Study On An Improved Expected Residual Method For Nonlinear Stochastic Variational Inequalities

Variational inequality theory is an important component in the operations research.It has been widely applied to mathematics,natural science,economic equilibrium and any other areas.Considering that there are many uncertain factors in practical problems,it is necessary to study the stochastic variational inequalities.In this paper,we propose an improved expectation residual method for nonlinear stochastic variational inequalities,and analyze the stochastic variational inequalities.The main results of this paper can be summarized as follows:(1)For the stochastic affine variational inequality problems with nonlinear perturbations,considering the convex combination of expectation and variance,the problem of improving the weighted expected residual minimization is obtained.On one hand,under some conditions,we study the differentiability of the objective function and the boundedness of the level set.On the anther hand,by using the Quasi Monte Carlo method,the discrete approximation problem of the improved expected residual minimization problem is obtained,and the optimal solution of the discrete approximation problem and the convergence of the stable solution are analyzed.(2)For the general stochastic nonlinear variational inequality,we first use the convex combination of expectation and variance to obtain an improved expected residual minimization problem.Secondly,when the sample space is compact,using the Quasi Monte Carlo method,the discrete approximation problem of the improved expected residual minimization problem is obtained,and the convergence of the discrete approximation problem is analyzed.Finally,when the sample space is noncompact,using a compact approximation approach,the compact approximation problem of the improved expectation residual minimization problem is obtained,and the convergence of the compact approximation problem is analyzed.