多目标约束向量优化问题的类拉格朗日乘数法

数学物理学报 ›› 2018, Vol. 38 ›› Issue (6): 1076-1094.

多目标约束向量优化问题的类拉格朗日乘数法

李润鑫¹(),黄辉²,尚振宏¹,曹宇¹,王红斌^1,*(),张晶¹

¹ 昆明理工大学信息工程与自动化学院昆明 650500
² 云南大学数学与统计学院昆明 650091

收稿日期:2016-12-30 出版日期:2018-12-26 发布日期:2018-12-27
通讯作者: 王红斌 E-mail:rxli@kmust.edu.cn;whbin2007@126.com
作者简介:李润鑫, E-mail:rxli@kmust.edu.cn
基金资助:
国家自然科学基金(11461080);国家自然科学基金(61562051);国家自然科学基金(61462052);国家自然科学基金(61462054);云南省人才培养基金(KKSY201603016)

Lagrange-Like Multiplier Rules for Weak Approximate Pareto Solutions of Multiobjective Constrained Vector Optimization Problems

Runxin Li¹(),Hui Huang²,Zhenhong Shang¹,Yu Cao¹,Hongbin Wang^1,*(),Jing Zhang¹

¹ Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming 650500
² Department of Mathematics, Yunnan University, Kunming 650091

Received:2016-12-30 Online:2018-12-26 Published:2018-12-27
Contact: Hongbin Wang E-mail:rxli@kmust.edu.cn;whbin2007@126.com
Supported by:
Supported by the NSFC(11461080);Supported by the NSFC(61562051);Supported by the NSFC(61462052);Supported by the NSFC(61462054);the Yunnan Provincial Foundation for Personnel Cultivation(KKSY201603016)

摘要/Abstract

摘要：

文献[21]给出了实希尔伯特空间中含有一个约束条件的向量优化问题的有关帕雷托解的拉格朗日乘数法.该文把文献[21]中的主要结果推广到了含有任意m个约束条件的多目标向量优化问题中，给出了实希尔伯特空间中，以proximal法锥和目标函数的coderivative刻画的多目标约束向量优化问题的类拉格朗日乘数法.

关键词: 向量优化, Proximal法锥, Coderivative, 弱ε-帕雷托解, 多目标约束向量优化问题

Abstract:

In real Hilbert space case, Zheng and Li[21] established a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems with only one constrained multifunction. In this paper, we improve and extend their main results to multiobjective constrained vector optimization problems' cases.

Key words: Vector optimization, Proximal normal cone, Coderivative, Weak ε-Pareto solution, Multiobjective constrained vector optimization problem

中图分类号:

李润鑫,黄辉,尚振宏,曹宇,王红斌,张晶. 多目标约束向量优化问题的类拉格朗日乘数法[J]. 数学物理学报, 2018, 38(6): 1076-1094.

Runxin Li,Hui Huang,Zhenhong Shang,Yu Cao,Hongbin Wang,Jing Zhang. Lagrange-Like Multiplier Rules for Weak Approximate Pareto Solutions of Multiobjective Constrained Vector Optimization Problems[J]. Acta mathematica scientia,Series A, 2018, 38(6): 1076-1094.

参考文献 25

1	Aubin J P , Frandowska H . Set-valued Analysis. Boston: Birkhauser, 1990
2	Bao T Q , Mordukhovich B S . Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern, 2007, 36: 531- 562
3	Bergstresser K , Yu P L . Domination structures and multicriteria problems in N-person games. Theory Decis, 1977, 8 (1): 5- 48 doi: 10.1007/BF00133085
4	Clarke F H . Optimization and Nonsmooth Analysis. New York: Wiley, 1983
5	Gotz A , Jahn J . The Lagrange multiplier rule in set-valued optimization. SIAM J Optim, 1999, 10: 331- 344
6	Gopfert A , Riahi H , Tamme C , Zalinescu C . Variational Methods in Partially Ordered Spaces. New York: Springer, 2003
7	Jahn J . Vector Optimization:Theory, Applications and Extensions. Berlin: Springer-Verlag, 2004
8	Luc D T . Theory of Vector Optimization. Berlin: Springer, 1989
9	Qu S J , Goh M , Souza R D , Wang T N . Proximal point algorithms for convex multi-criteria optimization with applications to supply chain risk management. J Optim Theory Appl, 2014, 163: 949- 956 doi: 10.1007/s10957-014-0540-8
10	Sun X K , Long X J , Chai Y . Sequential optimality conditions for fractional optimization with applications to vector optimization. J Optim Theory Appl, 2015, 164: 479- 499 doi: 10.1007/s10957-014-0578-7
11	Bao T Q , Mordukhovich B S . Relative Pareto minimizers for multiobjective problems:existence and optimality conditions. Math Program, 2011, 122: 301- 347
12	Bao T Q, Mordukhovich B S. Extended Pareto optimality in multiobjective problem//Ansari Q H, Yao J C, et al. Recent Advances in Vector Optimization, 2011: 467-515
13	Bao T Q , Mordukhovich B S . Necessary nondomination conditions in set and vector optimization with variable ordering structures. J Optim Theory Appl, 2014, 162: 350- 370 doi: 10.1007/s10957-013-0332-6
14	Helbig S , Pateva D . On several concepts for ε-efficiency. OR Spectrum, 1994, 16: 179- 186 doi: 10.1007/BF01720705
15	Isac G . Pareto optimization in infinite-dimensional spaces:The importance of nuclear cones. J Math Anal Appl, 1994, 182: 393- 404 doi: 10.1006/jmaa.1994.1093
16	Jameson G . Ordered Linear Spaces. New York: Springer, 1970
17	Guo X L , Li S J . Optimality conditions for vector optimization problems with difference of convex maps. J Optim Theory Appl, 2014, 162: 821- 844 doi: 10.1007/s10957-013-0327-3
18	Zhu Q J . Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints. SIAM J Control Optim, 2000, 39: 97- 112 doi: 10.1137/S0363012999350821
19	Zheng X Y , Ng K F . The Lagrange multilizer rule for multifuntions in Banach spaces. SIAM J Optim, 2006, 17: 1154- 1175
20	Zheng X Y , Ng K F . A unified separation theory for closed sets in a Banach space and optimality conditions for vector optimization. SIAM J Optim, 2011, 21: 886- 911 doi: 10.1137/100811155
21	Zheng X Y , Li R X . Lagrange multiplier rules for weak approximate pareto solutions of constrained vector optimization problems in Hilbert Spaces. J Optim Theory Appl, 2014, 162: 665- 679 doi: 10.1007/s10957-012-0259-3
22	Clarke F H , Ledyaev Y S , Stern R J , Wolenski P R . Nonsmooth Analysis and Control Theory. New York: Springer, 1998
23	Rockafellar R T , Wets R J B . Variational Analysis. Berlin: Springer, 1998
24	Schirotzek W . Nonsmooth Analysis. Berlin: Springer, 2007
25	Borwein J M , Preiss D . A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans Amer Math Soc, 1987, 303: 517- 527 doi: 10.1090/S0002-9947-1987-0902782-7