Speaker: Shu Lu

Title: Confidence regions and confidence intervals for stochastic variational inequalities via the normal map approach

Abstract:

Variational inequalities model a general class of equilibrium problems, and also arise as first-order conditions of nonlinear programs. This talk considers a stochastic variational inequality (SVI) defined over a polyhedron, with the function defining the variational inequality being an expectation function. A basic approach for solving the SVI is the sample average approximation (SAA) method, which replaces the expectation function by a sample average function, and solves the obtained SAA problem to use its solution as an estimate of the true solution. It is well known that under appropriate conditions the SAA solutions provide asymptotically consistent point estimators for the true solution to an SVI.

We consider the normal map formulation of the SVI and the SAA problems, and discuss methods to compute confidence regions and confidence intervals of prescribed level of significance for the true solution, given an SAA solution. These methods are based on the asymptotic distribution of SAA solutions, obtained from combining sensitivity analysis techniques with suitable central limit results. The discontinuity of certain quantities causes a difficulty in estimating a function that governs the asymptotic behavior of SAA solutions. The piecewise linear property of the latter function leads to an additional difficulty in computing confidence intervals. This talk presents methods proposed to overcome those difficulties, with numerical results.