Some Properties Of (F, K)-Convex Mappings

Optimization theory is the most important part of the optimization as well as an important theoretical basis in operational research. Convex set and convex mapping, as a basic property in optimization theory, is applied in many fields of mathematics. Accordingly, development of the convexity of sets and functions has practical significance. In 1999, Youness introduced the concept of E-convex set and E-convex function, then Yang Xinming and Chen Xiusu defined the semi-E-convex function and listed its properties, in addition, Hanson also defined preinvex function. In this thesis, based on the pre-concepts,the concepts and properties of the (F, K)-convex set, (F, K)-convex mapping, (F, K)-invariant convex set and (F, K)-preinvex set are introduced. In recent decades, set-valued analysis and convex analysis, as the tools of researching the optimized theorem, have made great progress, as well as provides new ideas, new methods for the generalized convexity study, and many useful conclusions, which also contributed to develop generalized convexity to be a focus for domestic and foreign scholars.This thesis consists of two parts:Firstly, based on the definition of E-convex sets, E-convex function the concepts of (F, K)-convex set, (F, K)-convex mapping and semi-(F, K)-convex mapping are proposed and some properties of them are investigated.Secondly, based on the concepts of semi-E-invariant convex set and semi-E-preinvex function, the concepts of the (F, K)-invariant convex set, (F, K)-preinvex mapping and semi-(F, K)-preinvex mapping are proposed and some properties of them are investigated.