Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions

Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points, when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifi...

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Vydáno v:Journal of optimization theory and applications Ročník 161; číslo 1; s. 257 - 284
Hlavní autoři: Sun, Hailin, Xu, Huifu, Wang, Yong
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.04.2014
Springer Nature B.V
Témata:
Applications of Mathematics
Approximation
Asymptotic methods
Asymptotic properties
Calculus of Variations and Optimal Control; Optimization
Convergence
Convex analysis
Deviation
Engineering
Expected values
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Random variables
Stochastic models
Stochasticity
Studies
Theory
Theory of Computation
Exponential convergence
Joint chance constraints
CVaR
DC-approximation
Almost H-calmness
Stationary point
ISSN:0022-3239, 1573-2878
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Shrnutí:Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points, when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifically, we prove under some moderate conditions that optimal solutions and stationary points, obtained from solving sample average approximated problems, converge with probability one to their true counterparts. Moreover, by exploiting the recent results on large deviation of random functions and sensitivity results for generalized equations, we derive exponential rate of convergence of stationary points. The discussion is also extended to the case, when CVaR approximation is replaced by a difference of two convex functions (DC-approximation). Some preliminary numerical test results are reported.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-012-0127-1