Clarke Generalized Derivative

In mathematics, the Clarke generalized derivatives are types generalized of derivatives that allow for the differentiation of nonsmooth functions. The Clarke derivatives were introduced by Francis Clarke in 1975.

Definitions

For a locally Lipschitz continuous function $f: \mathbb^ \rightarrow \mathbb,$ the ''Clarke generalized directional derivative'' of $f$ at $x \in \mathbb^n$ in the direction $v \in \mathbb^n$ is defined as
$f^ (x, v)= \limsup_ \frac,$
where $\limsup$ denotes the limit supremum.

Then, using the above definition of $f^$, the ''Clarke generalized gradient'' of $f$ at $x$ (also called the ''Clarke subdifferential'') is given as
inner product