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A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting

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  • Minh N. Dao

    (University of Newcastle)

  • Matthew K. Tam

    (Universität Göttingen)

Abstract

The Douglas–Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems; an observation which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas–Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:1:d:10.1007_s10898-018-0677-3
    DOI: 10.1007/s10898-018-0677-3
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. 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.
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    Cited by:

    1. 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.

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