On Douglas-Rachford operators that fail to be proximal mappings

TL;DR

This paper investigates the Douglas-Rachford operator in monotone operator theory, revealing that it often fails to be a proximal mapping even in symmetric linear relations, challenging assumptions about its structure.

Contribution

It proves that the Douglas-Rachford operator is generically not a proximal mapping within the class of symmetric linear relations, highlighting a fundamental limitation.

Findings

Douglas-Rachford operator often not a proximal mapping

Failure occurs generically in symmetric linear relations

Challenges assumptions about the operator's structure

Abstract

The problem of finding a zero of the sum of two maximally monotone operators is of central importance in optimization. One successful method to find such a zero is the Douglas-Rachford algorithm which iterates a firmly nonexpansive operator constructed from the resolvents of the given monotone operators. In the context of finding minimizers of convex functions, the resolvents are actually proximal mappings. Interestingly, as pointed out by Eckstein in 1989, the Douglas-Rachford operator itself may fail to be a proximal mapping. We consider the class of symmetric linear relations that are maximally monotone and prove the striking result that the Douglas-Rachford operator is generically not a proximal mapping.