Distributionally Robust Expected Residual Minimization for Stochastic Variational Inequality Problems

TL;DR

This paper introduces a distributionally robust extension of expected residual minimization for stochastic variational inequality problems, ensuring solutions are robust against distributional uncertainties by reformulating the problem as a convex semidefinite program.

Contribution

It extends ERM to a distributionally robust framework (DRERM) for SVIPs, allowing for robust solutions under uncertain distributions, and reformulates it as a convex nonlinear semidefinite program.

Findings

DRERM provides robust solutions under distributional ambiguity.

The reformulation as a convex semidefinite program avoids numerical integration.

The approach is applicable under suitable assumptions for SVIPs.

Abstract

The stochastic variational inequality problem (SVIP) is an equilibrium model that includes random variables and has been widely applied in various fields such as economics and engineering. Expected residual minimization (ERM) is an established model for obtaining a reasonable solution for the SVIP, and its objective function is an expected value of a suitable merit function for the SVIP. However, the ERM is restricted to the case where the distribution is known in advance. We extend the ERM to ensure the attainment of robust solutions for the SVIP under the uncertainty distribution (the extended ERM is referred to as distributionally robust expected residual minimization (DRERM), where the worst-case distribution is derived from the set of probability measures in which the expected value and variance take the same sample mean and variance, respectively). Under suitable assumptions, we demonstrate that the DRERM can be reformulated as a deterministic convex nonlinear semidefinite programming to avoid numerical integration.