向量优化问题Benson真有效解的稳定性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 35-44.

向量优化问题Benson真有效解的稳定性

曾静(),胡瑞婷*()

重庆工商大学数学与统计学院重庆 400067

收稿日期:2021-01-27 出版日期:2022-02-01 发布日期:2022-02-23
通讯作者: 胡瑞婷 E-mail:zengjing1983@ctbu.edu.cn;zengjing1983@ctbu.edu.cn; 943389111@qq.com
作者简介:曾静, E-mail: zengjing1983@ctbu.edu.cn
基金资助:
国家自然科学基金青年基金项目(12001445);重庆市自然科学基金（基础研究与前沿探索专项）面上项目(cstc2019jcyj-msxmX0605);重庆市教委科学技术研究项目(KJQN201800837)

Stability of Benson Proper Efficient Solutions for Vector Optimization Problems

Jing Zeng(),Ruiting Hu*()

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

Received:2021-01-27 Online:2022-02-01 Published:2022-02-23
Contact: Ruiting Hu E-mail:zengjing1983@ctbu.edu.cn;zengjing1983@ctbu.edu.cn; 943389111@qq.com
Supported by:
the NSFC(12001445);the General Project of Chongqing NSF(Special Project of Basic Research and Frontier Exploration)(cstc2019jcyj-msxmX0605);the Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJQN201800837)

摘要/Abstract

摘要：

首先利用非线性标量化技术，建立了向量优化问题Benson真有效解与一类标量优化问题解之间的等价关系.然后借助建立的等价性结果，在向量优化问题目标映射和约束条件均扰动的情况下，得到了向量优化问题Benson真有效点集和解集的抗干扰稳定性结果.该结果首次借助标量化技术，在扰动问题序列Painlevé-Kuratowski收敛于目标优化问题的情况下，研究了向量优化问题Benson真有效解的抗干扰性，该结果对数值计算分析有重要的理论价值.

关键词: Benson真有效解, Painlevé-Kuratowski收敛, 标量化, 稳定性

Abstract:

In this paper, at the beginning, we established the equivalent relationship between Benson proper efficient solutions of vector optimization problems and the solution of a class of scalar optimization problems by using nonlinear scaling technique. Besides, with the means of the equivalence result, we obtained the anti-interference stability results of Benson proper efficient point sets and the solution sets in the vector optimization problem when the objective function and the constraint conditions were perturbed. For the first time, by the means of the scalarization technique, we study the anti-interference of Benson proper efficient solutions of vector optimization problems under the condition that the disturbance problem sequence Painlevé-Kuratowski converges to the object optimization problem. And the results have important theoretical value for numerical calculation and analysis.

Key words: Benson proper efficient solution, Painlevé-Kuratowski convergence, Scalarization, Stability

中图分类号:

O224

曾静,胡瑞婷. 向量优化问题Benson真有效解的稳定性[J]. 数学物理学报, 2022, 42(1): 35-44.

Jing Zeng,Ruiting Hu. Stability of Benson Proper Efficient Solutions for Vector Optimization Problems[J]. Acta mathematica scientia,Series A, 2022, 42(1): 35-44.

