The Generalized Gradient Of Lipschitz Function And Its Applications

In the 1970s, a variety of generalized derivatives concepts appeared one after another. The famous one is generalized directional derivative and the generalized subgradient of Clarke's local Lipschitz function. But this concept has many limitations. In recent years, many scholars have made a considerable amount of research of it and have gotten a lot of meaningful results. Moreover, the applications of them in the optimization was discussed.Since Arivel has described the generalized convex of the function by means of Ben-Tal algebraic operations, and introduced the concepts of (h,φ)-convex function, the function and its related properties are studied. Studying on the optimization problem by the Ben-Tal generalized algebraic operations has been very attractive recently.This paper introduces the weak Lipschitz function and properties of its generalized gradient, and gives the examples to illustrate the applications of it, discusses value theorem, the necessary extreme conditions. On the basis of this, introduces regular weak,D regular weak Lipschitz function and properties of their generalized gradient, and gives a conclusion for a class of non-smooth planning optimization conditions. At last, with the help of Ben-Tal's generalized algebraic operations, a new kind of functions, termed (h,φ)-Lipschitz functions, is introduced. And discusses generalized directional derivative of (h,φ)-positive homogeneous,(h,φ)-sub-additive,lower semicontinuous of (h,φ)-Lipschitz functions. Discusses generalized gradient of non-empty,(h,φ)-convexity,closedness of bounded is also introduced. The relationship between generalized directional derivative and (h,φ)-Clarke contingent cone is presented.