# 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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## 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.

## Keywords

Joint chance constraints CVaR DC-approximation Almost H-calmness Stationary point Exponential convergence## Notes

### Acknowledgements

The work of H. Sun is carried out while he is visiting H. Xu in the School of Mathematics, University of Southampton sponsored by China Scholarship Council. The authors would like to thank Dr. Yi Yang for helpful discussions of the algorithm of the DC-approximation method. They would also like to thank two anonymous referees for insightful comments which have substantially helped improve the quality of the paper.

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