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On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces

Author

Listed:
  • Heinz H. Bauschke

    (University of British Columbia)

  • Minh N. Dao

    (University of British Columbia
    Hanoi National University of Education)

  • Dominikus Noll

    (Université de Toulouse)

  • Hung M. Phan

    (University of Massachusetts Lowell)

Abstract

The Douglas–Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater’s condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas–Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection–projection.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:65:y:2016:i:2:d:10.1007_s10898-015-0373-5
    DOI: 10.1007/s10898-015-0373-5
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    References listed on IDEAS

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    1. H. H. Bauschke & S. G. Kruk, 2004. "Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 503-531, March.
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    Cited by:

    1. Minh N. Dao, & Hung M. Phan, 2019. "Linear Convergence of Projection Algorithms," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 715-738, May.
    2. 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.
    3. 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.
    4. 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.

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