On the range of the Douglas-Rachford operator

TL;DR

This paper systematically studies the range of the Douglas-Rachford operator, revealing a simple formula for 3* monotone operators and exploring applications to various classes of operators and mappings.

Contribution

It provides a new, explicit formula for the range of the Douglas-Rachford operator for 3* monotone operators, advancing understanding in variational analysis.

Findings

Range nearly equals a set involving domains and ranges of operators

Formula applies to subdifferential operators and displacement vectors

Includes examples and counter-examples related to Brezis-Haraux theorem

Abstract

The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counter-examples are presented, including some concerning the celebrated Brezis-Haraux theorem.