Clarke generalized derivative

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

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 contininuous function

See also

References

  1. Clarke, F. H. (1975). "Generalized gradients and applications". Transactions of the American Mathematical Society. 205: 247. doi:10.1090/S0002-9947-1975-0367131-6. ISSN 0002-9947.
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