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A new projection method for finding the closest point in the intersection of convex sets

Author

Listed:
  • Francisco J. Aragón Artacho

    (University of Alicante)

  • Rubén Campoy

    (University of Alicante)

Abstract

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas–Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:1:d:10.1007_s10589-017-9942-5
    DOI: 10.1007/s10589-017-9942-5
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    References listed on IDEAS

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    1. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
    2. 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.
    3. 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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    Cited by:

    1. Francisco J. Aragón Artacho & Rubén Campoy, 2019. "Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 709-726, June.
    2. Dongying Wang & Xianfu Wang, 2019. "A parameterized Douglas–Rachford algorithm," Computational Optimization and Applications, Springer, vol. 73(3), pages 839-869, July.
    3. Bian, Fengmiao & Zhang, Xiaoqun, 2021. "A parameterized Douglas–Rachford splitting algorithm for nonconvex optimization," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    4. Francisco J. Aragón Artacho & Rubén Campoy & Matthew K. Tam, 2021. "Strengthened splitting methods for computing resolvents," Computational Optimization and Applications, Springer, vol. 80(2), pages 549-585, November.
    5. Rubén Campoy, 2022. "A product space reformulation with reduced dimension for splitting algorithms," Computational Optimization and Applications, Springer, vol. 83(1), pages 319-348, September.
    6. Yixuan Yang & Yuchao Tang & Chuanxi Zhu, 2019. "Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces," Mathematics, MDPI, vol. 7(2), pages 1-16, February.
    7. Aragón-Artacho, Francisco J. & Censor, Yair & Gibali, Aviv & Torregrosa-Belén, David, 2023. "The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    8. 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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