What are Capra-Convex Sets?
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918430282612736 |
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| author | Chancelier, Jean-Philippe de Lara, Michel Franc, Adrien Le Rakotomandimby, Seta |
| author_facet | Chancelier, Jean-Philippe de Lara, Michel Franc, Adrien Le Rakotomandimby, Seta |
| contents | This paper focuses on a specific form of abstract convexity known as Capra-convexity, where a constant along primal rays (Capra) coupling replaces the scalar product used in standard convex analysis to define generalized Fenchel conjugacies. A key motivating result is that the ${\ell}$0 pseudonorm - which counts the number of nonzero components in a vector - is equal to its Capra-biconjugate. This implies that ${\ell}$0 is a Capra-convex function, highlighting potential applications in statistics and machine learning, particularly for enforcing sparsity in models. Building on prior work characterizing the Capra-subdifferential of ${\ell}$0 and the role of source norms in defining the Capra-coupling, the paper provides a characterization of Capra-convex sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06392 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | What are Capra-Convex Sets? Chancelier, Jean-Philippe de Lara, Michel Franc, Adrien Le Rakotomandimby, Seta Optimization and Control This paper focuses on a specific form of abstract convexity known as Capra-convexity, where a constant along primal rays (Capra) coupling replaces the scalar product used in standard convex analysis to define generalized Fenchel conjugacies. A key motivating result is that the ${\ell}$0 pseudonorm - which counts the number of nonzero components in a vector - is equal to its Capra-biconjugate. This implies that ${\ell}$0 is a Capra-convex function, highlighting potential applications in statistics and machine learning, particularly for enforcing sparsity in models. Building on prior work characterizing the Capra-subdifferential of ${\ell}$0 and the role of source norms in defining the Capra-coupling, the paper provides a characterization of Capra-convex sets. |
| title | What are Capra-Convex Sets? |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2509.06392 |