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A SAA nonlinear regularization method for a stochastic extended vertical linear complementarity problem

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  • Zhang, Jie
  • He, Su-xiang
  • Wang, Quan

Abstract

In this paper we propose a new stochastic equilibrium model, a stochastic extended vertical linear complementarity problem, which arises in the stochastic generalized bimatrix games and includes stochastic linear complementarity problem as its special case. Based on the log-exponential function, a sample average approximation (SAA) regularization method is proposed for solving this problem. The analysis of this regularization method is carried out in two steps: first, under some mild conditions, the existence and convergence results to the proposed method are provided. Second, under conditions on row representative of matrices, the exponential convergence rate of this method is established. At last, the regularization method proposed is applied to finding a generalized Nash equilibrium pair for a stochastic generalized bimatrix game.

Suggested Citation

  • Zhang, Jie & He, Su-xiang & Wang, Quan, 2014. "A SAA nonlinear regularization method for a stochastic extended vertical linear complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 888-897.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:888-897
    DOI: 10.1016/j.amc.2014.01.121
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    References listed on IDEAS

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    1. Xiaojun Chen & Masao Fukushima, 2005. "Expected Residual Minimization Method for Stochastic Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 1022-1038, November.
    2. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    3. Gowda, M Seetharama & Sznajder, Roman, 1996. "A Generalization of the Nash Equilibrium Theorem on Bimatrix Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 1-12.
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