Quantification and convergence analysis of two-stage stochastic variational inequality problems

Abstract

The thesis is concerned with two-stage stochastic variational inequality problems. Then two topics are considered: 1. Quantitative analysis for a class of two-stage stochastic linear variational inequality problems. 2. Regularized two-stage stochastic variational inequality problems for Cournot-Nash equilibrium under uncertainty. For topic 1, we consider a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems owning a unique solution. We first give several conditions for the existence of solutions to both the original problem and its perturbed problem. Next we derive quantitative stability assertions of this two-stage stochastic problem under the total variation metric via the corresponding residual function. After that, we study its discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively. Finally, through solving a noncooperative two-stage stochastic game of multi-player, we numerically illustrate the obtained theoretical results. In view of the strong monotonicity of the second stage problem in topic 1, we relax this requirement to the monotonicity situation in topic 2. Specifically, for topic 2, we reformulate a convex two-stage non-cooperative multi-player game under uncertainty as a two-stage stochastic variational inequality problem where the second stage problem is just a monotone stochastic linear complementarity problem. Under standard assumptions, we provide sufficient conditions for the existence of solutions of the two-stage stochastic variational inequality problem and propose a regularized sample average approximation method for solving it. We prove the convergence of the method as the regularization parameter tends to zero and the sample size tends to infinity. Moreover, our approach is applied to a two-stage stochastic production and supply planning problem with homogeneous commodity in an oligopolistic market. Numerical results based on randomly generated data are presented to demonstrate the e.ectiveness of our convergence results.