A New Generalized Gradient And Applications In Optimization Problems

This paper, consisting of five sections, discusses a new generalized gradient and its applications in optimization problems.In Section 1, we first present our argument that the purpose of optimization is to search for the maximum value of a function. The major researches and recent development of the study of non-smooth analysis are reviewed. Also included in this section is our discussion of the theoretical importance and wide practical prospects in studying differential properties of a Lipschitz function.In Section 2, we introduce some definitions and notations. Let / be a function from Rn to R. Following the definitions of the generalized gradients proposed by Clarke and Xu Yihong, respectively, we define the D-regular weak Lipschitz function and propose a new generalized gradient as followswhere D_f(x; d) is the directional derivative of / in the direction d at the point x, namelySome properties are proposed. Let f(x) and f(x) stand for the generalized gradients proposed by Clarke and Xu Yihong respectively. Three of generalized gradients are compared and the inclusion relation between them isSo the set of the generalized gradient defined in this paper is smaller; they coincide when / is a convex function or f is differentiable. Examples are provide to demonstrate that it is easier to select the elements in thegeneralized gradient set defined in this paper. Finally, major objection to the generalized gradient defined in this paper is considered.Section 3 and Section 4 are the main parts of the paper. By employing the directional derivative and generalized gradient in the broad sense, as defined in this paper, the first order necessary condition and the first order sufficient condition of the single-objective non-smooth programming where the objective function is D-regular weak Lipschitz function and constrained functions are regular weak Lipschitz functions. As the directional derivative defined in this paper doesn't show convexity, when its properties are considered, it is necessary to search for new ways to reach the conclusion.In Section 5, Mond-Weir type dual and Wolfe type dual are introduced and the mixed type dual proposed by Xu Zengkun is discussed. Under the different conditions of the constrained functions, dividing constrained sets properly, and under generalized (F, ρ) convexity, the theorems of the weak duality, strong duality and strictly reverse duality are testified.