向量优化问题的E-真有效元及其标量化

E-properly efficient elements of vector optimization problems and their scalarization

本文研究实赋范空间中向量优化问题的真有效元.利用改进集,推广Benson-真有效元到EBenson-真有效元,推广Henig-真有效元到E-Henig-真有效元,并给出这两类E-真有效元之间的关系.在没有任何凸性假设下,利用一类非线性标量化函数,分别建立E-Benson-真有效元和E-Henig-真有效元的必要和充分最优性条件,给出它们的标量化刻画.本文所得结果推广了Kasimbeyli(2010)关于真有效元的相关结果.特别指出,本文所引入的E-Benson-真有效元的概念,称为Ⅱ-型E-Benson-真有效元,它改进了Zhao和Yang(2015)给出的相应的概念,因此能够保证向量优化问题的每一个E-Benson-真有效元都是E-有效元.

In this paper,we focus on properly efficient elements of vector optimization problems in the real-normed space.Applying improvement sets,we extend the concept of Benson properly efficient elements to E-Benson properly efficient elements,and Henig properly efficient elements to E-Henig properly efficient elements.The relationships between these two classes of E-properly efficient elements are given.Without any convexity assumption,we obtain necessary and sufficient conditions for the E-Benson properly efficient elements and the E-Henig properly efficient elements of vector optimization problems via nonlinear scalarization processes,respectively.The scalarization characterizations of them are given.The results obtained in this paper generalize the related results on properly efficient elements of vector optimization problems given by Kasimbeyli(2010).Specifically,the concept of E-Benson properly efficient elements of vector optimization problems introduced in this paper,which is called type-II E-Benson properly efficient elements,improves the corresponding concept proposed by Zhao and Yang(2015),and it can be guaranteed that every type-II E-Benson properly efficient element is an E-efficient element.