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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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    1. Gordon J. Alexander & Alexandre M. Baptista, 2004. "A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model," Management Science, INFORMS, vol. 50(9), pages 1261-1273, September.
    2. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
    3. Papageorgiou, Nikolaos S., 1985. "On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 17(2), pages 185-206, October.
    4. F. W. Meng & J. Sun & M. Goh, 2010. "Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 399-418, August.
    5. L. Jeff Hong & Yi Yang & Liwei Zhang, 2011. "Sequential Convex Approximations to Joint Chance Constrained Programs: A Monte Carlo Approach," Operations Research, INFORMS, vol. 59(3), pages 617-630, June.
    6. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    7. Antonio J. Conejo & Miguel Carrión & Juan M. Morales, 2010. "Decision Making Under Uncertainty in Electricity Markets," International Series in Operations Research and Management Science, Springer, number 978-1-4419-7421-1, December.
    8. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
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    Cited by:

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    2. Álvaro Porras & Concepción Domínguez & Juan Miguel Morales & Salvador Pineda, 2023. "Tight and Compact Sample Average Approximation for Joint Chance-Constrained Problems with Applications to Optimal Power Flow," INFORMS Journal on Computing, INFORMS, vol. 35(6), pages 1454-1469, November.
    3. Liu, Zhimin & Qu, Shaojian & Goh, Mark & Wu, Zhong & Huang, Ripeng & Ma, Gang, 2020. "Two-stage mean-risk stochastic optimization model for port cold storage capacity under pelagic fishery yield uncertainty," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    4. 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.
    5. Sojung Kim & Stefan Weber, 2020. "Simulation Methods for Robust Risk Assessment and the Distorted Mix Approach," Papers 2009.03653, arXiv.org, revised Jan 2022.
    6. Arash Gourtani & Tri-Dung Nguyen & Huifu Xu, 2020. "A distributionally robust optimization approach for two-stage facility location problems," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(2), pages 141-172, June.
    7. 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.

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