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Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

Author

Listed:
  • Francisco J. Aragón Artacho

    (University of Alicante)

  • Jonathan M. Borwein

    (University of Newcastle)

  • Matthew K. Tam

    (University of Newcastle)

Abstract

In recent times the Douglas–Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:65:y:2016:i:2:d:10.1007_s10898-015-0380-6
    DOI: 10.1007/s10898-015-0380-6
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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.
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    Citations

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    Cited by:

    1. 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.
    2. Veit Elser, 2017. "Matrix product constraints by projection methods," Journal of Global Optimization, Springer, vol. 68(2), pages 329-355, June.
    3. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    4. 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.
    5. 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.
    6. 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.
    7. Francisco J. Aragón Artacho & Rubén Campoy & Ilias Kotsireas & Matthew K. Tam, 2018. "A feasibility approach for constructing combinatorial designs of circulant type," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1061-1085, May.
    8. Chih-Sheng Chuang & Hongjin He & Zhiyuan Zhang, 2022. "A unified Douglas–Rachford algorithm for generalized DC programming," Journal of Global Optimization, Springer, vol. 82(2), pages 331-349, February.
    9. 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.
    10. Min Li & Zhongming Wu, 2019. "Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 535-565, November.
    11. 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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    6. 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.
    7. 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.
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