Duality and subdifferential for convex functions on complete metric spaces

✍ Scribed by Bijan Ahmadi Kakavandi; Massoud Amini

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 279 KB
• Volume: 73
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.07.033

✦ Synopsis

Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete CAT (0) (Hadamard) spaces.

For a Hadamard space X , its dual is a metric space X * which strictly separates non-empty, disjoint, convex closed subsets of X , provided that one of them is compact. If f : X → (-∞, +∞] is a proper, lower semicontinuous, convex function, then the subdifferential ∂f : X ⇒ X * is defined as a multivalued monotone operator such that, for any y ∈ X there exists some x ∈ X with -→ xy ∈ ∂f (x). When X is a Hilbert space, it is a classical fact that R(I + ∂f ) = X . Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential ∂ f (x) is non-empty, for any > 0 and any x in efficient domain of f . Our results generalize duality and subdifferential of convex functions in Hilbert spaces.