Leírás
Online Magyar Operációkutatási Szeminárium
Abstract: The question of defining a derivative-like object for nonsmooth functions
was initiated in the late fifties by Rockafellar in the seventies. Since then, derivative-like
objects for nonsmooth, in particular, for locally Lipschitz functions
$f:D\subseteq X\to Y$, where $X$ and $Y$ are normed spaces and $D$ is an open
set, have been the focus of intensive research.
When $X$ and $Y$ are finite dimensional normed spaces, Clarke introduced
in 1983 the notion of the {\it generalized Jacobian} of $f$ at $p\in D$ as
\[
\partial^c f(p)
:=\mbox{co}\,\big\{A\in L(X,Y)\mid\exists\,(x_i)_{i\in\mathbb{N}}
\mbox{ in }D :
\lim_{i\to\infty} x_i= p,\,\mbox{ and }
\lim_{i\to\infty} f'(x_i)= A\big\}.
\]
The nonemptiness of this set is a consequence of Rademacher's theorem on
the almost everywhere differentiability of locally Lipschitz functions
acting between finite dimensional spaces.
One could expect that the derivatives defined by way of operators have some
additional good properties. Such good properties should definitely be
translated as having ``tight'' calculus rules and computational utility in
the applications. Motivated by this problem, in our recent paper,
we have provided an extension of Clarke's generalized Jacobian to locally
Lipschitz functions from any normed space into a finite dimensional space.
Our generalized Jacobian, denoted by $\partial f(p)$, is a set of linear
operators from $X$ to $Y$. When $X$ is finite dimensional it coincides with
the Clarke's generalized Jacobian. On the other hand, when the domain is
infinite dimensional and the image space is the real line, $\partial f(p)$
coincides with Clarke's generalized gradient.
In some recent papers the nonemptiness, the $w^*$-compactness, the convexity,
and the upper semicontinuity property of this generalized Jacobian were
obtained. A chain rule for the composition of a smooth map between finite
dimensional spaces with a locally Lipschitz functions and, as consequences,
the sum and the product rules were proved. A computational rule for the
generalized Jacobian of piecewise smooth functions was also developed. In
our subsequent work, we obtained a complete characterization of
the set-valued map $\partial f(\cdot)$ and we proved that generalized
Jacobian is a strict Hadamard prederivative. Chain rules for the composition
of two nonsmooth functions were also established in this paper.
In the talk we review all these results and discuss possible generalizations
for functions with infinite dimensional range.
For Zoom access please contact E.-Nagy Marianna (marianna.eisenberg-nagy[at]uni-corvinus.hu).