Individual confidence intervals for true solutions to stochastic variational inequalities

TL;DR

This paper introduces three methods for constructing confidence intervals for true solutions of stochastic variational inequalities, enabling reliable estimation from a single sample approximation with theoretical guarantees and practical testing.

Contribution

It proposes three novel methods for confidence interval construction for SVI solutions, including direct and indirect approaches, with theoretical validation and implementation guidance.

Findings

All three methods produce valid confidence intervals under weak convergence.

The methods are applicable to real-world problems via numerical examples.

Intervals meet specified confidence levels in tested scenarios.

Abstract

Stochastic variational inequalities (SVI) provide a means for modeling various optimization and equilibrium problems where data are subject to uncertainty. Often it is necessary to estimate the true SVI solution by the solution of a sample average approximation (SAA) problem. This paper proposes three methods for building confidence intervals for components of the true solution, and those intervals are computable from a single SAA solution. The first two methods use an "indirect approach" that requires initially computing asymptotically exact confidence intervals for the solution to the normal map formulation of the SVI. The third method directly constructs confidence intervals for the true SVI solution; intervals produced with this method meet a minimum specified level of confidence in the same situations for which the first two methods are applicable. We justify the three methods theoretically with weak convergence results, discuss how to implement these methods, and test their performance using two numerical examples.