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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.17258 |
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| _version_ | 1866918031856238592 |
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| author | Barros, Pablo Behling, Roger Guigues, Vincent Santos, Luiz-Rafael |
| author_facet | Barros, Pablo Behling, Roger Guigues, Vincent Santos, Luiz-Rafael |
| contents | This paper introduces the Parallelized Circumcentered Reflection Method (P-CRM), a circumcentric approach that parallelizes the Circumcentered Reflection Method (CRM) for solving Convex Feasibility Problems in affine settings. Beyond feasibility, P-CRM solves the best approximation problem for any finite collection of affine subspaces; that is, it not only finds a feasible point but directly computes the projection of an initial point onto the intersection. Within a fully self-contained scheme, we also introduce the Framework for the Simultaneous Projection Method (F-SPM) which includes Cimmino's method as a special case. Theoretical results show that both P-CRM and F-SPM achieve linear convergence. Moreover, P-CRM converges at a rate that is at least as fast as, and potentially superior to, the best convergence rate of F-SPM. As a byproduct, this also yields a new and simplified convergence proof for Cimmino's method. Numerical experiments show that P-CRM is competitive compared to CRM and indicate that it offers a scalable and flexible alternative, particularly suited for large-scale problems and modern computing environments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parallelizing the Circumcentered-Reflection Method Barros, Pablo Behling, Roger Guigues, Vincent Santos, Luiz-Rafael Optimization and Control 90C25, 90C30, 90C60 This paper introduces the Parallelized Circumcentered Reflection Method (P-CRM), a circumcentric approach that parallelizes the Circumcentered Reflection Method (CRM) for solving Convex Feasibility Problems in affine settings. Beyond feasibility, P-CRM solves the best approximation problem for any finite collection of affine subspaces; that is, it not only finds a feasible point but directly computes the projection of an initial point onto the intersection. Within a fully self-contained scheme, we also introduce the Framework for the Simultaneous Projection Method (F-SPM) which includes Cimmino's method as a special case. Theoretical results show that both P-CRM and F-SPM achieve linear convergence. Moreover, P-CRM converges at a rate that is at least as fast as, and potentially superior to, the best convergence rate of F-SPM. As a byproduct, this also yields a new and simplified convergence proof for Cimmino's method. Numerical experiments show that P-CRM is competitive compared to CRM and indicate that it offers a scalable and flexible alternative, particularly suited for large-scale problems and modern computing environments. |
| title | Parallelizing the Circumcentered-Reflection Method |
| topic | Optimization and Control 90C25, 90C30, 90C60 |
| url | https://arxiv.org/abs/2505.17258 |