The circumcentered-reflection method achieves better rates than alternating projections

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 o...

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Bibliographic Details
Published in:Computational optimization and applications 2021-06, Vol.79 (2), p.507-530
Main Authors: Arefidamghani, Reza, Behling, Roger, Bello-Cruz, Yunier, Iusem, Alfredo N., Santos, Luiz-Rafael
Format: Article
Language:English
Subjects:
Asymptotic methods
Convergence
Convex and Discrete Geometry
Convexity
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Statistics
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Summary: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.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-021-00275-6