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A convergent relaxation of the Douglas–Rachford algorithm

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  • Nguyen Hieu Thao

    (Delft University of Technology
    Can Tho University)

Abstract

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.

Suggested Citation

  • Nguyen Hieu Thao, 2018. "A convergent relaxation of the Douglas–Rachford algorithm," Computational Optimization and Applications, Springer, vol. 70(3), pages 841-863, July.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9989-y
    DOI: 10.1007/s10589-018-9989-y
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    References listed on IDEAS

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    1. Alexander Y. Kruger & Nguyen H. Thao, 2015. "Quantitative Characterizations of Regularity Properties of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 41-67, January.
    2. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    3. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
    4. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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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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