In this article, a new approach is presented to survey the validity of the nonlinear and nonsmooth inequalities on a compact domain using optimization. Here, an optimization problem corresponding with the considered inequality is proposed and by solving of which, the validity of the inequality will be determined. The optimization problem, in smooth and nonsmooth forms, is solved by a linearization approach. The efficiency of presented approach is illustrated in some examples.

In this paper, we first propose a new generalized derivative for non-smooth functions and then we utilize this generalized derivative to convert a class of non-smooth optimal control problem to the corresponding smooth form. In the next step, we apply the discretization method to approximate the obtained smooth problem to the nonlinear programming problem. Finally, by solving the last problem, we obtain an approximate optimal solution for main problem.

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.