On the circumcentered-reflection method for the convex feasibility problem

The ancient concept of circumcenter has recently given birth to the circumcentered-reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection-based schemes, namely, the D...

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Bibliographic Details
Published inNumerical algorithms Vol. 86; no. 4; pp. 1475 - 1494
Main Authors Behling, Roger, Bello-Cruz, Yunier, Santos, Luiz-Rafael
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2021
Springer Nature B.V
Subjects
Algebra
Algorithms
Computer Science
Convexity
Feasibility
Intersections
Iterative methods
Numeric Computing
Numerical Analysis
Original Paper
Subspaces
Theory of Computation
Accelerating convergence
65B99
Douglas-Rachford method
Convex feasibility problem
65K05
Reflection method
Method of alternating projections
90C25
49M27
Circumcenter
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Summary:The ancient concept of circumcenter has recently given birth to the circumcentered-reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection-based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-020-00941-6