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The Douglas–Rachford algorithm for a hyperplane and a doubleton

Author

Listed:
  • Heinz H. Bauschke

    (University of British Columbia)

  • Minh N. Dao

    (University of Newcastle)

  • Scott B. Lindstrom

    (University of Newcastle)

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.

Suggested Citation

  • Heinz H. Bauschke & Minh N. Dao & Scott B. Lindstrom, 2019. "The Douglas–Rachford algorithm for a hyperplane and a doubleton," Journal of Global Optimization, Springer, vol. 74(1), pages 79-93, May.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:1:d:10.1007_s10898-019-00744-7
    DOI: 10.1007/s10898-019-00744-7
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    References listed on IDEAS

    as
    1. Joël Benoist, 2015. "The Douglas–Rachford algorithm for the case of the sphere and the line," Journal of Global Optimization, Springer, vol. 63(2), pages 363-380, October.
    2. Minh N. Dao & Hung M. Phan, 2018. "Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems," Journal of Global Optimization, Springer, vol. 72(3), pages 443-474, November.
    3. Heinz H. Bauschke & Minh N. Dao & Dominikus Noll & Hung M. Phan, 2016. "On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces," Journal of Global Optimization, Springer, vol. 65(2), pages 329-349, June.
    4. Francisco J. Aragón Artacho & Jonathan M. Borwein & Matthew K. Tam, 2016. "Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem," Journal of Global Optimization, Springer, vol. 65(2), pages 309-327, June.
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    Cited by:

    1. Francisco J. Aragón Artacho & Rubén Campoy & Matthew K. Tam, 2020. "The Douglas–Rachford algorithm for convex and nonconvex feasibility problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(2), pages 201-240, April.

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