Some Research On Stochastic Variational Inequality Problems

Variational inequality theory is an important branch in operations research,which has been extensively applied to economic equilibrium,transportation networks theory,engineering management and many other areas.However,it is often necessary to consider uncertain factors in practice.For example,in the market equilibrium problem,the market demand is usually uncertain.Therefore,the stochastic variational inequality problem has attracted more and more attention.This thesis mainly focuses on the reformulation and the quantitative stability of the stochastic variational inequality problem,which is divided into the following six chapters:In Chapter 1,the background,academic meaning and research status of stochastic variational inequality problems are briefly introduced.Then,the motivations and main content of this thesis are presented.In Chapter 2,some notations,definitions,and basic properties are given.The basic theory of variational inequality and probability are presented.In Chapter 3,an unconstrained optimization reformulation for a class of stochastic nonlinear complementarity problems is presented via the D-gap function.Firstly,the existence of a solution to the unconstrained optimization reformulation is discussed via the discussing of the level set.Then,the sample average approximation method is used to solving the reformulation problem,and the convergence analysis of minimizers or stationary points of the discrete approximation problems is obtained.Finally,the application of the unconstrained optimization formulation to the traffic equilibrium problem is presented.In Chapter 4,the quantitative stability analysis of a class of stochastic linear variational inequalities is considered.Firstly,the expected residual minimization(ERM)formulation for the stochastic linear variational inequality is defined by using the residual function.The existence of solutions of the ERM formulation and its distribution perturbed problem is discussed.Then,the quantitative stability of the ERM formulation is derived under suitable probability metric.Finally,the sample average approximation problem of the ERM formulation is studied,and the rates of convergence of optimal solution sets of the approximation problem is discussed.In Chapter 5,the quantitative stability analysis of a class of two-stage stochastic linear variational inequality problems is considered.Firstly,the existence of solutions of this two-stage stochastic linear variational inequality problems and its distribution perturbed problems is discussed.Then,the quantitative stability is derived under suitable probability metric.Finally,the convergence of optimal solution sets is obtained.In Chapter 6,we simply summarize the thesis,and provide some problems for further consideration.