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Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints

Author

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  • Gui-Hua Lin

    (Shanghai University)

  • Mei-Ju Luo

    (Liaoning University)

  • Jin Zhang

    (Hong Kong Baptist University)

Abstract

We consider a stochastic non-smooth programming problem with equality, inequality and abstract constraints, which is a generalization of the problem studied by Xu and Zhang (Math Program 119:371–401, 2009) where only an abstract constraint is considered. We employ a smoothing technique to deal with the non-smoothness and use the sample average approximation techniques to cope with the mathematical expectations. Then, we investigate the convergence properties of the approximation problems. We further apply the approach to solve the stochastic mathematical programs with equilibrium constraints. In addition, we give an illustrative example in economics to show the applicability of proposed approach.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:66:y:2016:i:3:d:10.1007_s10898-016-0413-9
    DOI: 10.1007/s10898-016-0413-9
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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.
    2. J. J. Ye & X. Y. Ye, 1997. "Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 977-997, November.
    3. Alexander, S. & Coleman, T.F. & Li, Y., 2006. "Minimizing CVaR and VaR for a portfolio of derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 583-605, February.
    4. D. Ralph & H. Xu, 2005. "Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 673-699, March.
    5. N. D. Yen, 1997. "Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 199-225, April.
    6. 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.
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    Cited by:

    1. Min Li & Chao Zhang, 2020. "Two-Stage Stochastic Variational Inequality Arising from Stochastic Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 324-343, July.

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