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The circumcentered-reflection method achieves better rates than alternating projections

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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.

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Acknowledgements

We thank the anonymous referees for their valuable suggestions which significantly improved this manuscript.

Funding

RB was partially supported by the Brazilian Agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grants 304392/2018-9 and 429915/2018-7; YBC was partially supported by the National Science Foundation (NSF), Grant DMS—1816449.

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Authors and Affiliations

  1. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, 22460-320, Brazil

    Reza Arefidamghani & Alfredo N. Iusem

  2. School of Applied Mathematics, Fundação Getúlio Vargas, Rio de Janeiro, RJ, 22250-900, Brazil

    Roger Behling

  3. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115-2828, USA

    Yunier Bello-Cruz

  4. Department of Mathematics, Federal University of Santa Catarina, Blumenau, SC, 88040-900, Brazil

    Luiz-Rafael Santos

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  1. Reza Arefidamghani
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  2. Roger Behling
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  3. Yunier Bello-Cruz
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  4. Alfredo N. Iusem
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  5. Luiz-Rafael Santos
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Correspondence to Luiz-Rafael Santos.

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Arefidamghani, R., Behling, R., Bello-Cruz, Y. et al. The circumcentered-reflection method achieves better rates than alternating projections. Comput Optim Appl 79, 507–530 (2021). https://doi.org/10.1007/s10589-021-00275-6

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