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Pré-Publication, Document De Travail Année : 2020

Enlargements of the Moreau-Rockafellar Subdifferential

Résumé

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.
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Dates et versions

hal-02961001 , version 1 (08-10-2020)
hal-02961001 , version 2 (24-01-2021)
hal-02961001 , version 3 (12-03-2021)

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  • HAL Id : hal-02961001 , version 1

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Malek Abbasi, Alexander y Kruger, Michel Théra. Enlargements of the Moreau-Rockafellar Subdifferential. 2020. ⟨hal-02961001v1⟩
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