参考文献

1	Oppezzi P , Rossi A M . A convergence for vector valued functions. Optimization, 2007, 57 (3): 435- 448
2	李萌, 余国林. 不确定凸优化问题鲁棒近似解的最优性. 数学物理学报, 2020, 40 (4): 1007- 1017
2	Li M , Yu G L . Optimality of robust approximation solutions for uncertain convex optimization problems. Acta Math Sci, 2020, 40 (4): 1007- 1017
3	杨新民. Benson真有效解与Borwein真有效解的等价性. 应用数学, 1994, 7 (2): 246- 247
3	Yang X M . Equivalence between Benson proper efficient solution and Borwein proper efficient solution. Math Appl, 1994, 7 (2): 246- 247
4	周志昂. 集值优化问题Borwein真有效解与Benson真有效解的等价性. 数学的实践与认识, 2012, 42 (1): 247- 250
4	Zhou Z A . The equivalence of Borwein properly efficient solution and Benson properly efficient solution of set-valued optimization problem. Math Practice Theory, 2012, 42 (1): 247- 250
5	刘三阳, 盛宝怀. 非凸向量集值优化Benson真有效解的最优性条件与对偶. 应用数学学报, 2003, 26 (2): 337- 344
5	Liu S Y , Sheng B H . The optimality conditions and duality of nonconvex vector set-valued optimization with Benson proper efficiency. Acta Math Appl Sin, 2003, 26 (2): 337- 344
6	盛宝怀, 刘三阳. Benson真有效意义下集值优化的广义最优性条件. 数学学报, 2003, 28 (3): 611- 620
6	Sheng B H , Liu S Y . Generalized optimality conditions for set-valued optimization of Benson proper efficient. Acta Math Sin, 2003, 28 (3): 611- 620
7	盛宝怀, 刘三阳. 多目标主从向量集值优化Benson真有效解的连通性. 西安电子科技大学学报, 2001, 28 (1): 79- 82
7	Sheng B H , Liu S Y . Connectivity of Benson proper efficient solution for multi-objective master-slave vector set-valued optimization. Journal of Xidian University, 2001, 28 (1): 79- 82
8	袁春红. 集值映射多目标半定规划问题Benson真有效解集的连通性. 内蒙古师范大学学报(自然科学汉文版), 2016, 45 (1): 9- 12
8	Yuan C H . Connectedness of Benson proper efficient solution set of multiobjective semidefinite programming problems with set-valued maps. Journal of Inner Mongolia Normal University(Natural Science Edition), 2016, 45 (1): 9- 12
9	徐义红, 杨赟. 集值优化Benson真有效元的二阶刻画. 运筹学学报, 2016, 20 (2): 88- 96
9	Xu Y H , Yang Y . Second-order characterizations on Benson proper efficient element of set-valued optimization. Operations Research Transactions, 2016, 20 (2): 88- 96
10	徐义红, 熊卫芝, 汪涛. 集值优化问题的Benson次梯度及其应用. 南昌大学学报(理科版), 2010, 34 (4): 326- 331
10	Xu Y H , Xiong W Z , Wang T . Benson-subgradient of set-valued optimization and its applications. Journal of Nanchang University(Natural Science), 2010, 34 (4): 326- 331
11	Gutiérrez C , Jiménez B , Novo V . On approximate solutions in vector optimization problems via scalarization. Comput Optim Appl, 2006, 35: 305- 324
12	Sawaragi Y , Nakayama H , Tanino T . Theory of Multiobjective Optimization. Orlando: Academic Press, 1985
13	Ansari Q H, K?bis E, Yao J C. Vector Variational Inequalities and Vector Optimization: Theory and Applications. Switzerland: Springer Cham, 2018
14	Huang X X . Stability in vector-valued and set-valued optimization. Math Methods Oper Res, 2000, 52: 185- 193
15	Chen G Y, Huang X X, Yang X Q. Vector Optimization: Set-valued and Variational Analysis. Berlin: Springer-Verlag, 2005
16	Dontchev A L, Zolezzi T. Well-Posed Optimization Problems. Berlin: Springer-Verlag, 1993
17	Zeng J , Li S J , Zhang W Y . Stability results for convex vector-valued optimization problems. Positivity, 2011, 15: 441- 453
18	Lucchetti R E , Miglierina E . Stability for convex vector optimization problems. Optimization, 2004, 53 (5/6): 517- 528
19	G?pfert A, Riahi H, Tammer C, Zǎlinescu C. Variational Methods in Partially Ordered Spaces. New York: Springer-Verlag, 2003
20	Bertsekas D P, Nedi? A, Ozdaglar A E. Convex Analysis and Optimization. Belmont Massachusetts: Athena Scientific, 2006
21	Rockafellar R T, Wets R J. Variational Analysis. Berlin: Springer-Verlag, 2004

[1]	王春, 许天周. 空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子[J]. 数学物理学报, 2025, 45(5): 1381-1391.
[2]	邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.
[3]	吕东霆. 两种群空间非均匀反应扩散竞争模型的全局渐近稳定性[J]. 数学物理学报, 2025, 45(4): 1171-1183.
[4]	梁青. 两类带 Markov 切换的随机比例泛函微分方程全局解的存在唯一性和稳定性[J]. 数学物理学报, 2025, 45(4): 1184-1205.
[5]	刘佳, 包雄雄. 非局部时滞扩散方程棱锥形波前解的渐近稳定性[J]. 数学物理学报, 2025, 45(1): 44-53.
[6]	肖江龙, 宋永利, 夏永辉. 一个扩散模型中恐惧效应与集群行为协同诱导的时空动力学研究[J]. 数学物理学报, 2024, 44(6): 1577-1594.
[7]	张咏雪, 贾文生. 有限理性下广义多目标多主多从博弈受控系统解的稳定性[J]. 数学物理学报, 2024, 44(6): 1689-1702.
[8]	樊天娇, 冯立超, 杨艳梅. 不等式的推广以及在加性时变时滞系统中的应用[J]. 数学物理学报, 2024, 44(5): 1335-1351.
[9]	杨红, 张晓光. 单纯复形上的随机 SIS 传染病模型[J]. 数学物理学报, 2024, 44(5): 1392-1399.
[10]	郭燕, 徐小川. 基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性[J]. 数学物理学报, 2024, 44(4): 859-870.
[11]	潘丽君, 吕顺, 翁莎莎. 耦合Aw-Rascle-Zhang模型的Riemann解及其稳定性[J]. 数学物理学报, 2024, 44(4): 885-895.
[12]	章春国, 孙宝楠, 付煜之, 于欣. 具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定[J]. 数学物理学报, 2024, 44(4): 946-959.
[13]	高彩霞, 赵东霞. 带干扰项的 ARZ 交通流模型的时滞控制与 ISS 稳定性[J]. 数学物理学报, 2024, 44(4): 960-977.
[14]	徐瑞, 周凯娟, 白宁. 一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型[J]. 数学物理学报, 2024, 44(3): 771-782.
[15]	邱仰聪, 王其如. 一阶非线性时标动态方程的 Hyers-Ulam-Rassias 稳定性[J]. 数学物理学报, 2024, 44(2): 465-475.