Clarke generalized derivative explained

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

f:Rⁿ → R,

the Clarke generalized directional derivative of

x\inRⁿ

in the direction

v\inRⁿ

is defined as

f^ (x, v)= \limsup_ \frac,

where

\limsup

denotes the limit supremum.

Then, using the above definition of

f^\circ

, the Clarke generalized gradient of

(also called the Clarke subdifferential) is given as

\langle ⋅ , ⋅ \rangle

represents an inner product of vectors in

Note that the Clarke generalized gradient is set-valued—that is, at each

x\inR^n,

the function value

\partial^\circf(x)

is a set.

More generally, given a Banach space

and a subset

Y\subsetX,

the Clarke generalized directional derivative and generalized gradients are defined as above for a locally Lipschitz contininuous function

f:Y\toR.

References

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

Notes and References

Clarke. F. H. . Transactions of the American Mathematical Society . Generalized gradients and applications . 205 . 247 . 1975 . 0002-9947 . 10.1090/S0002-9947-1975-0367131-6. free .

Clarke generalized derivative explained

Definitions

See also

References

Notes and References