Approximate Henig Efficient Solutions Of Set-Valued Optimization Problems

Efficient solution of vector optimization problem is solution in the sense of non-inferiority with respect to partial order, generally speaking, the set of solutions is very large, some of these solution set have poor properties. Then, they has been efforts to find effcient solutions which have the better properties-called proper effcient solution. It has the following two advantages:Firstly, proper efficient solution can be dense on the effcient solution set, thus when the effcient solution set is reduced to proper efficient solution, do not lost some effcient solution:Secondly, proper efficient solution can be given in the form of scalarization. so that we can easily ensure the proper efficient solution. In recent years, people have got the different sense with the true effective solution, for example, Henig proper solution, Benson proper solution, Super efficient solution and so on. But, the existence condition of super effcient solution is very strong. At the same time, for discussing the scalarization theorem in the sense of the Benson proper effeiency, we have to demand that the ordering cone has the compact or weak-compact base. In many situations, they are unable to achieve them. Henig proper solution has many desirable properties, its existence condition is weaker than that of super efficient solutions and the ordering cone only requires a base. So far, there are the few papers which deal with Henig proper solution. Thus, the research for this topic has important theoretical value and practical signifcance.The main contents are as follows: Firstly, we give the defnition of the approximate effective point for vector optimization problem with set-valued maps in the locally convex space, we obtain some equivalent conditions of the approximate efficient point, we discuss the relationship among the approximate efficient point, approximate Benson effcient point, approximate super effcient point and approximate Hcnig efficient point.Secondly, we give the scalar characterization of the approximate Henig efficient solution and existence theorem for vector optimization problem with set-valued maps. At the same time, we discuss the topological properties of the approxi-mate Henig efficient solution, such as compactness, closed and the connectivity of approximate Henig effcient solution sets.Finally, We study the subdiffcrential of the set-valued mapping and prove its existence under certain conditions. At the same time, we discuss its related properties.