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The Douglas–Rachford algorithm for a hyperplane and a doubleton

  • Heinz H. Bauschke
  • Minh N. DaoEmail author
  • Scott B. Lindstrom
Article
  • 23 Downloads

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 

Mathematics Subject Classification

Primary 47H10 49M27 Secondary 65K05 65K10 90C26 

Notes

Acknowledgements

The authors thank two anonymous referees for their careful reading and constructive comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2018-03703). MND was partially supported by the Australian Research Council (Grant No. DP160101537).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.CARMAUniversity of NewcastleCallaghanAustralia

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