Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2019
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909115430731776 |
|---|---|
| author | Bergmann, Ronny Herzog, Roland Louzeiro, Maurício Silva Tenbrinck, Daniel Vidal-Núñez, José |
| author_facet | Bergmann, Ronny Herzog, Roland Louzeiro, Maurício Silva Tenbrinck, Daniel Vidal-Núñez, José |
| contents | This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_02022 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds Bergmann, Ronny Herzog, Roland Louzeiro, Maurício Silva Tenbrinck, Daniel Vidal-Núñez, José Numerical Analysis Optimization and Control This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature. |
| title | Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/1908.02022 |