Generalized Convexity And Generalized Monotonicity

Convexity and generalized convexity play an important role for optimization problems,equilibrium problems and variational inequality problems.Because the local extremum of convex function on convex set si global extremum.But the limitations of convex function are also obvious.Because a number of functions abotu practical problems are non-convex functions.For the need of solving practical problems,People has generalized the convexity from different point of view. Therefore,studing the generalized forms of convexity and application in optimization theroy is very important and interesting.In the paper,we mainly study the following things:Chapter 1 acts as the general introduction to the significance and current situation in the study of the generalized convex functions and generalized monotonicities.Chapter 2,because X.M.Yang etc.derivate the concepts of generalized invariant nonotonicity in referce[8],then some relations of generalized invariant monotonicities and generalized invariant convexities.In the chapter we will show that some examples in[8]are incorrect by means of eight counterexamles.And nine available examples are presented,the errors of Remark 2.5 in[8]will be corrected.In chapter 3,because M.A.Noor etc.put forward concepts ofα-invex,α-preinvex,αη-monotone in reference[9].In the chapter,by means of a series of counterexamples,we study in a systematic way the relationships among (pseudo,quasi)α-preinvexity,(strict,strong,pseudo,equasi)α-invexity and (strict,strong,pseudo,equasi)αη-monotonicity.Chapter 4,the definition of strictly semi-B-preinvexity is firstly introduced on the base of reference[6].Then,some properties of strictly semi-B-preinvex function are discussed.Our results are an generalization of those presented in reference[5,14].