An inertial iteratively regularized extragradient method for bilevel variational inequality problems

TL;DR

This paper introduces an inertial regularized extragradient method for solving complex bilevel variational inequality problems, providing theoretical convergence bounds and preliminary numerical validation.

Contribution

It proposes a novel inertial variant of the regularized extragradient method tailored for bilevel VI problems, extending previous work with momentum and improved analysis.

Findings

Establishes iteration-complexity bounds for non-strongly monotone cases.

Derives improved convergence results under strong monotonicity.

Provides preliminary numerical experiments demonstrating method behavior.

Abstract

We study a bilevel variational inequality problem where the feasible set is itself the solution set of another variational inequality. Motivated by the difficulty of computing projections onto such sets, we consider a regularized extragradient method, as proposed by Samadi and Yousefian (2025), which operates over a simpler constraint set. Building on this framework, we introduce an inertial variant (called IneIREG) that incorporates momentum through extrapolation steps. We establish iteration-complexity bounds for the general (non-strongly monotone) case under both constant and diminishing regularization, and derive improved results under strong monotonicity assumptions. Our analysis extends and refines the results of the previous work by capturing both inertial and regularization effects within a unified framework. Preliminary numerical experiments are also presented to illustrate the behavior of the proposed method.