The Block-wise Circumcentered-Reflection Method

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection sche...

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Bibliographic Details
Published inarXiv.org
Main Authors Behling, Roger, J -Yunier Bello-Cruz, Luiz-Rafael Santos
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 26.03.2021
Subjects
Algorithms
Hyperplanes
Mathematics - Optimization and Control
Reflection
Subspaces
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Summary:The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection scheme by the Douglas--Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of alternating projections. Linear convergence for this novel block-wise circumcenter framework is derived and illustrated numerically. Furthermore, we prove that the original circumcentered-reflection method essentially finds the best approximation solution in one single step if the given affine subspaces are hyperplanes.
ISSN:2331-8422
DOI:10.48550/arxiv.1902.10866