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Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints

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Abstract

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of \(\epsilon \)-optimal values and \(\epsilon \)-optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.

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Acknowledgments

The authors are grateful to the three anonymous referees for their insightful comments which have significantly helped improve the quality of the paper. They are also thankful to Professor Henry Wolkowicz for organizing an effective review.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Hailin Sun

  2. School of Engineering and Mathematical Sciences, City University London, Northampton Square, London, EC1V 0HB, UK

    Huifu Xu

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  1. Hailin Sun
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  2. Huifu Xu
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Correspondence to Huifu Xu.

Additional information

Dedicated to Professor Jon Borwein on the occasion of his 60th birthday.

The first author was supported by China Scholarship Council, the National Natural Science Foundation of China No. 11171159 and the Specialized Research Fund of Doctoral Program of Higher Education of China No. 20103207110002.

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Sun, H., Xu, H. Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints. Math. Program. 143, 31–59 (2014). https://doi.org/10.1007/s10107-013-0711-7

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