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 the Clarke generalized directional derivative of at in the direction is defined as
Then, using the above definition of , the Clarke generalized gradient of at (also called the Clarke
subdifferential) is given as
where represents an
inner product of vectors in
Note that the Clarke generalized gradient is set-valued—that is, at each the function value is a set.
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 contininuous function
See also
Subgradient method — Class of optimization methods for nonsmooth functions.