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The Douglas–Rachford algorithm for the case of the sphere and the line

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  • Joël Benoist

Abstract

In this paper, we solve a conjecture proposed by Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, Springer optimization and its applications, 2011 ) in a Hilbert space setting. For the simple non-convex example of the sphere and the line, the sequence of Douglas–Rachford iterates converges in norm to a point of the intersection except when the initial value belongs to the hyperplane of symmetry. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:2:p:363-380
    DOI: 10.1007/s10898-015-0296-1
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    Citations

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    Cited by:

    1. Hoa T. Bui & Scott B. Lindstrom & Vera Roshchina, 2019. "Variational Analysis Down Under Open Problem Session," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 430-437, July.
    2. Ohad Giladi & Björn S. Rüffer, 2019. "A Lyapunov Function Construction for a Non-convex Douglas–Rachford Iteration," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 729-750, March.
    3. Minh N. Dao & Matthew K. Tam, 2019. "A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting," Journal of Global Optimization, Springer, vol. 73(1), pages 83-112, January.
    4. Francisco J. Aragón Artacho & Rubén Campoy, 2018. "A new projection method for finding the closest point in the intersection of convex sets," Computational Optimization and Applications, Springer, vol. 69(1), pages 99-132, January.
    5. 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.
    6. Scott B. Lindstrom, 2022. "Computable centering methods for spiraling algorithms and their duals, with motivations from the theory of Lyapunov functions," Computational Optimization and Applications, Springer, vol. 83(3), pages 999-1026, December.
    7. Francisco J. Aragón Artacho & Rubén Campoy & Veit Elser, 2020. "An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm," Journal of Global Optimization, Springer, vol. 77(2), pages 383-403, June.
    8. 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.
    9. 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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