Two Classes Of Generalized Convex Functions And Their Applications To Optimization

Convexity and generalized convexity play a central role in mathema -tical economics,engineering, management science, and optimization theo -ry.Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming.In this paper, we mainly make research about two classes of genera- lized convex functions. First, by using semi-preinvexity functions and (p,r)-preinvexity functions, a class of new generalized convex functions called semi (p,r)- preinvex functions is defined. It is showed with the aid of some examples that it is real generalization of the semi-preinvexity functions and (p,r)-preinvexity functions, thus it is the generalization of well known convexity functions and invexity functions. Therefore, the introduction of semi (p,r)-preinvex functions has many theory signifi cance. In this paper, we study this class of generalized convex functions from the following aspects: (1)obtain some basic properties of semi (p,r)-preinvex functions; (2) consider the multiobjective programming problems, in which the objective and the constraint functions are nondifferentiable and differentiable respectively and establish some optimality conditions; (3)consider the multiobjective fractional program- ming problems and establish some saddle optimality conditions; (4) estab -lish Wolfe-type duality for multiobjective fractional programming prob- lems and obtain some weak duality,strong duality and strict converse dua- lity theorems. Second, locally invex set is defined and based on this, a class of new generalized convex functions called semilocallyλ-subinvex function is defined. It is showed with the aid of some examples that it is real generalization of the semilocallyλ-sub-convex function, thus it is the generalization of well known convexity function,λ-subconvexity function and locally convexity functions. (1)some important properties of the type function are derived; (2) some applications to extreme problem are studied.