Sample Average Approximation Method For Solving A Deterministic Formulation For Box Constrained Stochastic Variational Inequality Problems

In this paper, we discuss the Expected Residual Minimization (ERM) method, which is to minimize the expected residue of some merit function for box constrained stochastic variational inequality problems (BSVIPs). This method provides a deterministic model, which formulates BSVIPs as an optimization problem. We first study the conditions under which the level sets of the ERM problem are bounded. Then, we show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in BSVIPs. Since the integrality involved in the ERM problem is difficult to compute generally, we then employ sample average approximation method to solve it. Finally, we show that the global optimal solutions and generalized KKT points of the approximate problems converge to their counterparts of the ERM problem. On the other hand, as an application, we consider the model of European natural gas market under price uncertainty. Preliminary numerical experiments indicate that the proposed approach is applicable.