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

PDF (PC)

116

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
[1]WARBURTON A R. Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives[J]. Journal of Optimization Theory and Applications, 1983, 40(4): 537-557.
[2]HELBIG S. On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces[J]. Journal of Optimization Theory and Applications, 1990, 65(2): 257-270.
[3]HU Y D, SUN E J. Connectedness of the efficient set in strictly quasiconcave vector maximization[J]. Journal of Optimization Theory and Applications, 1993, 78(3): 613-622.
[4]CHEN G Y, LI S J. Existence of solutions for a generalized vector quasivariational inequality[J]. Journal of Optimization Theory and Applications, 1996, 90(2): 321-334.
[5]GONG X H, YAO J C. Connectedness of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138(2): 189-196.
[6]曹敏, 陈剑尘, 高洁. 集值优化问题强有效解集的连通性[J]. 数学的实践与认识, 2014, 44(9): 253-258.
[7]巨兴兴,陈加伟,张俊容,等.含参广义向量均衡问题近似解集的连通性[J]. 应用数学和力学, 2018, 39(10): 1206-1212.
[8]XU Y D, ZHANG P P. Connectedness of solution sets of strong vector equilibrium problems with an application[J]. Journal of Optimization Theory and Applications, 2018, 178(1): 131-152.
[9]李科科,彭再云,赵勇,等.含参广义集值强向量平衡问题的稳定性[J]. 数学学报,2019,62(4): 653-662.
[10]FLORES-BAZAN F, VERA C. Characterization of the nonemptiness and compactness of solution sets in convex and nonconvex vector optimization[J]. Journal of Optimization Theory and Applications, 2006, 130(2): 185-207.
[11]CHEN Z. Asymptotic analysis in convex composite multiobjective optimization problems[J]. Journal of Global Optimization, 2013, 55(3): 507-520.
[12]LEE G M, BU I J. On solution sets for affine vector variational inequality[J]. Nonlinear Analysis: Theory Methods Applications, 2005, 63(5/6/7): 1847-1855.
[13]AUBIN J P, EKELAND I.Applied nonlinear analysis[M].New York:Wiley, 1984.
[14]LEE G M, KIM D S, LEE B S, et al. Vector variational inequality as a tool for studying vector optimization problems[J]. Nonlinear Analysis: Theory Methods Applications, 1998, 34(5): 745-765.
[15]AUSLENDER A, TEBOULLE M. Asymptotic cones and functions in optimization and variational inequalities[M]. New York: Springer, 2003.
[16]DENG S. Characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems in real Banach spaces[J]. Journal of Optimization Theory and Applications, 2009, 140(1): 1-7.
[17]SALINETTI R, WETS J B. On the relations between two types of convergence for convex functions[J]. Journal of Mathematical Analysis and Applications, 1977, 60(1): 211-226.
[1] 陈燕, 吴文康, 梁俊斌. Sink移动无线传感网中高安全性密钥预分配方案[J]. 广西师范大学学报（自然科学版）, 2013, 31(3): 164-168.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 庄枫红, 马姜明, 张雅君, 苏静, 于方明. 中华水韭对不同光照条件的生理生态响应[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 93 -100 .
[2] 滕志军, 吕金玲, 郭力文, 许媛媛. 基于改进粒子群算法的无线传感器网络覆盖策略[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 9 -16 .
[3] 刘铭, 张双全, 何禹德. 基于改进SOM神经网络的异网电信用户细分研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 17 -24 .
[4] 苗新艳, 张龙, 罗颜涛, 潘丽君. 一类交替变化的竞争—合作混杂种群模型研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 25 -31 .
[5] 邓亚彬,蒋品群,宋树祥. 新型低压微功耗伪差分跨导放大器设计[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 17 -24 .
[6] 黄兵方,闻炳海,邱文,赵琬玲,陈燕雁. 基于晶格Boltzmann方法的接触角实时测量研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 34 -43 .
[7] 戴喜生,李光,周星宇. 大坝—河流渠道灌溉系统的迭代学习控制[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 53 -60 .
[8] 呼文军,马忠军,马梅. 领导—跟随多智能体系统在分布式自适应控制下的滞后一致性[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 70 -75 .
[9] 张军文,李成思,卢永昌,史生辉. MEKC测定麻花艽中齐墩果酸和熊果酸含量[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 99 -104 .
[10] 谢静,唐贺,林万华,孙华英,吴桂生,和晓明,邓科,罗怀容. 小鼠杏仁中央核逆向投射的研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(1): 149 -157 .
版权所有 © 广西师范大学学报（自然科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发