# Enlargements of the Moreau-Rockafellar Subdifferential.

Abstract:

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 $\ell^2(\mathbb{N})$, whose Moreau–Rockafellar subdifferential is empty at every point of its domain.

This talk  proposes some enlargements of  the Moreau—Rockafellar subdifferential: the sup$^\star$-sub\-differential,  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 subdifferential: convexity, weak$^*$-closedness, weak$^*$-com\-pactness and, under some additional assumptions, possess certain calculus rules.

The sup$^\star$ 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.

Date: Jul 15, 2020 at 10:00:00 h