The Douglas-Rachford algorithm for a hyperplane and a doubleton

Title: The Douglas-Rachford algorithm for a hyperplane and a doubleton
Creator: Bauschke, Heinz H.; Dao, Minh N.; Lindstrom, Scott B.
Relation: ARC.DP160101537 http://purl.org/au-research/grants/arc/DP160101537
Relation: Journal of Global Optimization Vol. 74, Issue 1, p. 79-93
Publisher Link: http://dx.doi.org/10.1007/s10898-019-00744-7
Publisher: Springer
Resource Type: journal article
Date: 2019
Description: The Douglas-Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which-somewhat surprisingly-depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.
Subject: closed-form expressions; cycling; Douglas-Rachford algorithm; feasibility problem; finite set; hyperplane
Identifier: http://hdl.handle.net/1959.13/1467042
Identifier: uon:47723
Identifier: ISSN:0925-5001
Language: eng
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