Solvability of Multistage Pseudomonotone Stochastic Variational Inequalities

TL;DR

This paper investigates the solvability of multistage pseudomonotone stochastic variational inequalities, establishing theoretical properties and proposing an elicited progressive hedging algorithm with numerical validation for solving such problems.

Contribution

It introduces a theoretical framework linking pseudomonotone SVIs to deterministic cases and proposes a novel elicited PHA for nonmonotone SVIs.

Findings

Theoretical results on existence and properties of solutions for pseudomonotone SVIs.

Development of sufficient conditions for applying elicited PHA.

Numerical experiments demonstrating the efficiency of the proposed method.

Abstract

This paper focuses on the solvability of multistage pseudomonotone stochastic variational inequalities (SVIs). On one hand, some known solvability results of pseudomonotone deterministic variational inequalities cannot be directly extended to pseudomonotone SVIs, so we construct the isomorphism between both and then establish theoretical results on the existence, convexity, boundedness and compactness of the solution set for pseudomonotone SVIs via such an isomorphism. On the other hand, the progressive hedging algorithm (PHA) is an important and effective method for solving monotone SVIs, but it cannot be directly used to solve nonmonotone SVIs. We propose some sufficient conditions on the elicitability of pseudomonotone SVIs, which opens the door for applying elicited PHA to solve pseudomonotone SVIs. Numerical results on solving a pseudomonotone two-stage stochastic market optimization problem and randomly generated two stage pseudomonotone linear complementarity problems are presented to show the efficiency of the elicited PHA for solving pseudomonotone SVIs.