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

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.

[1]  Xing-Si Li,et al.  AN AGGREGATE FUNCTION METHOD FOR NONLINEAR PROGRAMMING , 1991 .

[2]  E. S. Kuh,et al.  Piecewise-Linear Theory of Nonlinear Networks , 1972 .

[3]  Roman Sznajder,et al.  A generalization of the Nash equilibrium theorem on bimatrix games , 1996 .

[4]  Gül Gürkan,et al.  Sample-path solution of stochastic variational inequalities , 1999, Math. Program..

[5]  Xiaojun Chen,et al.  Expected Residual Minimization Method for Stochastic Linear Complementarity Problems , 2005, Math. Oper. Res..

[6]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[7]  S. Min.  Singular control problems in bounded intervals , 1987 .

[8]  Ji-Ming Peng,et al.  A non-interior continuation method for generalized linear complementarity problems , 1999, Math. Program..

[9]  Jie Zhang,et al.  On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Variational Inequality , 2012 .

[10]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[11]  R. Sznajder,et al.  Generalizations of P0- and P-properties; extended vertical and horizontal linear complementarity problems , 1995 .

[12]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[13]  Li Xingsi,et al.  AN ENTROPY-BASED AGGREGATE METHOD FOR MINIMAX OPTIMIZATION , 1992 .

[14]  Peter Kall,et al.  Stochastic Programming , 1995 .

[15]  Li-Zhi Liao,et al.  A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems , 1999, SIAM J. Matrix Anal. Appl..

[16]  Houyuan Jiang,et al.  Stochastic Approximation Approaches to the Stochastic Variational Inequality Problem , 2008, IEEE Transactions on Automatic Control.

[17]  Huifu Xu,et al.  Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications , 2009, Math. Program..

[18]  R. Tyrrell Rockafellar,et al.  Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming , 1993, Math. Oper. Res..

[19]  R. Sznajder,et al.  The Generalized Order Linear Complementarity Problem , 1994, SIAM J. Matrix Anal. Appl..

[20]  Jie Zhang,et al.  On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Generalized Equation , 2011 .