Several Characterizations Of E-Preinvex Functions

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 farther research about two classes of generalized convex functions.First,a class of functions called semistrictly-E-preinvex functions,which is a generalization of the semi-E-convex and E-preinvex,is introduced.Then the introduction of semistictly,E-preinvex functions has many theory significance.In the paper,we study this class of generalized convex functions from the following aspects:1) we give example to show that there exist functions which are semistrictly-E-preinvex functions but are neither semi-E-convex functions nor E-preinvex functions;2) we obtain some basic properties semistrictly-E-preinvex functions.On the other hand,in the paper,three classes of functions called E-semilocally prequasi-invex functions,semistrictly-E-semilocally prequasi-invex functions and strictly-E-semilocally prequasi-invex functions which are the generalization of semilocally prequasi-invex functions,s emistrictly semilocally prequasi-invex functions strictly and semilocally prequasi-invex functions are introduced.In the paper,we study these classes of generalized convex functions from the following aspects:1) definitions of E-semilocally prequasi-invex functions,semistrictly-E-semilocally prequasi-invex functions and strictly-E-semilocally prequasi-invex functions are introduced,we give example to show the strictly E-Semilocally prequasi-invex functions not necessarily imply the E-semilocally prequasi-invex functions,the semistric E-Semilocally prequasi-invex functions does not imply the strictly -E-semilocally prequasi-invex functions;2) we obtain some basic properties three classes of functions;3) some results for the mathematical programming problem which the objective functions is three classes of functions are presented.