The Research Of The Generalized E-convexity And Related Optimization Questions

Generalized convexity analysis is widely applied in optimization theory,control theory,decision theory and other disciplines.As an important carrier of convex analysis,convex sets and convex functions play important roleas in the research of optimization theory.But many sets and functions in the practical problems are nonconvex that limit the solution of the problem,thus the concept of generalized convex sets and generalized convex functions appear.There are many scholars that have been taken into the study of generalized convex sets and generalized convex function,and also obtained rich results.All these constantly enrich and perfect the study on the theory of generalized convex.On the basis of predecessors' study results of generalized convexity,we mainly make in-deepth study of generalized convex conversity and generalized E-conversity about their concepts,properties,and own some new properties and conclusions;At last,we study the generalized E-convexity in the optimization problem and gain some new conclusions.The work of this paper can expand research on generalized E-convexity and optimization applied theory in a certain extent.The main content of this paper is divided into the following chapters:The first chapter is introduction.Firstly,we introduce the research status of convexity,generalized convexity and their optimization applications.At last we introduce the main research work and meaning about this paper.The second chapter is about the basic knowledge of E-convexity.We sum up the main related basic concepts,properties,theorems and conclusions of E-convex sets,E-convex functions,E-convexity optimization.All these work can be ready for the later study.The third chapter is about the equivalent theorem of E-convex functions.First of all,we make in-depth study of E-convex functions' elements,and then from the point of set,by defining the convexity of a function in arbitrary sets,we gain a new equivalent theorem of E-convex functions,all these work can further expand the understanding of E-convex functions.The fourth chapter is about some new conclusions of quasi-semi-E-convex functions.Firstly,we gain some new properties of quasi-semi-E-convex functions.Then,we study the relationships between quasi-E-convex functions and semi-E-convex functions.Finally,we give a judgement theorem of quasi-semi-E-convex functions and have a criterion of it in the case of low half continuous.The fifth chapter is about the optimality of generalized E-convex programming.We mainly research multi-objective programming problem under E-convex and generalized E-convexity conditions,study the relevant theorems and conclusions of optimal solution under undifferential and E-differentiable and at last research the sufficient optimality conditions for the existence of the optimal solution.The sixth chapter is about summary and prospect.We summarize the main work of this paper and explain the shortcomings of the article.As well we point out the feasible problem and direction which will be mainly studied in the future.