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.

Posted on Jul 13, 2020 in Optimization and Equilibrium, Seminars