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
editFor 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
edit- Subgradient method — Class of optimization methods for nonsmooth functions.
- Subderivative
References
edit- ^ 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.
- Clarke, F. H. (January 1990). Optimization and Nonsmooth Analysis. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971309. ISBN 978-0-89871-256-8.
- Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, R. R. (1998). Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics. Vol. 178. Springer. doi:10.1007/b97650. ISBN 978-0-387-98336-3.