The α-Preinvex Function

Convexity and generalized convexity play a central role in mathematical economics, engineering, management science, and optimization theory. The research on convexity and generalized convexity is one of the most important aspects in mathematical programming. In this paper, we extend the classes of invex functoin toα—preinvex functions.In the first part, we introduce the relative-backgroud on this paper and some preparation knowledgethe of this paper.In the second part, we give examples to show that there exist functions which areα—preinvex functions and the relations ofα-preinvex functions, quasiα—preinvex functions, and pseudoα—preinvex functions. Some properties of theα-preinvex functions are obtained. Further more, we give the application of theα-preinvex functions in optimizational theory.In the thrid part, we study a new class of invex monotone functions,αη-monotone, quasiαη-monotone, and pseudoαη-monotone, and give examples to present their relations. The sufficient and necessary conditions betweenα—preinvex functions andαη—monotone, quasiα—preinvex functions and quasiαη—monotone are derived.