A New Approach for the Generalized First Derivative and Extension It to the Generalized Second Derivative of Nonsmooth Functions

Abstract

Abstract-In this paper, first derivative of smooth function is defined by the optimal solution of a special optimization problem. In the next step, by using this optimization problem for nonsmooth function, we obtain an approximation for first derivative of nonsmooth function which it is called generalized first derivative. We then extend it to define generalized second derivative for nonsmooth function. Finally, we show the efficiency of our approach by evaluating derivative and generalized first and second derivative of some smooth and nonsmooth functions, respectively.

Solving a Class of Non-Smooth Optimal Control Problems

“…Despite existence of these methods, solving of nonsmooth optimal control problems is difficult and often an impossible act. To solve these problems, the [4,5,6] for generalized derivative of non-smooth functions and in addition, we used it for non-smooth ordinary differential equations, non-smooth optimization problems and system of non-smooth equations (see [7,8]).…”

On the Validity of Nonlinear and Nonsmooth Inequalities

“…A similar approach is g iven [13] to solve the nonsmooth optimizat ion problems. Also, some other types of GDs are presented in [14,15].…”