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A convergent relaxation of the Douglas–Rachford algorithm
This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Converge...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer US
2018
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6445491/ https://www.ncbi.nlm.nih.gov/pubmed/31007390 http://dx.doi.org/10.1007/s10589-018-9989-y |
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author | Thao, Nguyen Hieu |
author_facet | Thao, Nguyen Hieu |
author_sort | Thao, Nguyen Hieu |
collection | PubMed |
description | This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems. |
format | Online Article Text |
id | pubmed-6445491 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2018 |
publisher | Springer US |
record_format | MEDLINE/PubMed |
spelling | pubmed-64454912019-04-17 A convergent relaxation of the Douglas–Rachford algorithm Thao, Nguyen Hieu Comput Optim Appl Article This paper proposes an algorithm for solving structured optimization problems, which covers both the backward–backward and the Douglas–Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the corresponding operator is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point iterations are established. When applied to nonconvex feasibility including potentially inconsistent problems, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets. In this special case, we refine known linear convergence criteria for the Douglas–Rachford (DR) algorithm. As a consequence, for feasibility problem with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems. Springer US 2018-03-06 2018 /pmc/articles/PMC6445491/ /pubmed/31007390 http://dx.doi.org/10.1007/s10589-018-9989-y Text en © The Author(s) 2018 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Article Thao, Nguyen Hieu A convergent relaxation of the Douglas–Rachford algorithm |
title | A convergent relaxation of the Douglas–Rachford algorithm |
title_full | A convergent relaxation of the Douglas–Rachford algorithm |
title_fullStr | A convergent relaxation of the Douglas–Rachford algorithm |
title_full_unstemmed | A convergent relaxation of the Douglas–Rachford algorithm |
title_short | A convergent relaxation of the Douglas–Rachford algorithm |
title_sort | convergent relaxation of the douglas–rachford algorithm |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6445491/ https://www.ncbi.nlm.nih.gov/pubmed/31007390 http://dx.doi.org/10.1007/s10589-018-9989-y |
work_keys_str_mv | AT thaonguyenhieu aconvergentrelaxationofthedouglasrachfordalgorithm AT thaonguyenhieu convergentrelaxationofthedouglasrachfordalgorithm |