Convergence Results of the Restricted ERM Model for Nonsmooth Stochastic Generalized Nash Equilibrium Problems

Abstract

SYNOPTIC ABSTRACTIn this study, we consider stochastic generalized Nash equilibrium problems (SGNEP). They have a wide range of application and have drawn significant attention of researchers recently. First, we use the first order optimality conditions of SGNEP and the nonlinear complementary function to present an expected residual minimization (ERM) model for the case in which the involved function is not continuously differentiable. But this ERM model might have many solutions. It is not enough to find just an arbitrary solution, because other solutions could also provide reasonable outcomes of SGNEP. To this end, we introduce a restricted ERM model of SGNEP in which some additional conditions are imposed on the Lagrange multipliers for the dependent constraints. Then, we will employ a smoothing function to the restricted ERM model to deal with the noncontinuously differentiable case. We further show that the solutions of this smoothing restricted ERM converge to the true problem. Because the restricted ERM formulation contains an expectation, we further propose a sample average approximate method to deal with this expectation. Moreover, we show that the global optimal solutions of these approximate problems converge to the global optimal solution of the restricted ERM problem. Note that there are two ways to converge: one is to fix the smoothing parameter, the other is to let the smoothing parameter tend to zero as the sample size increases. In fact, by changing the smoothing parameter and sample size simultaneously, it is difficult to obtain the convergence results.