|
|
Pure Mathematics 2024
集值均衡问题的-Benson真有效解的非线性标量化
|
Abstract:
众所周知，在最优化理论中，集值均衡问题是一个关键组成部分，它在数理经济和交通系统等实际应用中具有重要的研究意义和理论价值。许多学者从不同的角度提出了集值均衡问题不同类型的解。然而，如何推广和改进集值均衡问题的解是有意义的。本文，我们利用改进集和回收锥这两种研究最优化理论的重要工具，研究了带约束集值均衡问题的E∞-Benson真有效解，建立了集值均衡问题的非线性标量化定理。
As is well-known in optimization theory, the problem of set-valued equilibrium is a key component. It holds significant research significance and theoretical value in practical applications such as mathematical economics and transportation systems. Many scholars have proposed various types of solutions to the set-valued equilibrium problem from different perspectives. However, the meaningful task lies in generalizing and improving the solutions to the set-valued equilibrium problem. In this paper, utilizing the improvement set and the recession cone which are two important tools to study optimization theory, we investigateE∞-Benson properly efficient solution of the set-valued equilibrium problem. Furthermore, we establish the nonlinear scalarization theorems of the set-valued equilibrium problem.
| [1] | Dhingra, M. and Lalitha, C.S. (2017) Approximate Solutions and Scalarization in Set-Valued Optimization. Optimization, 66, 1793-1805. https://doi.org/10.1080/02331934.2016.1271419 |
| [2] | Gao, Y., Hou, S.H. and Yang, X.M. (2012) Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems. Journal of Optimization Theory and Applications, 152, 97-120. https://doi.org/10.1007/s10957-011-9891-6 |
| [3] | Zhou, Z.A., Yang, X.M. and Peng, J.W. (2012) ε-Strict Subdifferentials of Set-Valued Maps and Optimality Conditions. Nonlinear Analysis: Theory, Methods & Applications, 75, 3761-3775. https://doi.org/10.1016/j.na.2012.01.030 |
| [4] | Zhou, Z.A., Yang, X.M. and Peng, J.W. (2014) ε-Optimality Conditions of Vector Optimization Problems with set-Valued Maps Based on the Algebraic Interior in Real Linear Spaces. Optimization Letters, 8, 1047-1061. https://doi.org/10.1007/s11590-013-0620-y |
| [5] | Zhou, Z.A. and Peng, J.W. (2012) Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces. Journal of Optimization Theory and Applications, 154, 830-841. https://doi.org/10.1007/s10957-012-0045-2 |
| [6] | Zhou, Z.A. and Yang, X.M. (2014) Scalarization of ε-Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces. Journal of Optimization Theory and Applications, 162, 680-693. https://doi.org/10.1007/s10957-014-0565-z |
| [7] | Gutiérrez, C., Jiménez, B. and Novo, V. (2006) A Unified Approach and Optimality Conditions for Approximate Solutions of Vector Optimization Problems. SIAM Journal on Optimization, 17, 688-710. https://doi.org/10.1137/05062648X |
| [8] | Kuhn, H.W. and Tucker, A.W. (1951) Nonlinear Analysis. In: David, F.N. and Neyman, J., Eds., Proceeding of the Second Berley Symposium on Mathematical Statistics and Probability, University of California Press, California, 481-492. https://doi.org/10.1525/9780520411586-036 |
| [9] | Hartley, R. (1978) On Cone-Efficiency, Cone-Convexity and Cone-Compactness. SIAM Journal of Applied Mathematics, 34, 211-222. https://doi.org/10.1137/0134018 |
| [10] | Benson, H.P. (1979) An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones. Journal of Mathematical Analysis and Applications, 71, 232-241. https://doi.org/10.1016/0022-247X(79)90226-9 |
| [11] | Borwein, J.M. (1980) The Geometry of Pareto Efficiency Over Cones. Optimization, 11, 236-248. https://doi.org/10.1080/02331938008842650 |
| [12] | Henig, M.L. (1982) Proper Efficiency with Respect to Cones. Journal of Optimization Theory and Applications, 36, 387-407. https://doi.org/10.1007/BF00934353 |
| [13] | Borwein, J.M. and Zhuang, D.M. (1991) Super Efficiency in Convex Vector Optimization. Methods and Models of Operations Research, 35, 175-184. https://doi.org/10.1007/BF01415905 |
| [14] | Borwein, J.M. and Zhuang, D.M. (1993) Super Efficiency in Vector Optimization of Set-Valued Maps. Transaction of American Mathematical Society, 338, 105-122. https://doi.org/10.1090/S0002-9947-1993-1098432-5 |
| [15] | Yang, X.M., Yang, X.Q. and Chen, G.Y. (2000) Theorems of the Alternative and Optimization with Set-Valued Maps. Journal of Optimization and Applications, 107, 627-640. https://doi.org/10.1023/A:1026407517917 |
| [16] | 杨新民. Benson真有效解与Borwein真有效解的等价性[J]. 应用数学, 1994, 7(2): 246-247. |
| [17] | Zhao, K.Q., Yang, X.M. and Chen, G.Y. (2015) Approximate Proper Efficiency in Vector Optimization. Optimization, 64, 1777-1793. https://doi.org/10.1080/02331934.2014.979818 |
| [18] | 付科程. 向量优化中基于回收锥的真有效解的性质研究[D]: [硕士学位论文]. 重庆: 重庆师范大学, 2015. |
| [19] | 赵克全, 戎卫东, 杨新民. 新的非线性分离定理及其在向量优化中的应用[J]. 中国科学: 数学, 2017, 47(4): 533-544. |
| [20] | 徐义红, 龙鑫灿, 黄斌. 集值均衡问题近似Benson真有效解的非线性刻画[J]. 运筹学学报, 2021, 25(4): 80-90. |
| [21] | Chicco, M., Mignanego, F., Pusillo, L. and Tijs, S. (2011) Vector Optimization Problems via Improvement Sets. Journal of Optimization of Theory and Applications, 150, 516-529. https://doi.org/10.1007/s10957-011-9851-1 |
| [22] | Zhao, K.Q. and Yang, X.M. (2015) E-Benson Proper Efficiency in Vector Optimization. Optimization, 64, 739-752. https://doi.org/10.1080/02331934.2013.798321 |
| [23] | Mishra, S.K., Wang, S.Y. and Lai, K.K. (2009) Generalized Convexity and Vector Optimization. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85671-9 |
| [24] | Gutiérrez, C., Jiménez, B. and Novo, V. (2012) Improvement Sets and Vector Optimization. European Journal of Operational Research, 223, 304-311. https://doi.org/10.1016/j.ejor.2012.05.050 |
| [25] | Zhou, Z.A., Liang, K.H. and Ansari, Q.H. (2020) Optimality Conditions for Benson Proper Efficiency of Set-Valued Equilibrium Problems. Journal of Inequalities and Applications, 2020, Article No. 87. https://doi.org/10.1186/s13660-020-02352-6 |
| [26] | Zhao, K.Q., Xia, Y.M. and Yang, X.M. (2015) Nonlinear Scalarization Characterizations of E-Efficiency in Vector Optimization. Taiwanese Journal of Mathematics, 19, 455-466. https://doi.org/10.11650/tjm.19.2015.4360 |
