Adaptive Methods or Variational Inequalities with Relatively Smooth and Reletively Strongly Monotone Operators

TL;DR

This paper develops and analyzes adaptive algorithms for variational inequalities involving relatively smooth and strongly monotone operators, providing convergence guarantees and numerical validation for specific problems.

Contribution

It introduces an adaptive variant of the proximal extragradient method with proven convergence rates for this class of problems.

Findings

Convergence rate estimates for the adaptive method.

Numerical experiments on ridge regression.

Numerical experiments on box-simplex games.

Abstract

The article is devoted to some adaptive methods for variational inequalities with relatively smooth and relatively strongly monotone operators. Starting from the recently proposed proximal variant of the extragradient method for this class of problems, we investigate in detail the method with adaptively selected parameter values. An estimate of the convergence rate of this method is proved. The result is generalized to a class of variational inequalities with relatively strongly monotone generalized smooth variational inequality operators. Numerical experiments have been performed for the problem of ridge regression and variational inequality associated with box-simplex games.