A New Definition for Generalized First Derivative of Nonsmooth Functions

Abstract

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

Keywords

Generalized Derivative, Smooth and Nonsmooth Functions, Fourier analysis, Linear Programming, Functional Optimization

Generalized Derivative, Smooth and Nonsmooth Functions, Fourier analysis, Linear Programming, Functional Optimization

Cite this paper

nullA. Kamyad, M. Skandari and H. Erfanian, "A New Definition for Generalized First Derivative of Nonsmooth Functions,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1252-1257. doi: 10.4236/am.2011.210174.

nullA. Kamyad, M. Skandari and H. Erfanian, "A New Definition for Generalized First Derivative of Nonsmooth Functions,"

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