Subdifferentials with respect to dualities

AbstractLetX andW be two sets andΔ: ¯RX → ¯RW 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 pointxo∈ 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].