Warped Proximal Iterations for Monotone Inclusions

TL;DR

This paper introduces warped resolvents, a generalized concept for set-valued operators, providing a unified framework for analyzing and developing algorithms to solve complex monotone inclusion problems.

Contribution

It proposes warped resolvents, explores their properties, and develops convergence principles, offering a new approach to design and analyze splitting algorithms for monotone inclusions.

Findings

Warped resolvents generalize classical resolvents.

Convergence principles for warped resolvents are established.

New solution methods for challenging inclusion problems are derived.

Abstract

Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.