广西师范大学学报（自然科学版） ›› 2021, Vol. 39 ›› Issue (3): 62-68.doi: 10.16088/j.issn.1001-6600.2020091101

• • 上一篇    下一篇

凸向量优化问题弱有效解集的连通性

钟丽明, 范江华*   

  1. 广西师范大学 数学与统计学院, 广西 桂林 541006
  • 收稿日期:2020-09-11 修回日期:2020-10-12 发布日期:2021-05-13
  • 通讯作者: 范江华(1967—),男,湖南邵阳人,广西师范大学教授,博士。E-mail: jhfan@gxnu.edu.cn
  • 基金资助:
    国家自然科学基金(71561004); 广西研究生教育创新计划(XYCSZ2020062)

Connectedness of Weakly Effective Solution Sets for Convex Vector Optimization Problems

ZHONG Liming, FAN Jianghua*   

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541006, China
  • Received:2020-09-11 Revised:2020-10-12 Published:2021-05-13

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摘要： 研究向量优化问题解集的连通性。利用标量化方法,讨论了无界闭凸集上凸向量优化问题弱有效解集的连通性。在向量值函数为锥下半连续、锥凸时,运用极锥的紧凸基的连通性,证明了解集映射是上半连续映射,从而得到解集的连通性;在向量值函数为锥下半连续、锥严格凸时,得到了凸向量优化问题弱有效解集的道路连通性;得到了复合多目标规划问题的弱有效解集与仿射向量变分不等式问题弱有效解集的连通性。

关键词: 向量优化问题, 弱有效解, 非空紧致集, 连通性, 道路连通性

Abstract: The connectedness of solution sets for vector optimization problems is studied in this paper. By means of the scalarization method, the connectedness of weakly efficient solutions sets for convex vector optimization problems on the unbounded, closed and convex sets are discussed. When the vector valued mapping is cone-lowersemicontinuous and cone-convex, the mapping of solution sets is proved to be upper semicontinuous and the connectedness of solution set by using the connectedness of the compact and convex base of the polar cone is established. When the vector valued mapping is cone-lowersemicontinuous and strictly cone-convex, the path-connectedness of solution sets of convex vector optimization problems are obtained. Furthermore, the connectedness of weakly efficient solutions sets of composite multiobjective optimization problems and the connectedness of solution sets of affine vector variational inequality problems are obtained.

Key words: vector optimization problem, weakly efficient solutions, nonempty and compact set, connectedness, path-connectedness

中图分类号: 

  • O224.1
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