Strongly nonexpansive mappings revisited: uniform monotonicity and operator splitting

TL;DR

This paper explores the relationship between uniformly monotone operators and strongly nonexpansive mappings, providing new insights into operator splitting methods like Douglas-Rachford and Peaceman-Rachford.

Contribution

It offers a comprehensive analysis of uniformly monotone operators, their reflected resolvents, and their connection to strongly nonexpansive operators, with applications to splitting algorithms.

Findings

Reflected resolvents of uniformly monotone operators are closely related to strongly nonexpansive operators.

Connections between duality and inverse operators are systematically established.

Applications to convergence analysis of Douglas-Rachford and Peaceman-Rachford methods are provided.

Abstract

The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of the class of uniformly monotone operators and their corresponding reflected resolvents. We show that the latter is closely related to the class of the strongly nonexpansive operators introduced by Bruck and Reich. Connections to duality via inverse operators are systematically studied. We provide applications to Douglas-Rachford and Peaceman-Rachford methods. Examples that illustrate and tighten our results are presented.