Subdifferentials with respect to dualities

Let X and W be two sets and Δ: -RX → -RW a duality (i.e., a mapping {Mathematical expression} such that {Mathematical expression} for all {Mathematical expression} and all index sets I). We introduce and study the subdifferential {Mathematical expression} of a function {Mathematical expression} at a point xo∈ X, with respect to Δ. We also consider the particular cases when Δ is a (Fenchel-Moreau) conjugation, or a ∨ -duality, or a ⊥-duality, in the sense of [8]. © 1995 Physica-Verlag.