On Generalized High Order Derivatives of Nonsmooth Functions

Samaneh Soradi Zeid, Ali Vahidian Kamyad

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

DOI: 10.4236/ajcm.2014.44028                

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

Conflicts of Interest

The authors declare no conflicts of interest.

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