The Douglas–Rachford algorithm for a hyperplane and a doubleton

Heinz H. Bauschke (), Minh N. Dao () and Scott B. Lindstrom ()
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Heinz H. Bauschke: University of British Columbia
Minh N. Dao: University of Newcastle
Scott B. Lindstrom: University of Newcastle

Journal of Global Optimization, 2019, vol. 74, issue 1, No 5, 79-93

Abstract: Abstract 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.

Keywords: Closed-form expressions; Cycling; Douglas–Rachford algorithm; Feasibility problem; Finite set; Hyperplane; Method of alternating projections; Projector; Reflector; Primary 47H10; 49M27; Secondary 65K05; 65K10; 90C26 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-019-00744-7

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