Strong Convergence of Relaxed Inertial Inexact Progressive Hedging Algorithm for Multi-stage Stochastic Variational Inequality Problems

TL;DR

This paper introduces a Halpern-type relaxed inertial inexact progressive hedging algorithm for multi-stage stochastic variational inequalities, proving its strong convergence and demonstrating acceleration through numerical examples.

Contribution

It develops a novel algorithm that combines relaxation, inertia, and inexact subproblem solving for stochastic variational inequalities, with proven convergence.

Findings

The algorithm converges strongly under certain conditions.

Over-relaxation and inertial terms accelerate convergence.

Numerical examples confirm the effectiveness of the proposed method.

Abstract

A Halpern-type relaxed inertial inexact progressive hedging algorithm (PHA) is proposed for solving multi-stage stochastic variational inequalities in general probability spaces. The subproblems in this algorithm are allowed to be calculated inexactly. It is found that the Halpern-type relaxed inertial inexact PHA is closely related to the Halpern-type relaxed inertial inexact proximal point algorithm (PPA). The strong convergence of the Halpern-type relaxed inertial inexact PHA is proved under appropriate conditions. Some numerical examples are given to indicate that the over-relaxed parameter and the inertial term can accelerate the convergence of the algorithm.