向量优化问题Benson真有效解的稳定性

Abstract

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

CLC Number:

O224

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.

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References 21

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 doi: 10.3969/j.issn.1003-3998.2020.04.016
	Li M , Yu G L . Optimality of robust approximation solutions for uncertain convex optimization problems. Acta Math Sci, 2020, 40 (4): 1007- 1017 doi: 10.3969/j.issn.1003-3998.2020.04.016
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
	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 doi: 10.3969/j.issn.1000-0984.2012.01.034
	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 doi: 10.3969/j.issn.1000-0984.2012.01.034
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 doi: 10.3321/j.issn:0254-3079.2003.02.016
	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 doi: 10.3321/j.issn:0254-3079.2003.02.016
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 doi: 10.3321/j.issn:0583-1431.2003.03.026
	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 doi: 10.3321/j.issn:0583-1431.2003.03.026
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 doi: 10.3969/j.issn.1001-2400.2001.01.019
	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 doi: 10.3969/j.issn.1001-2400.2001.01.019
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 doi: 10.3969/j.issn.1001-8735.2016.01.003
	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 doi: 10.3969/j.issn.1001-8735.2016.01.003
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
	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 doi: 10.3969/j.issn.1006-0464.2010.04.003
	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 doi: 10.3969/j.issn.1006-0464.2010.04.003
11	Gutiérrez C , Jiménez B , Novo V . On approximate solutions in vector optimization problems via scalarization. Comput Optim Appl, 2006, 35: 305- 324 doi: 10.1007/s10589-006-8718-0
12	Sawaragi Y , Nakayama H , Tanino T . Theory of Multiobjective Optimization. Orlando: Academic Press, 1985
13	Ansari Q H, Köbis 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 doi: 10.1007/s001860000085
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 doi: 10.1007/s11117-010-0093-5
18	Lucchetti R E , Miglierina E . Stability for convex vector optimization problems. Optimization, 2004, 53 (5/6): 517- 528
19	Göpfert A, Riahi H, Tammer C, Zǎlinescu C. Variational Methods in Partially Ordered Spaces. New York: Springer-Verlag, 2003
20	Bertsekas D P, Nedić 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

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Stability of Benson Proper Efficient Solutions for Vector Optimization Problems

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