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.^[1]

Definitions

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

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

where ${\displaystyle \limsup }$ denotes the limit supremum.

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

$${\displaystyle \partial ^{\circ }\!f(x):=\{\xi \in \mathbb {R} ^{n}:\langle \xi ,v\rangle \leq f^{\circ }(x,v),\forall v\in \mathbb {R} ^{n}\},}$$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ represents an inner product of vectors in ${\displaystyle \mathbb {R} .}$ Note that the Clarke generalized gradient is set-valued—that is, at each ${\displaystyle x\in \mathbb {R} ^{n},}$ the function value ${\displaystyle \partial ^{\circ }\!f(x)}$ is a set.

More generally, given a Banach space ${\displaystyle X}$ and a subset ${\displaystyle Y\subset X,}$ the Clarke generalized directional derivative and generalized gradients are defined as above for a locally Lipschitz contininuous function ${\displaystyle f:Y\to \mathbb {R} .}$

See also

• Subgradient method — Class of optimization methods for nonsmooth functions.
• Subderivative

References

• Clarke, F. H. (January 1990). Optimization and Nonsmooth Analysis. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971309. ISBN 978-0-89871-256-8.
• Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, R. R. (1998). Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics. Vol. 178. Springer. doi:10.1007/b97650. ISBN 978-0-387-98336-3.