On the Douglas-Rachford algorithm

TL;DR

This paper advances understanding of the Douglas-Rachford algorithm, especially in inconsistent cases, by proving full weak convergence in convex feasibility problems and providing broader conditions for convergence.

Contribution

It provides a complete proof of weak convergence in the inconsistent convex feasibility setting and introduces a more general convergence condition.

Findings

Weak convergence of the shadow sequence is established in the convex feasibility setting.

A more general sufficient condition for weak convergence in the inconsistent case is proposed.

Examples illustrate the theoretical results.

Abstract

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results.