Survey: Sixty Years of Douglas--Rachford

TL;DR

This survey reviews sixty years of research on the Douglas--Rachford method, highlighting its theoretical foundations, applications in convex and nonconvex problems, and its surprising effectiveness in solving complex feasibility problems like Sudoku puzzles.

Contribution

It provides a comprehensive overview of the method's development, applications, and recent advances, emphasizing its performance in nonconvex contexts and commemorating Borwein's contributions.

Findings

Well-understood in convex settings

Surprisingly effective in nonconvex problems

Notable success in solving Sudoku puzzles

Abstract

The Douglas--Rachford method is a splitting method frequently employed for finding zeroes of sums of maximally monotone operators. When the operators in question are normal cones operators, the iterated process may be used to solve feasibility problems of the form: Find $x \in \bigcap_{k=1}^N S_k.$ The success of the method in the context of closed, convex, nonempty sets $S_1,\dots,S_N$ is well-known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less understood yet surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg, and Thibault's success in applying the method for solving Sudoku Puzzles, began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein's celebrated contributions to the area.