Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds

TL;DR

This paper introduces an inexact proximal point method for variational inequalities on Hadamard manifolds, utilizing the concept of enlargement of monotone vector fields to extend linear operator techniques to Riemannian geometry, with applications to constrained optimization.

Contribution

It extends the concept of enlargement of monotone operators to Riemannian manifolds and develops an inexact proximal point method for variational inequalities in this setting.

Findings

Proves convergence of the proposed method.

Generalizes linear operator concepts to Riemannian manifolds.

Provides an application to constrained optimization.

Abstract

In this paper an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced and studied its convergence properties. The main tool used for presenting the method is the concept of enlargement of monotone vector fields, which generalizes the concept of enlargement of monotone operators from the linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained.