A Proximal-Projection Method for Finding Zeros of Set-Valued Operators

TL;DR

This paper introduces a proximal-projection iterative algorithm for finding zeros of non-monotone set-valued operators in Banach spaces, extending convergence results to broader classes of operators and applications.

Contribution

It presents a new convergence analysis for the proximal-projection method, generalizing existing results and applying it to ill-posed problems and convex optimization.

Findings

Proximal-projection method converges for non-monotone operators.

Extension of convergence results to operators with weakly closed graphs.

Application to variational inequalities and convex optimization problems.

Abstract

In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method is, essentially, a fixed point procedure and our convergence results are based on new generalizations of Lemma Opial. We show how the proximal-projection method can be applied for solving ill-posed variational inequalities and convex optimization problems with data given or computable by approximations only. The convergence properties of the proximal-projection method we establish also allow us to prove that the proximal point method (with Bregman distances), whose convergence was known to happen for maximal monotone operators, still converges when the operator involved in it is monotone with sequentially weakly closed graph.