带变序结构锥约束向量优化问题的对偶理论
Duality Theory of the Cone-Constrained Vector Optimization Problem with Variable Ordering Structures
摘要: 带变序结构的锥约束向量优化问题在工程分析、资源分配、偏好建模等问题中有着重要的应用。文章研究带变序结构的锥约束向量优化问题的对偶理论，用非线性的标量化泛函定义了与所研究问题相关的拉格朗日函数，讨论了这个拉格朗日函数鞍点的性质，并且根据拉格朗日函数的鞍点来刻画给定锥约束向量优化问题的Benson型真极小解。此外，还根据拉格朗日函数定义了给定锥约束向量优化问题相应的对偶集，并在稳定性假设下证明了相关的强对偶结果。
Abstract: The problem of cone-constraint vector optimization with variable ordering structures has important applications in engineering analysis, resource allocation, and preference modeling. This paper studies the dual theory of the cone-constrained vector optimization problem with variable ordering structures. The Lagrangian associated with the nonlinear scalar function is defined. The saddle point’s properties of this Lagrangian are discussed, and the properly minimal solution in the sense of Benson is characterized according to the saddle point of the Lagrangian. Furthermore, the corresponding dual set of the given cone-constrained vector optimization problem is defined according to the Lagrangian, and the associated strong dual results are proved under the assumption of stability.
文章引用：马俊涛, 游曼雪. 带变序结构锥约束向量优化问题的对偶理论[J]. 应用数学进展, 2025, 14(2): 93-102. https://doi.org/10.12677/aam.2025.142055

参考文献

[1] Wacker, M. (2008) Multikriterielle Optimierung bei der Registrierung medizinischer Daten. University of Erlangen Nürnberg.
[2] Fischer, B., Haber, E., Moderstitzki, J. (2008) Mathematics Meets Medicine, an Optimal Alignment. SIAG/OPT Views and News, 19, 1-7.
[3] Wacker, M. and Deinzer, F. (2009) Automatic Robust Medical Image Registration Using a New Democratic Vector Optimization Approach with Multiple Measures. Medical Image Computing and Computer-Assisted Intervention—MICCAI 2009, London, 20-24 September 2009, 590-597. [Google Scholar] [CrossRef] [PubMed]
[4] Engau, A. (2007) Variable Preference Modeling with Ideal-Symmetric Convex Cones. Journal of Global Optimization, 42, 295-311. [Google Scholar] [CrossRef] 
[5] Wiecek, M.M. (2007) Advances in Cone-Based Preference Modeling for Decision Making with Multiple Criteria. Decision Making in Manufacturing and Services, 1, 153-173. [Google Scholar] [CrossRef] 
[6] Yu, P.L. (1985) Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions. Springer. [Google Scholar] [CrossRef] 
[7] Chew, K.L. (1979) Domination Structures in Abstract Spaces. Proceedings of the First Franco-Southeast Asian Mathematical Conference, Singapore, 14 May-1 June 1979, 190-204.
[8] Yu, P.L. (1974) Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives. Journal of Optimization Theory and Applications, 14, 319-377. [Google Scholar] [CrossRef] 
[9] Bergstresser, K., Charnes, A. and Yu, P.L. (1976) Generalization of Domination Structures and Nondominated Solutions in Multicriteria Decision Making. In: Leitmann, G., Ed., Multicriteria Decision Making and Differential Games, Springer, 73-83. [Google Scholar] [CrossRef] 
[10] Baatar, D. and Wiecek, M.M. (2006) Advancing Equitability in Multiobjective Programming. Computers & Mathematics with Applications, 52, 225-234. [Google Scholar] [CrossRef] 
[11] Eichfelder, G. (2011) Cone-Valued Maps in Optimization. Applicable Analysis, 91, 1831-1846. [Google Scholar] [CrossRef] 
[12] Rubinov, A.M. and Gasimov, R.N. (2004) Scalarization and Nonlinear Scalar Duality for Vector Optimization with Preferences That Are Not Necessarily a Pre-Order Relation. Journal of Global Optimization, 29, 455-477. [Google Scholar] [CrossRef] 
[13] Nowak, M. (2007) Aspiration Level Approach in Stochastic MCDM Problems. European Journal of Operational Research, 177, 1626-1640. [Google Scholar] [CrossRef] 
[14] Eichfelder, G. (2011) Optimal Elements in Vector Optimization with a Variable Ordering Structure. Journal of Optimization Theory and Applications, 151, 217-240. [Google Scholar] [CrossRef] 
[15] Eichfelder, G. and Kasimbeyli, R. (2013) Properly Optimal Elements in Vector Optimization with Variable Ordering Structures. Journal of Global Optimization, 60, 689-712. [Google Scholar] [CrossRef] 
[16] Kasimbeyli, R. (2010) A Nonlinear Cone Separation Theorem and Scalarization in Nonconvex Vector Optimization. SIAM Journal on Optimization, 20, 1591-1619. [Google Scholar] [CrossRef] 
[17] Eichfelder, G. (2014) Variable Ordering Structures in Vector Optimization. Springer Science & Business Media.
[18] 刘彩平, 杨新民. 向量优化问题的E-真有效元及其标量化[J]. 中国科学: 数学, 2023, 53(10): 1397-1408.
[19] 范聪. 带变动序结构的不确定优化问题若干理论研究[D]: [硕士学位论文]. 重庆: 重庆交通大学, 2024.
[20] 熊佳琪. 变动集优化问题的适定性与稳定性研究[D]: [硕士学位论文]. 南昌: 南昌大学, 2022.
[21] Eichfelder, G. and Gerlach, T. (2015) Characterization of Properly Optimal Elements with Variable Ordering Structures. Optimization, 65, 571-588. [Google Scholar] [CrossRef] 
[22] Peng, J.W., Wei, W.B., Ghosh, D., Yao, J.C. (2024) Characterization of E-Benson Proper Efficient Solutions of Vector Optimization Problems with Variable Ordering Structures in Linear Spaces. Journal of Nonlinear & Variational Analysis, 8, 659-680.
[23] You, M. and Li, G. (2022) Approximate Properly Solutions of Constrained Vector Optimization with Variable Coradiant Sets. Optimization Letters, 17, 721-738. [Google Scholar] [CrossRef] 
[24] Ekeland, I. and Temam, R. (1976) Convex Analysis and Variational Problems: Convex Analysis and Variational Problems. Elsevier.

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