Some Methods for Relatively Strongly Monotone Variational Inequalities

TL;DR

This paper develops and analyzes new numerical methods, including modified Mirror Descent and adaptive Proximal Mirror algorithms, for solving variational inequalities with relatively strongly monotone and smooth operators, achieving improved convergence rates.

Contribution

It introduces modified and adaptive algorithms tailored for variational inequalities with relatively bounded and relatively smooth operators, enhancing convergence efficiency.

Findings

Modified Mirror Descent improves convergence for relatively bounded operators.

Adaptive Proximal Mirror achieves linear convergence for relatively smooth, strongly monotone problems.

Algorithms outperform existing methods in convergence speed and robustness.

Abstract

The article is devoted to the development of numerical methods for solving variational inequalities with relatively strongly monotone operators. We consider two classes of variational inequalities related to some analogs of the Lipschitz condition of the operator that appeared several years ago. One of these classes is associated with the relative boundedness of the operator, and the other one with the analog of the Lipschitz condition (namely, relative smoothness). For variational inequalities with relatively bounded and relatively strongly monotone operators, we introduce a modification of the Mirror Descent method, which optimizes the convergence rate. We also propose the adaptive Proximal Pirror algorithm and its restarted version with a linear convergence rate for problems with relatively smooth and relatively strongly monotone operators.