The circumcentered-reflection method achieves better rates than alternating projections

• Published in: Computational optimization and applications Vol. 79; no. 2; pp. 507 - 530
• Main Authors: Arefidamghani, Reza, Behling, Roger, Bello-Cruz, Yunier, Iusem, Alfredo N., Santos, Luiz-Rafael
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Asymptotic methods; Convergence; Convex and Discrete Geometry; Convexity; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Statistics; Alternating projections; 65B99; Convex feasibility problem; 65K05; 90C25; Circumcentered-reflection method; Convergence rate; 49M27
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-021-00275-6

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.