Enlargements of the Moreau-Rockafellar Subdifferential

Malek Abbasi (1) , Alexander y Kruger (2) , Michel Théra (3, 2)

1 University of Isfahan
2 Federation University Australia Ballarat
3 XLIM - XLIM

The Moreau-Rockafellar subdifferential is a highly important notion in convex analysis and optimization theory. But there are many functions which fail to be subdifferentiable at certain points. In particular, there is a continuous convex function defined on 2 (N), whose Moreau-Rockafellar subdifferential is empty at every point of its domain. This paper proposes some enlargements of the Moreau-Rockafellar subdifferential: the sup-subdifferential, sup-subdifferential and symmetric subdifferential, all of them being nonempty for the mentioned function. These enlargements satisfy the most fundamental properties of the Moreau-Rockafellar sub-differential: convexity, weak *-closedness, weak *-compactness and, under some additional assumptions, possess certain calculus rules. The sup and sup subdifferentials coincide with the Moreau-Rockafellar subdifferential at every point at which the function attains its minimum, and if the function is upper semi-continuous, then there are some relationships for the other points. They can be used to detect minima and maxima of arbitrary functions.

Mots clés: Moreau-Rockafellar subdifferential; Convex function; Normal cone; Directional derivative; Optimality conditions Mathematics Subject Classification (2000) 49J52; 49J53; 90C30

Domaines: Optimisation et contrôle [math.OC]

Malek Abbasi, Alexander y Kruger, Michel Théra. Enlargements of the Moreau-Rockafellar Subdifferential. 2021. ⟨hal-02961001v3⟩