| Generalized convexity is an important theoretical basis and a useful tool to study of mathematical programming, calculus of variations, optimization theory and other disciplines. However, a large number of functions in practical problems are non-convex functions. In recent years, For further discussion of non-smooth multi-objective programming issues, the concepts of convexity have various forms of promotion.Using the discussions to sub-differential and generalized gradient, differential convex functions had been have extended to the local Lipschitz function, in which invariant convex function is a very important promotional form.Based on some scholars'results and referred to the methods of their study about the Definitions and Natures, Using the upper right derivative and the Clarke generalized gradient, this paper gives a further promotion of the B-convex function. With the help of some definitions of the connected B-vex functions, it defines connected B-invariant vex, connected B-invariant pseudoconvex, and connected B-invariant quasiconvex. These connected B-invariant functions are applied to non-smooth multi-objective programming. At last the author gives some corresponding sufficient conditions and the duality of Mond-Weir type. These complements and extend previous results improve Multi-objective Programming Optimality conditions for the existence and the corresponding duality theory. Optimization algorithms which are the cornerstone of the algorithm provide a strong theoretical basis. |
