Fenchel duality and a separation theorem on hadamard manifolds
In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the ot...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
May 10, 2022
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| In: |
SIAM journal on optimization
Year: 2022, Volume: 32, Issue: 2, Pages: 854-873 |
| ISSN: | 1095-7189 |
| DOI: | 10.1137/21M1400699 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1137/21M1400699 Verlag, lizenzpflichtig, Volltext: https://epubs.siam.org/doi/10.1137/21M1400699 
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| Author Notes: | Maurício Silva Louzeiro, Ronny Bergmann, and Roland Herzog |
| Summary: | In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the other hand, this new definition still possesses properties known to hold in the Euclidean case. It even yields a broader interpretation of the Fenchel conjugate in the Euclidean case itself. Most prominently, our definition of the Fenchel conjugate provides a Fenchel--Moreau theorem for geodesically convex, proper, lower semicontinuous functions. In addition, this framework allows us to develop a theory of separation of convex sets on Hadamard manifolds, and a strict separation theorem is obtained. |
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| Item Description: | Gesehen am 29.07.2022 |
| Physical Description: | Online Resource |
| ISSN: | 1095-7189 |
| DOI: | 10.1137/21M1400699 |