An inexact proximal point method for variational inequality on Hadamard manifolds

TL;DR

This paper introduces an inexact proximal point algorithm for solving variational inequality problems on Hadamard manifolds, analyzing its convergence and extending it to various constrained and nonlinear optimization problems.

Contribution

It proposes a novel inexact proximal point method tailored for Hadamard manifolds, with convergence analysis and applications to multiple problem types.

Findings

Convergence of the proposed inexact method is established.

The method effectively handles constrained and nonlinear optimization on manifolds.

Applicability to equilibrium problems demonstrated.

Abstract

In this paper we present an inexact proximal point method for variational inequality problem on Hadamard manifolds and study its convergence properties. The proposed algorithm is inexact in two sense. First, each proximal subproblem is approximated by using the enlargement of the vector field in consideration and then the next iterated is obtained by solving this subproblem allowing a suitable error tolerance. As an application, we obtain an inexact proximal point method for constrained optimization problems, equilibrium problems and nonlinear optimization problems on Hadamard manifolds.