In mathematics, the Clarke generalized derivatives are generalized types of derivatives that allow for the differentiation of nonsmooth functions. The Clarke derivatives were introduced by Francis Clarke in 1975.
Definitions For a locally Lipschitz continuous function the Clarke generalized directional derivative of at in the direction is defined as
where denotes the limit supremum.
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 continuous function
See also Subgradient method — Class of optimization methods for nonsmooth functions. Subderivative Dini derivative
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