Confidence Regions for Stochastic Variational Inequalities

The sample average approximation (SAA) method is a basic approach for solving stochastic variational inequalities (SVI). It is well known that under appropriate conditions the SAA solutions provide asymptotically consistent point estimators for the true solution to an SVI. It is of fundamental interest to use such point estimators along with suitable central limit results to develop confidence regions of prescribed level of significance for the true solution. However, standard procedures are not applicable because the central limit theorem that governs the asymptotic behavior of SAA solutions involves a discontinuous function evaluated at the true solution of the SVI. This paper overcomes such a difficulty by exploiting the precise geometric structure of the variational inequalities and by appealing to certain large deviations probability estimates, and proposes a method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solutions. We justify this method theoretically by establishing a precise limit theorem, apply it to complementarity problems, and test it with a linear complementarity problem.