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] For a locally Lipschitz continuous function
the Clarke generalized directional derivative of
in the direction
is defined as
lim sup
lim sup
denotes the limit supremum.
, the Clarke generalized gradient of
(also called the Clarke subdifferential) is given as
represents an inner product of vectors in
Note that the Clarke generalized gradient is set-valued—that is, at each
More generally, given a Banach space
and a subset
the Clarke generalized directional derivative and generalized gradients are defined as above for a locally Lipschitz continuous function