A unified approach to a family of optimization problems in Banach spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929681833394176 |
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| author | Paul, Kallol Roy, Saikat Sain, Debmalya Sohel, Shamim |
| author_facet | Paul, Kallol Roy, Saikat Sain, Debmalya Sohel, Shamim |
| contents | Our principal aim is to illustrate that the concept Birkhoff-James orthogonality can be applied effectively to obtain a unified approach to a large family of optimization problems in Banach spaces. We study such optimization problems from the perspective of Birkhoff-James orthogonality in certain suitable Banach spaces. In particular, we demonstrate the duality between the Fermat-Torricelli problem and the Chebyshev center problem which are important particular cases of the least square problem. We revisit the Fermat-Torricelli problem for three and four points and solve it using the same technique. We also investigate the behavior of the Fermat-Torricelli points under the addition or replacement of a new point, and present several new results involving the locations of the Fermat-Torricelli point and the Chebyshev center. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unified approach to a family of optimization problems in Banach spaces Paul, Kallol Roy, Saikat Sain, Debmalya Sohel, Shamim Functional Analysis 46N10, 47N10, 51N20, 46B20 Our principal aim is to illustrate that the concept Birkhoff-James orthogonality can be applied effectively to obtain a unified approach to a large family of optimization problems in Banach spaces. We study such optimization problems from the perspective of Birkhoff-James orthogonality in certain suitable Banach spaces. In particular, we demonstrate the duality between the Fermat-Torricelli problem and the Chebyshev center problem which are important particular cases of the least square problem. We revisit the Fermat-Torricelli problem for three and four points and solve it using the same technique. We also investigate the behavior of the Fermat-Torricelli points under the addition or replacement of a new point, and present several new results involving the locations of the Fermat-Torricelli point and the Chebyshev center. |
| title | A unified approach to a family of optimization problems in Banach spaces |
| topic | Functional Analysis 46N10, 47N10, 51N20, 46B20 |
| url | https://arxiv.org/abs/2501.10718 |