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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Abstract	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.
AbstractList	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. 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.[PUBLICATION ABSTRACT]
Author	Xu, Huifu Wang, Yong Sun, Hailin
Author_xml	– sequence: 1 givenname: Hailin surname: Sun fullname: Sun, Hailin email: mathhlsun@gmail.com organization: Department of Mathematics, Harbin Institute of Technology – sequence: 2 givenname: Huifu surname: Xu fullname: Xu, Huifu organization: School of Mathematics, University of Southampton – sequence: 3 givenname: Yong surname: Wang fullname: Wang, Yong organization: Department of Mathematics, Harbin Institute of Technology
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Cites_doi	10.1287/opre.1100.0910 10.1016/j.jmaa.2010.03.021 10.1007/978-3-642-02431-3 10.1016/S0031-3203(99)00020-5 10.1007/s10957-010-9676-3 10.1016/j.spl.2007.05.026 10.1007/978-1-4419-7421-1 10.1080/02331930801954177 10.1214/aop/1176996275 10.1007/978-1-4612-1394-9 10.1007/BFb0087685 10.1287/moor.21.3.513 10.1287/moor.1110.0506 10.21314/JOR.2000.038 10.1137/050622328 10.1016/j.jmaa.2011.07.007 10.1007/s10107-004-0523-x 10.1007/BF01442544 10.1007/s10957-012-0127-1 10.1007/s10107-013-0711-7 10.1007/s10107-008-0214-0 10.1142/6478 10.1017/CBO9780511804441 10.1016/0022-247X(65)90049-1 10.1007/PL00011380 10.1016/0047-259X(85)90078-8 10.1287/mnsc.1040.0201
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Snippet	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...
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SubjectTerms	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
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Title	Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions
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