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.{{cite journal | last=Clarke|first=F. H. | journal=Transactions of the American Mathematical Society | title=Generalized gradients and applications | volume=205 | pages=247 | date= 1975 | issn=0002-9947 | doi=10.1090/S0002-9947-1975-0367131-6| doi-access=free }}

Definitions

For a locally Lipschitz continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R},$ the Clarke generalized directional derivative of $f$ at $x \in \mathbb{R}^n$ in the direction $v \in \mathbb{R}^n$ is defined as

$f^{\circ} (x, v)= \limsup_{y \rightarrow x, h \downarrow 0} \frac{f(y+ hv)-f(y)}{h},$

where $\limsup$ denotes the limit supremum.

Then, using the above definition of $f^{\circ}$ , the Clarke generalized gradient of $f$ at $x$ (also called the Clarke subdifferential) is given as