Algorithms for solving variational inequalities and saddle point problems with some generalizations of Lipschitz property for operators

TL;DR

This paper develops accelerated numerical algorithms for saddle point problems and variational inequalities, relaxing smoothness requirements and introducing new boundedness conditions to improve efficiency and applicability.

Contribution

It introduces accelerated methods for strongly convex-concave saddle point problems and an analogue of mirror descent for operators with generalized boundedness conditions.

Findings

Improved complexity estimates for algorithms

Introduction of an operator boundedness analogue

Optimal mirror descent method for generalized operators

Abstract

The article is devoted to the development of numerical methods for solving saddle point problems and variational inequalities with simplified requirements for the smoothness conditions of functionals. Recently there were proposed some notable methods for optimization problems with strongly monotone operators. Our focus here is on newly proposed techniques for solving strongly convex-concave saddle point problems. One of the goals of the article is to improve the obtained estimates of the complexity of introduced algorithms by using accelerated methods for solving auxiliary problems. The second focus of the article is introducing an analogue of the boundedness condition for the operator in the case of arbitrary (not necessarily Euclidean) prox structure. We propose an analogue of the mirror descent method for solving variational inequalities with such operators, which is optimal in the considered class of problems.