Tow Kinds Of Generalized Convex Functions And Properties

In this thesis, two kinds of generalized convex functions and some properties arediscussed. In chapter 1, semi- E -convex functions and E -quasiconvex functionsdefined on an E -convex set and their properties are discussed. On paper [15], we giveda new proof of Th12 from the aspect of variational inequality. Wang.J.Y extendE -convex functions to E -quasiconvex functions and discussed some importantproperties of E -quasiconvex functions, refer to several incorrect results of this paper,we gived several counterexamples and modified the incorrect results by compressmapping and extend mapping. We also established the equivalent relation betweenE -quasiconvex functions and corresponding E -level sets. Afterwards, we also definedE -pseudoconvex functions and (E,F) -quasiconvex functions on an E-convex set.Discussed the relation of E -pseudoconvex functions and E -convex functions, E -quasiconvex functions, as well as discussed some properties and criteria of (E,.F)-quasiconvex functions.The chapter 2 main involves two parts: invex functions and preinvex functions. Firstly, on the condition of quadratic differential, generalized invex Hessian matrix are defined. We get some criteria of optimal solution of invex functions and strictly invex functions. Second, based on the definition of G -invex sets, strictly G -invex sets and relative strictly G -invex sets are defined. As well as the relation of preinvex functions, strictly preinvex functions and explicitly invex functions and epigraphs, corresponding G -invex sets, and extreme point are discussed. Meanwhile we get some necessary conditions of preinvex functions, strictly preinvex functions and explicitly invex functions.