(Quasi-)Monte Carlo Methods And Their Convergence For Stochastic Variational Inequality Problems

Variational inequality theory is an important branch in operations research. It has extensive applications in natural science, engineering calculation, economic equilibrium, and many other areas. However, some elements may involve uncertain data in practice. For example, the demands are generally difficult to be determined in supply chain network for it varies with the change of income level, personal preferences, and other factors. Hence, in this paper, we mainly consider stochastic variational inequality problems (SVIP) and some related topics. The main results of this dissertation can be summarized as follows:1. Because of existence of the random elements, the SVIP may have no solution in general. Therefore, how to construct a reasonable deterministic model is very important in the study of SVIP. Using the so-called regularized gap function, in chapter 3, we give an expected residual minimization (ERM) model for SVIP. Then, we discuss the properties of the ERM problem including the boundedness of level sets, error bounds and the differentiability of the objective function. Since the ERM formulation contains an expectation, which is usually difficult to compute, we then propose a quasi-Monte Carlo method to solve the ERM problem. If the function involved in SVIP is affine, we show that, under some conditions, any accumulation point of global optimal solutions or stationary points of the corresponding approximation problems is optimal or stationary to the ERM problem. When the function involved is not affine, we investigate convergence properties of the approximation problems for the case where the underlying sample space is compact. Furthermore, we suggest a compact approximation approach for the case where the sample space is noncompact.2. In chapter 4, we consider stochastic variational inequality problems with addi-tional constraints. Similarly, because of existence of the random elements, the problems may have no solution in general. Hence, we use the so-called regularized gap function to give an ERM formulation for the considered problem. Since the ERM formulation contains an expectation, we then propose a quasi-Monte Carlo method to give approximation prob-lems and investigate its convergence properties. As an application, we present a model of supply chain network with restricted output and random demands. We formulate the equilibrium conditions of the supply chain network into a stochastic variational inequality problem with additional constraints. Preliminary numerical experiments indicate that the proposed approach is applicable.3. In chapter 5, we mainly discuss the box constrained stochastic variational in-equality problems (BSVIP). Since the regularized gap function is relatively complex to evaluate for the above problems, in this chapter, we use the merit function presented by Sun and Womersley to give an ERM model for BSVIP. Furthermore, we verify that the level sets of this ERM formulation is bounded under mild conditions. In addition, under some condition, we show that solutions of the ERM formulation have a minimum sensitivity with respect to random parameter variations in BSVIP. Since the expectation involved in the ERM problem is difficult to compute generally, we then employ sample average approach, which is based on the Monte Carlo methods, to give approximation problems for the ERM problem. Finally, we show that, with probability one, any accu-mulation point of the global optimal solutions or stationary points of the approximation is optimal or stationary to the ERM problem under mild conditions.