, Volume 42, Issue 1, pp 109-125

Subdifferentials with respect to dualities

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Abstract

LetX 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 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].