Fenchel duality and a separation theorem on hadamard manifolds

• Verfasst von: Silva Louzeiro, Maurício [VerfasserIn]
• Verfasst von: Bergmann, Ronny [VerfasserIn]
• Verfasst von: Herzog, Roland [VerfasserIn]
• Titel: Fenchel duality and a separation theorem on hadamard manifolds
• Verf.angabe: Maurício Silva Louzeiro, Ronny Bergmann, and Roland Herzog
• E-Jahr: 2022
• Jahr: May 10, 2022
• Umfang: 20 S.
• Fussnoten: Gesehen am 29.07.2022
• Titel Quelle: Enthalten in: Society for Industrial and Applied MathematicsSIAM journal on optimization
• Ort Quelle: Philadelphia, Pa. : SIAM, 1991
• Jahr Quelle: 2022
• Band/Heft Quelle: 32(2022), 2, Seite 854-873
• ISSN Quelle: 1095-7189
• DOI: doi:10.1137/21M1400699
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 26B25
• Sach-SW: 49N15
• Sach-SW: 49Q99
• Sach-SW: 90C25
• Sach-SW: convex analysis
• Sach-SW: Fenchel conjugate function
• Sach-SW: Hadamard manifold
• Sach-SW: Riemannian manifold
• K10plus-PPN: 1811957714

Abstract

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.