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

Author

Listed:
  • Hailin Sun

    (Harbin Institute of Technology)

  • Huifu Xu

    (University of Southampton)

  • Yong Wang

    (Harbin Institute of Technology)

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.

Suggested Citation

  • Hailin Sun & Huifu Xu & Yong Wang, 2014. "Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 257-284, April.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:1:d:10.1007_s10957-012-0127-1
    DOI: 10.1007/s10957-012-0127-1
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    References listed on IDEAS

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    3. Kim, Sojung & Weber, Stefan, 2022. "Simulation methods for robust risk assessment and the distorted mix approach," European Journal of Operational Research, Elsevier, vol. 298(1), pages 380-398.
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    5. Lukáš Adam & Martin Branda & Holger Heitsch & René Henrion, 2020. "Solving joint chance constrained problems using regularization and Benders’ decomposition," Annals of Operations Research, Springer, vol. 292(2), pages 683-709, September.
    6. Lukáš Adam & Martin Branda, 2016. "Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 419-436, August.
    7. Sojung Kim & Stefan Weber, 2020. "Simulation Methods for Robust Risk Assessment and the Distorted Mix Approach," Papers 2009.03653, arXiv.org, revised Jan 2022.

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