Conditional Value-at-Risk Model For Solving Stochastic Structured Monotone Variational Inequality Problems And Their Convergence Analysis

Structured monotonic variational inequality problem(SMVIP)is a kind of special variational inequality problem that contains special structure.And it has important applications in many fields such as economy,traffic balance,engineering,etc.In the real world,the problems we encounter are generally in a dynamic environment with random factors.If we choose to ignore the impact of these uncertain factors to deal with the stochastic structured monotonic variational inequality problem,then the decision is very likely to be inconsistent with the actual situation,and will cause irreparable losses more seriously.In view of this,it is particularly important for scholars to study the stochastic structured monotonic variational inequality problem.Usually,because of the existence of stochastic variables,the solution of the stochastic structured monotonic variational inequality problem usually doesn't exist.In view of this,this paper will construct and solve the CVaR model corresponding to the stochastic structured monotonic variational inequality problem,and give the corresponding approximation problem,and then further discuss the convergence of the approximation problem.This paper mainly includes the following contents.Firstly,the origin,development and research status of variational inequality problems and stochastic variational inequality problems are briefly introduced,and then the research status and significance of structured monotonic variational inequality problems and stochastic structured monotonic variational inequality problems are introduced.Secondly,some preliminary knowledge related to this paper is briefly introduced,mainly including some related concepts,Jensen inequality,Cauchy-Schwarz inequality,related lemmas,and some related theoretical knowledge of value function,conditional value-at-risk model and sample average approximation method.Thirdly,this paper deals with the stochastic structured monotonic variational inequality problem through the value function,and converts it into its equivalent stochastic optimization problem,and then proposes the CVaR model for solving the stochastic structured monotonic variational inequality problem.Then,through the smoothing method,the non-smooth part of the objective function in the CVaR model is smoothed,so as to obtain the smooth approximation problem of the CVaR model.However,the model still contains mathematical expectations,and it isn't easy to be directly worked out.Therefore,the sample average approximation method is used to further give the smooth sample average approximation problem of the CVaR model.Since the solution set isn't necessarily bounded,the boundedness of the level set of the model is studied.Further,this paper analyses the convergence of the global optimal solution of the smooth approximation problem and the smooth sample average approximation problem of the CVaR model.In addition,the convergence of corresponding stationary point of the smooth approximation problem and the smooth sample average approximation problem is given.Finally,the research content of this paper is summarized,and the next research plan for the stochastic structured monotonic variational inequality problem is given.