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The circumcentered-reflection method achieves better rates than alternating projections

Author

Listed:
  • Reza Arefidamghani

    (Instituto de Matemática Pura e Aplicada)

  • Roger Behling

    (Fundação Getúlio Vargas)

  • Yunier Bello-Cruz

    (Northern Illinois University)

  • Alfredo N. Iusem

    (Instituto de Matemática Pura e Aplicada)

  • Luiz-Rafael Santos

    (Federal University of Santa Catarina)

Abstract

We study the convergence rate of the Circumcentered-Reflection Method (CRM) for solving the convex feasibility problem and compare it with the Method of Alternating Projections (MAP). Under an error bound assumption, we prove that both methods converge linearly, with asymptotic constants depending on a parameter of the error bound, and that the one derived for CRM is strictly better than the one for MAP. Next, we analyze two classes of fairly generic examples. In the first one, the angle between the convex sets approaches zero near the intersection, so that the MAP sequence converges sublinearly, but CRM still enjoys linear convergence. In the second class of examples, the angle between the sets does not vanish and MAP exhibits its standard behavior, i.e., it converges linearly, yet, perhaps surprisingly, CRM attains superlinear convergence.

Suggested Citation

  • Reza Arefidamghani & Roger Behling & Yunier Bello-Cruz & Alfredo N. Iusem & Luiz-Rafael Santos, 2021. "The circumcentered-reflection method achieves better rates than alternating projections," Computational Optimization and Applications, Springer, vol. 79(2), pages 507-530, June.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:2:d:10.1007_s10589-021-00275-6
    DOI: 10.1007/s10589-021-00275-6
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    References listed on IDEAS

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    1. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
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    Cited by:

    1. Hui Ouyang & Xianfu Wang, 2021. "Bregman Circumcenters: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 252-280, October.

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