Subdifferentials with respect to dualities

Abstract

LetX andW be two sets andΔ: ¯R^X → ¯R^W a duality (i.e., a mapping$\Delta :f \in \bar R^X \to f^\Delta \in \bar R^W$ such that$\left( {\mathop {\inf f_i }\limits_{i \in I} } \right)^\Delta = \mathop {\sup }\limits_{i \in I} f_i^\Delta$ for all$\{ f_i \} _{i \in I} \subseteq \bar R^X$ and all index setsI). We introduce and study the subdifferential$\partial ^\Delta f(x_0 )$ of a function$f \in \bar R^X$ at a pointx _o ∈ X, with respect toΔ. We also consider the particular cases whenΔ is a (Fenchel-Moreau) conjugation, or a ∨ -duality, or a ⊥-duality, in the sense of [8].