An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities
Journal
SIAM Journal on Optimization
Journal Volume
35
Journal Issue
3
Start Page
1469
End Page
1497
ISSN
1052-6234
1095-7189
Date Issued
2025-07-03
Author(s)
Zhang, Shuzhong
Abstract
In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as the Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem, where the associated operator consists of an approximation operator satisfying a pth-order Lipschitz bound with respect to the original mapping, and the gradient of a (p+1)th-order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a pth-order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is a broad scheme, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds.
Subjects
composite operators
extra-gradient method
high-order methods
variational inequality
SDGs
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Type
journal article