An Extragradient-type Algorithm for Variational Inequality on Hadamard Manifolds

TL;DR

This paper introduces an extragradient algorithm for solving variational inequalities on Hadamard manifolds, utilizing $\

Contribution

It develops a novel extragradient method for variational inequalities on Hadamard manifolds using $\\epsilon$-enlargement of vector fields, establishing convergence properties in this setting.

Findings

Convergence of the proposed method is proven on Hadamard manifolds.

The $\\epsilon$-enlargement of maximal monotone vector fields is shown to be lower-semicontinuous.

The method extends variational inequality solutions to non-Euclidean geometric contexts.

Abstract

The aim of this paper is to present an extragradient method for variational inequality associated to a point-to-set vector field in Hadamard manifolds and to study its convergence properties. In order to present our method the concept of $\epsilon$-enlargement of maximal monotone vector fields is used and its lower-semicontinuity is stablished in order to obtain the convergence of the method in this new context.