Generalized Conveity And Generalized Monotonicity With Applications

Convexity and generalized convexity play a central role in mathematical economics, engineering, management science, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming. In this paper, we mainly make fruther research about two classes of generalized convex functions. First, a class of functions called semi-B-preinvex functions, which is a generalization of the semipreinvex functions and B-vex function , is introduced. Then the introduction of semi-B-preinvex functions has many theory significance. In the paper, we study this class of generalized convex functions from the following aspects: (l)give examples to show that there exist functions which are semi-B-preinvex functions but are neither semipreinvex nor B-vex; (2)obtain some basic properties of semi-B-preinvex functions; (3)some results for the extremum problem which the objective function is semi-B-preinvex functions are presented. Univex functions is the second class generalized functions we considered in this paper, which was presented by Bector, Duneja and Gupta [22] and which is a generalization of invex functions and v-invex functions. Optimization and duality results are also obtained for a nonlinear multiobjective programming problem in [22]. In this paper, we are considered the multiobjective fractional programming problem. Some duality theorems for multiobjective fractional programming problems with univex functions are obtained. Furthermor, we consider their nondiferentiable situation, we define nonsmooth univex functions for Lipschitz functions by using Clarke generalized directional derivative and study nonsmooth multiobjective fractional programming with the new convexity. We establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality condition and prove weak, strong and strict converse duality theorems for nonsmooth multiobjective fractional programming problems containing univex functions.On the other hand, a concept closely related to the conveity is the monotonicity. It is well known that the convexity of a real-valued function is equivalent to the monotonicity of the corresponding gradient function. It is worth noting that monotonicity played a very important role in studying the existence and the sensitivity analysis of solutions for variational inequality, variational inclusions and complementarity problems. Theorefore, the last section of this paper we study the sensitivity of the solution of completely generalized strongly nonlinear implicit quasi-variational...