Enlargements of the Moreau–Rockafellar Subdifferential

This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup ⋆ - and symmetric subdifferentials. They satisfy the most fundamental properties of the conventional subdifferential: convexity, weak ∗ -closedness and, if the function is bounded, weak ∗ -compactness. Moreover, they are nonempty for the Rainwater function, possess certain calculus rules, and provide necessary and sufficient optimality conditions for not necessarily convex functions.