The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911905712439296 |
|---|---|
| author | Censor, Yair Mansour, Rafiq Reem, Daniel |
| author_facet | Censor, Yair Mansour, Rafiq Reem, Daniel |
| contents | Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets Censor, Yair Mansour, Rafiq Reem, Daniel Optimization and Control Numerical Analysis Functional Analysis 41A50, 41A65, 90C25, 46N10, 47N10, 41A52, 47H09, 47H10, 46C05 G.1.6; G.1.2; I.3.5 Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra. |
| title | The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets |
| topic | Optimization and Control Numerical Analysis Functional Analysis 41A50, 41A65, 90C25, 46N10, 47N10, 41A52, 47H09, 47H10, 46C05 G.1.6; G.1.2; I.3.5 |
| url | https://arxiv.org/abs/2304.09600 |