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The Douglas–Rachford algorithm for convex and nonconvex feasibility problems

Author

Listed:
  • Francisco J. Aragón Artacho

    (University of Alicante)

  • Rubén Campoy

    (University of Alicante)

  • Matthew K. Tam

    (University of Göttingen)

Abstract

The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “(m, n)-queens problem”, and the problem of constructing a probability distribution with prescribed moments.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:91:y:2020:i:2:d:10.1007_s00186-019-00691-9
    DOI: 10.1007/s00186-019-00691-9
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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.
    2. 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.
    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.
    5. Jonathan M. Borwein & Matthew K. Tam, 2014. "A Cyclic Douglas–Rachford Iteration Scheme," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 1-29, January.
    6. Francisco Aragón Artacho & Jonathan Borwein, 2013. "Global convergence of a non-convex Douglas–Rachford iteration," Journal of Global Optimization, Springer, vol. 57(3), pages 753-769, November.
    7. Nguyen Hieu Thao, 2018. "A convergent relaxation of the Douglas–Rachford algorithm," Computational Optimization and Applications, Springer, vol. 70(3), pages 841-863, July.
    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.
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