The Optimality Conditions For Vector Optimization Problems

As an important research aspect of optimization theory and applications,the research on vector optimization not only involves many disciplines such as:nonlinear analysis,convex analysis,nonsmooth analysis,and so on.It is also widely used in many fields,such as:financial investment,engineering design,environmental protection,health and medical community,transportation,etc.Therefore,the research on this topic has both theoretical value and practical significance.This whole thesis is divided into three chapters and we mainly study the theory of vector optimization in two aspects:scalarization for approximate solutions of vector optimization problems on real linear spaces and optimality condition for quasiconvex multiobjective optimization problem.The main results,obtained in this dissertation,may be summarized as follows:In chapter 1,we give a brief introduction about research significance and contents of vector optimization problems.And we sum up the development history and current situation of vector optimization as well as three aspects:solution of optimization prob-lems,optimization problems on real linear spaces and quasiconvex optimization problem related to this article.Finally,we state the main contents studied in this thesis.Chapter 2 is committed to study scalarization for approximate solutions of vector optimization problems on real linear spaces.First,we point out that the approximate proper efficiency solution given in the existing literature can deduce the exact efficiency solution and it is also exact Benson proper efficiency solution under certain conditions.At the same time,we point out the conditions of the main results in this paper are not reasonable for point convex cones.Then,we introduce a new concept of approximate weakly effective solution,effective solution and approximate proper efficiency solutions by using the concepts and properties of algebra interior,vector closure,algebraic dual cone and other related concepts on real linear spaces,then we present linear scalarizations for corresponding approximate solutions under the condition of generalized convexity.Chapter 3 aims at optimality condition for quasiconvex multiobjective optimization problems.First,we point out the definition of the subdifferential of vector-valued func-tions in the existing literature is not reasonable,derive optimality conditions for weak ef-ficient solutions,efficient solutions and proper efficient solutions of(MOP)with abstract set constraints by using the existing subdifferential of numerical function.Lastly,we consider special inequality constraints,present the corresponding optimality conditions.In particular,for some quasiconvex vector-valued functions,the exact subdifferentials we define may not exist.Therefore,for quasiconvex vector-valued functions,the approximate subdifferentials and the optimality conditions of the corresponding approximate solutions are given.