On Generalized High Order Derivatives of Nonsmooth Functions
On Generalized High Order Derivatives of Nonsmooth Functions作者机构：Department of Mathematics Ferdowsi University of Mashhad Mashhad Iran
出 版 物：《American Journal of Computational Mathematics》 (美国计算数学期刊（英文）)
年 卷 期：2014年第4卷第4期
页 面：317-328页
学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学]
主 题：Generalized Derivative Smooth and Nonsmooth Functions Nonsmooth Optimization Problem Linear Programming
摘 要：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.
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