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A smoothing SAA method for a stochastic mathematical program with complementarity constraints

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Abstract

A smoothing sample average approximation (SAA) method based on the log-exponential function is proposed for solving a stochastic mathematical program with complementarity constraints (SMPCC) considered by Birbil et al. (S. I. Birbil, G. Gürkan, O. Listes: Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res. 31 (2006), 739–760). It is demonstrated that, under suitable conditions, the optimal solution of the smoothed SAA problem converges almost surely to that of the true problem as the sample size tends to infinity. Moreover, under a strong second-order sufficient condition for SMPCC, the almost sure convergence of Karash-Kuhn-Tucker points of the smoothed SAA problem is established by Robinson’s stability theory. Some preliminary numerical results are reported to show the efficiency of our method.

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Authors and Affiliations

  1. Institute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P.R.China

    Jie Zhang & Li-wei Zhang

  2. School of Management, University of Southampton, Highfield Southampton, Southampton, SO17 1BJ, UK

    Yue Wu

Authors
  1. Jie Zhang
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  2. Li-wei Zhang
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  3. Yue Wu
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Correspondence to Jie Zhang.

Additional information

This research was supported by the National Natural Science Foundation of China under project No. 11071029 and the Fundamental Research Funds for the Central Universities.

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Zhang, J., Zhang, Lw. & Wu, Y. A smoothing SAA method for a stochastic mathematical program with complementarity constraints. Appl Math 57, 477–502 (2012). https://doi.org/10.1007/s10492-012-0028-5

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  • DOI: https://doi.org/10.1007/s10492-012-0028-5

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