On Generalized High Order Derivatives of Nonsmooth Functions - American Journal of Computational Mathematics

AJCM > Vol.4 No.4, September 2014

On Generalized High Order Derivatives of Nonsmooth Functions ()

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DOI: 10.4236/ajcm.2014.44028 3,045 Downloads 3,952 Views Citations

Author(s)

Samaneh Soradi Zeid, Ali Vahidian Kamyad

Affiliation(s)

Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.

ABSTRACT

In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.

KEYWORDS

Generalized Derivative, Smooth and Nonsmooth Functions, Nonsmooth Optimization Problem, Linear Programming

Share and Cite:

Zeid, S. and Kamyad, A. (2014) On Generalized High Order Derivatives of Nonsmooth Functions. American Journal of Computational Mathematics, 4, 317-328. doi: 10.4236/ajcm.2014.44028.

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