Two Classes Of New Models And Their Solving Methods For Box-constrained Stochastic Variational Inequality Problems

The variational inequality problem as an equilibrium problem with general signifi-cance, it is widely used in many fields, such as supply chain, transportation, inventory and other issues which meet the requirements, preferences, weather and other uncertain factors. If these uncertainties are ignored, we will obtain disastrous consequences, which result in that in recent years the research of the box constrained variational inequality problem (SVI(l,u, F)) becomes a hot issue, and make this problem either in theory or in algorithm has fruitful results. On the basis of previous researches, this article mainly aims at the related researches on the two methods for solving SVI(l,u,F), and the research results are summarized as follows:Firstly, motivated by Sun and Womersley, who presented a continuously differentiable merit function, we construct, the Expected Value (EV) model for solving SVI(l,u, F).Furthermore, under mild conditions, we show that the level set of the EV problem is bounded. Since the objective function of the EV problem contains an expectation, it is not easy to calculate generally. Then we employ sample average approximation method,which is based on the Monte Carlo methods to give approximation problems for the EV problem. In theory, we consider the convergence results of global solution sequences and stationary point sequences for the corresponding approximation problems.Secondly, when the random variable fluctuates, even if SVI(l,u, F) has a solution,the solution obtained by the EV method will deviate greatly from the actual solution. For this reason, we use the method that minimize the maximum residual function to construct the robust optimization problem, which is equivalent to SLVI(l,u,F). Since solving this problem is an optimization problem that containing the maximum and minimum func-tions,it is usually not easy to solve. Therefore, we give several uncertain sets, then the robust optimization problem can be transformed into a tractable robust reformulation. It is worth noting that the transformation also can be applied to find a robust solution of non-monotone SLVI(l. u, F).