Characterization of regularity via variational stability of alternating projections sequences
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914526790680576 |
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| author | Battistoni, Francesco Daniilidis, Aris De Bernardi, Carlo Alberto Miglierina, Enrico |
| author_facet | Battistoni, Francesco Daniilidis, Aris De Bernardi, Carlo Alberto Miglierina, Enrico |
| contents | The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07953 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of regularity via variational stability of alternating projections sequences Battistoni, Francesco Daniilidis, Aris De Bernardi, Carlo Alberto Miglierina, Enrico Optimization and Control 47J25 (Primary) 90C25, 90C48 (Secondary) The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded. |
| title | Characterization of regularity via variational stability of alternating projections sequences |
| topic | Optimization and Control 47J25 (Primary) 90C25, 90C48 (Secondary) |
| url | https://arxiv.org/abs/2511.07953 |