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Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

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Abstract

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example.

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Authors and Affiliations

  1. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    Gui-Hua Lin

  2. School of Mathematics, University of Southampton, Highfield Southamptom, UK

    Huifu Xu

  3. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Masao Fukushima

Authors
  1. Gui-Hua Lin
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  2. Huifu Xu
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  3. Masao Fukushima
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Corresponding author

Correspondence to Gui-Hua Lin.

Additional information

The first author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science and SRF for ROCS, SEM. The second author’s work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council grant. The third author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.

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Lin, GH., Xu, H. & Fukushima, M. Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints. Math Meth Oper Res 67, 423–441 (2008). https://doi.org/10.1007/s00186-007-0201-x

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  • DOI: https://doi.org/10.1007/s00186-007-0201-x

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