Approximate Solutions And Scalarization Methods To Vector Optimization Problems

Vector optimization is an important branch of mathematical programming.The theoretical research of vector optimization has made rich achievements,which mainly involve the concepts of various solutions,the optimality conditions,the scalarizations of solutions,the algebraic and topological properties of solution sets,and the variational inequalities closely related to vector optimization problems.In this paper,we considet the approximate solutions of vector optimization problem based on normal cone,and study the scalarizations and optimality conditions of the approximate solutions under convex and nonconvex conditions.The main contents of this paper are as follows:In the first part,we study the approximate proper efficient solutions of multiobjective optimization problems.First,we propose the concept of approximate proper efficient of multiobjective optimization problem by using the proximal normal cone and the co-radiant set.And the relationship betwen approximate efficient,approximate Benson proper efficient and approximate Borwein proper efficient of optimization problems is studied.Then,under the local starshaped conditions,we got the linear scalarization results of the approximate proper efficient solutions.In the second part,we investigate the properties of Gerstewitz nonlinear scalar func-tions in the real linear spaces.Firstly,the subdifferential of the function are given by the special properties of Gerstewitz scalar function with the co-radiant set.Then,based on the nonlinear scalarization function and the sub differential results of the functional,the limiting cone is used to describe the approximate efficient of the nonconvex set,and the optimality condition of the approximate solutions of vector optimization problem is studied in Asplund space.Finally,the nonlinear scalarization function is applied to the risk measure function to study its applications.