solvers
solution of NLPτ is always feasible to the CCOPT
Disadvantages:
Objective and constraints may require the evaluation of highdimensional integrals
convexity structures of CCOPT may not be preserved in the newapproximation
currently, the approximation works only for single chanceconstraints
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Conclusions
Analytic approximation facilitates tractability of nonlinear chanceconstrained optimization problems.
For fast processes, linearization and analytic approximation mightbe imperative.
Large-scale problems require faster evaluation of multidimensionalprobability integrals. This can be achieved through: efficient numerical integration techniques (like QMC,
sparse-grid, etc. , integration ) the design of new numerical integration technique parallel computation, etc.
Chance constrained optimization - applications, properties and numerical issues
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References:
Andrieu, L.; Henrion, R.; Romisch, W. A model for dynamicchance constraints in hydro power reservoir management.European j. Oper. Res. 207, 579 – 589, 2010.Liu, B. Theory and practice of uncertain programming.Physica-Verlag, Heidelberg, 2002.Ben-Tal A.; Nemirovski, A. Robust convex optimization. Math.of OR 23, 769 - 805.Birge, J. R.; Lauveaux, F. Introduction of stochasticprogramming. Springer Verlag, New York, 1997.Califore, G.; Campi, M. C. Uncertain convex programs:randomized solutions and confidence levels. Math. Prog. Ser. A,102, 25 – 46, 2005.Campi, M. C.; Garetti, S. A sampling-and-discarding-approach tochance-constrained optimization: feasibility and optimality. J.Optim. Theory and Appl. 148, 257 – 280, 2011.
Chance constrained optimization - applications, properties and numerical issues
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References:
Celik, A. N.; Makkawi, A.; Muneer, T. Critical evaluation of windspeed frequency distribution functions. Journal of Renewable andSustainable Energy, 2, 013102, 2010.Charnes, A.; Cooper, W. Chance constrained programming.Management Science, 6, 73 – 79, 1959.Charnes, A.; Cooper, W.; Symonds, G. H. Cost horizons andcertainty equivalence: An approach in stochastic programming ofheating oil. Management Science, 4, 235 – 263, 1958.Geletu, A.; Kloppel, M.; Zhag, H.; Li, P. Advances andapplications of chance-constrained approaches to systemsoptimization under uncertainty. Inter. J. of Sys. Sc., 2012,accepted.Geletu, A.; Hoffmann, A.; Kloppel, M.; Li, P. Monotony analysisand sparse-grid integration for nonlinear chance constrainedprocess optimization. Eng. Optim., 43(10), 1019 – 1041, 2011.
Chance constrained optimization - applications, properties and numerical issues
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References:
Henrion, R. Some remarks on value-at-risk optimization. Inter. J.Man. Sc. and Eng. Man., 1, 111 – 118, 2006.
Henrion, R.; Kuchler, C.; Romsch, W. Scenario reduction instochastic programming with respect to discrepancy distances.Comput. Optim. Applic., 43, 67 – 93, 2009.
Henrion, R.; Strugarek, C. Convexity of chance constraints withindependent random variables. Comput. Optim. Applic. 41,262-276, 2008.
Kibzun, A.I.; Kan, Y.S. Stochastic programming problems: withprobability and quantile functions. John Wiley & Sons Ltd.,Chichester, England, 1996.
Chance constrained optimization - applications, properties and numerical issues
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References:
Kloppel, M.; Geletu, A.; Hoffmann, A.; Li, P. Using sparse-gridmethods to improve computation efficiency in solving dynamicnonlinear chance-constrained optimization problems. Ind. Eng.Chem. Res., 50, 5693 – 5704, 2011.
Marti, K. Differentiation formulas for probability functions: Thetransformation method. Math. Prog. 75, 201 – 220, 1996.
Nemirovski, A. On safe tractable approximation of chanceconstraints. EJOR, 219, 707 – 718, 2012.
Nemirovski, A.; Shapiro, A. Convex approximation of chanceconstrained programs. SIAM J. Optim., 17, 969 – 996, 2006.
Ouarda, T. B. M. J.; Labadie, J. W. Chance-constrained optimalcontrol for multireservoir system optimization and risk analysis.Stochastic Environmental Research and Risk Analysis, 15,185–204, 2001.
Chance constrained optimization - applications, properties and numerical issues
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References:
Pagnoncelli, B. K.; Ahmed, S.; Shapiro, A. Sample averageapproximation method for chance constrained programming:theory and applications. JOTA, 142, 399 – 416, 2009.
Pinter, J. Deterministic approximation of probability inequalities.Oper. Res. 33, 219 – 239, 1989.
Prekopa, A.; Yoda, K.; Subasi, M. M. Uniform quasi-convexity inprobabilistic constrained stochastic programming. OR Letters, 39,188 – 192, 2011.
Prekopa, A. On the concavity of multivariate probabilitydistribution functions. OR Letters, 29, 1 – 4, 2001.
Prekopa, A. Stochastic programming. Kluwer AcademicPublishers, Dordrecht, The Netherlands, 1995.
Rockafellar, R. T.; Uryasev, S. Optimization of conditionalvalue-at-risk. J. of Risk, 2, 21–41, 2000.
Chance constrained optimization - applications, properties and numerical issues
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References:
Wendt, M.; Li, P.; Wozny, G. Nonlinear chance-constrainedprocess optimization under uncertainty. Ind. Eng. Chem. Res., 41,3621 – 3629, 2002.
Wozabal, D. Value-at-Risk optimization using the difference ofconvex algorithm. OR Spectrum, Published Online, 2010.
Wozabal, D.; Hochreiter, R.; Pflug, G. Ch. A difference of convexformulation of value-at-risk constrained optimization. J. Math.Prog. and OR, 59, 377–400, 2010.
Uryasev, S. Derivatives of probability functions and someapplications. Ann Oper Res, 56, 287–311, 1995.
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