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.

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Abstract. Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals.

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Stochastic variational inequalities model a large class of equilibrium problems subject to data uncertainty. The true solution to such a problem is usually estimated by a solution to its sample average approximation (SAA) problem. This paper proposed a new method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solution.

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Sparse regression techniques have been popular in recent years because of their ability in handling high dimensional data with built-in variable selection.The lasso is perhaps one of the most well-known examples. Despite intensive work in this direction, how to provide valid inference for sparse regularized methods remains a challenging statistical problem. We take a unique point of view of this problem and propose to make use of stochastic variational inequality techniques in optimization to derive confidence intervals and regions for the lasso. Some theoretical properties of the procedure are obtained. Both simulated and real data examples are used to demonstrate the performance of the method.

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