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SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD

Published online by Cambridge University Press:  20 February 2020

SCOTT B. LINDSTROM*
Affiliation:
CARMA, University of Newcastle, Australia
BRAILEY SIMS
Affiliation:
CARMA, University of Newcastle, Australia

Abstract

The Douglas–Rachford method is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. When the operators in question are normal cone operators, the iterated process may be used to solve feasibility problems of the following form: Find $x\in \bigcap _{k=1}^{N}S_{k}$. The success of the method in the context of closed, convex, nonempty sets $S_{1},\ldots ,S_{N}$ is well known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less well understood, yet it is surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg and Thibault’s success in applying the method to 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.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by G. Willis

This work is dedicated to the memory of Jonathan M. Borwein our greatly missed friend, mentor and colleague. His influence on both the topic at hand, as well as his impact on the present authors personally, cannot be overstated.

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