Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

TL;DR

This paper addresses the existence of solutions for inexact resolvent inclusion problems in nonlinear analysis, demonstrating convergence properties, stability, and generalizations, thereby advancing the understanding of inexact algorithms in optimization.

Contribution

It proves the existence of solutions for inexact resolvent inclusion problems and extends convergence and stability results to broader classes of functions and operators.

Findings

Positive existence results for inexact resolvent inclusion solutions

Introduction of fully Legendre functions with stability properties

Generalization of strong monotonicity concept

Abstract

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.