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

Author

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  • Gui-Hua Lin

  • Huifu Xu

  • Masao Fukushima

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. Copyright Springer-Verlag 2008

Suggested Citation

  • Gui-Hua Lin & Huifu Xu & Masao Fukushima, 2008. "Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(3), pages 423-441, June.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:3:p:423-441
    DOI: 10.1007/s00186-007-0201-x
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
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    Cited by:

    1. Yongchao Liu & Huifu Xu & Jane J. Ye, 2011. "Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 670-694, November.
    2. C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 461-478, March.
    3. Yongchao Liu & Huifu Xu & Gui-Hua Lin, 2012. "Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 537-555, February.
    4. Gui-Hua Lin & Mei-Ju Luo & Jin Zhang, 2016. "Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 487-510, November.

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