ε-真有效解的非线性标量化  

Nonlinear Scalarization ofε-Properly Efficient Solutions

作　　者：夏远梅 张万里[1] 赵克全[1] 

机构地区：[1]重庆师范大学数学学院,重庆401331

出　　处：《重庆师范大学学报（自然科学版）》2015年第1期12-15,共4页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.11301574);第二批重庆市高等学校青年骨干教师资助计划;重庆师范大学博士基金(No.13XLB029)

摘　　要：本文利用基于点闭凸锥的经典非线性标量化函数Δ-K对向量优化问题ε-真有效解的非线性标量化性质进行了研究。首先证明了向量优化问题(VP)的ε-真有效解蕴含标量化问题(Py)的dε+K(0)-近似解,并通过例子说明了这一结论的逆不一定成立。进一步,证明了标量化问题(Py)的严格β-近似解蕴含向量优化问题(VP)的ε-真有效解,并举例说明了如果集合f(S)+ε+K-f(x)的锥包不是闭集,这一结论不一定成立以及标量化问题(Py)的β-近似解不一定蕴含向量优化问题(VP)的ε-真有效解。In this paper, we study some nonlinear scalarization characterizations of e-properly efficient solutions for vector optimization problems via the classical nonlinear scalarization function Δ-κ obtained by using a pointed closed convex cone. We first prove that s-properly efficient solutions of the vector optimization problem （VP） implies dε＋n （0）-approximate solutions of the scalarization problem （Py） and give an example to illustrate the fact that the converse of this conclusion may not be valid. Furthermore, we also prove that strictly/9-approximate solutions of the scalarization problem （Py） implies e-properly efficient solutions of the vector optimization problem （VP）, and propose some examples to illustrate the facts that this conclusion may not be true if the cone hull of the set f（S）＋ε＋K-f（^-x） is not closed, and β-approximate solutions of the scalarization problem （Py） does not imply e-properly efficient solutions of the vector optimization problem （VP）.

关 键 词：向量优化 ε-真有效解 非线性标量化 

分 类 号：O221.6[理学—运筹学与控制论]