Solving Methods And Their Convergence Of Stochastic Generalized Nash Equilibrium Problems

The generalized Nash equilibrium problems (GNEP) have been widely applied in practice. However, there are some uncertain factors in real life. Ignoring these factors will lead to decision-making errors. Due to the introduction of random factors, the GNEP become more complex and more widely applied. Hence, in this paper, we study stochastic generalized Nash equilibrium problems (SGNEP). Since the existence of random variables, the SGNEP may have no solutions in general. It is necessary to construct a reasonable deterministic model, then to solve the deterministic model and to regard the solution of the deterministic model as the solution of SGNEP. We first consider the non-smooth case of SGNEP. In order to give deterministic model of the SGNEP, we present expected residual minimization (ERM) model by using the first order optimality conditions of SGNEP and the nonlinear complementary function. Because of the objective function of the model is non-convex, which makes the obtained solutions by optimization algorithm more than one. It may not be the best if we take an arbitrary solution from the obtained solutions as the solution of SGNEP. To this end, we introduce a restricted ERM model of SGNEP. Since the model is not smooth, we give the smoothing problem of the model by smoothing method and analyze convergence of optimal solution of the smoothing problem. Due to the objective function of the smoothing problem contains expectation, and the expectation is not easy to get. Therefore, we further apply sample average approximation (SAA) method to obtain approximate problems of the smoothing problem and give the convergence of optimal solutions of the approximate problems. Secondly, we mainly discuss the smooth SGNEP. We construct a low risk model of the SGNEP by applying first order necessary conditions of the SGNEP and conditional value-at-risk (CVaR). Since the low risk model contains non-smooth constraints and mathematical expectation, we then use smoothing method and penalized SAA technique to present approximation problems of the model and analyze the convergence of optimal solutions of the approximation problems. The last, we show some numerical examples to verify the feasibility of the proposed method in this paper.