Nonlinear Fenchel Conjugates
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909310568628224 |
|---|---|
| author | Schiela, Anton Herzog, Roland Bergmann, Ronny |
| author_facet | Schiela, Anton Herzog, Roland Bergmann, Ronny |
| contents | The classical concept of Fenchel conjugation is tailored to extended real-valued functions defined on linear spaces. In this paper we generalize this concept to functions defined on arbitrary sets that do not necessarily bear any structure at all. This generalization is obtained by replacing linear test functions by general nonlinear ones. Thus, we refer to it as nonlinear Fenchel conjugation. We investigate elementary properties including the Fenchel-Moreau biconjugation theorem. Whenever the domain exhibits additional structure, the restriction to a suitable subset of test functions allows further results to be derived. For example, on smooth manifolds, the restriction to smooth test functions allows us to state the Fenchel-Young theorem for the viscosity Fréchet subdifferential. On Lie groups, the restriction to real-valued group homomorphisms relates nonlinear Fenchel conjugation to infimal convolution and yields a notion of convexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear Fenchel Conjugates Schiela, Anton Herzog, Roland Bergmann, Ronny Functional Analysis Differential Geometry Optimization and Control The classical concept of Fenchel conjugation is tailored to extended real-valued functions defined on linear spaces. In this paper we generalize this concept to functions defined on arbitrary sets that do not necessarily bear any structure at all. This generalization is obtained by replacing linear test functions by general nonlinear ones. Thus, we refer to it as nonlinear Fenchel conjugation. We investigate elementary properties including the Fenchel-Moreau biconjugation theorem. Whenever the domain exhibits additional structure, the restriction to a suitable subset of test functions allows further results to be derived. For example, on smooth manifolds, the restriction to smooth test functions allows us to state the Fenchel-Young theorem for the viscosity Fréchet subdifferential. On Lie groups, the restriction to real-valued group homomorphisms relates nonlinear Fenchel conjugation to infimal convolution and yields a notion of convexity. |
| title | Nonlinear Fenchel Conjugates |
| topic | Functional Analysis Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2409.04492 |