The Study On Preinvexity

Convexity and generalized convexity theory are important theoretical foundations and powerful tools in mathematical subject,such as mathematical programming,variational principle and optimization theory.Preinvexity is a generalization of convexity,and is an important generalized convexity.This dissertation studies some problems in preinvexity and their applicationsThe first chapter briefly introduces the research status of convex functions and preinvex functions.The second chapter uses a new method to study the properties of the midpoint preinvex function,and proves that the midpoint preinvex function is(0,1)?Q-preinvex.This result is applied to discuss the minimization problems and the multi objective programming problems.The third chapter studies the Jensen inequality of the preinvex function and the upper bound of its error.Two results have been obtained.First,by introducing a generalized convex linear combination suitable for invariant convex sets and preinvex functions,the Jensen inequality for preinvex functions is obtained;second,the error of Jensen's inequality generated by two points is discussed,and an upper bound of the error is obtained.In chapter four,we present two conjectures.