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.