On the order of the operators in the Douglas-Rachford algorithm

TL;DR

This paper systematically studies the two variants of the Douglas-Rachford algorithm, revealing how the order of operators affects fixed points and providing formulas under certain conditions, with illustrative examples.

Contribution

It offers a detailed analysis of the order dependence in Douglas-Rachford operators, including fixed point relations and explicit formulas under specific assumptions.

Findings

Reflectors act as bijections between fixed point sets of the two operators

Explicit formulas for fixed points under additional assumptions

Examples illustrating the theoretical results

Abstract

The Douglas-Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas-Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas-Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas-Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.