Tính liên thông của tập nghiệm hữu hiệu yếu cho bài toán tối ưu tập không lồi
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Abstract
This paper considers a nonconvex set optimization problem and discusses connectedness conditions for its weakly efficient solution set. Firstly, various concepts of connectedness for a set-valued map are proposed. Secondly, sufficient conditions for the connectedness of an extension of the oriented distance of Hiriart-Urruty are formulated. Finally, the connectedness properties of a weakly efficient solution set to such problem are investigated via the extension of the oriented distance of Hiriart-Urruty.
Tóm tắt
Bài báo này xem xét một bài toán tối ưu tập không lồi và thảo luận các điều kiện liên thông cho tập nghiệm hữu hiệu yếu của nó. Đầu tiên, các khái niệm khác nhau về tính liên thông cho ánh xạ có giá trị tập được đề xuất. Thứ hai, các điều kiện đủ cho tính liên thông cho một dạng mở rộng của hàm khoảng cách định hướng của Hiriart-Urruty được trình bày. Cuối cùng, tính liên thông của tập nghiệm hữu hiệu yếu cho bài toán trên được nghiên cứu thông qua dạng mở rộng của hàm khoảng cách định hướng của Hiriart-Urruty.
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Tài liệu tham khảo
Alonso, M., & Rodríguez-Marín, L. (2005). Set-relations and optimality conditions in set-valued maps. Nonlinear Analysis: Theory, Methods & Applications, 63(8), 1167-1179. https://doi.org/10.1016/j.na.2005.06.002
Anh, L. Q., Anh. N. T., Duoc, P. T., Khanh, L. T. V., & Thu, P. T. A. (2022a). The connectedness of weakly and strongly efficient solution sets of nonconvex vector equilibrium problems. Applied Set-Valued Analysis and Optimization, 4(1), 109-127.
https://doi.org/10.23952/asvao.4.2022.1.08
Anh, L. Q., Duoc, P. T., & Duong, T. T. T. (2022b). Connectedness properties of the efficient sets and the nondominated sets to vector optimization problems. Optimization Letters, 1-12. https://doi.org/10.1007/s11590-021-01841-x
Anh, N. T., Dược, P. T., Khánh, L. T. V., & Thư, P. T. A. (2022). Tính liên thông của tập nghiệm hữu hiệu yếu cho bài toán tối ưu vector không lồi. Tạp chí Khoa học Trường Đại học Cần Thơ, 58(Giáo dục Đồng bằng sông Cửu Long), 1-9. https://doi.org/10.22144/ctu.jvn.2022.145
Araya, Y. (2012). Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Analysis: Theory, Methods & Applications, 75(9), 3821-3835. https://doi.org/10.1016/j.na.2012.02.004
Avriel, M., & Zang, I. (1980). Generalized arcwise-connected functions and characterizations of local-global minimum properties. Journal of Optimization Theory and Applications, 32(4), 407-425.
https://doi.org/10.1007/BF00934030
Gong, X. (1994). Connectedness of the efficient solution set of a convex vector optimization in normed spaces. Nonlinear Analysis: Theory, Methods & Applications, 23(9), 1105-1114. https://doi.org/10.1016/0362-546X(94)90095-7
Gong, X. H. (2007). Connectedness of the solution sets and scalarization for vector equilibrium problems. Journal of Optimization Theory and Applications, 133(2), 151-161. https://doi.org/10.1007/s10957-007-9196-y
Gopfert, A., Riahi, H., Tammer, C., & Zalinéscu, C. (2003). Variational Methods in Partially Ordered Spaces. Springer. Berlin. https://doi.org/10.1007/b97568
Gutiérrez, C., Miglierina, E., Molho, E., & Novo, V. (2012). Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 1822-1833.
Han, Y., & Huang, N. J. (2016). Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 12(3), 1135. https://doi.org/10.3934/jimo.2016.12.1135
Han, Y. (2020). Connectedness of weak minimal solution set for set optimization problems. Operations Research Letters, 48(6), 820-826. https://doi.org/10.1016/j.orl.2020.10.002
Hiriart-Urruty, J. B. (1979). Tangent cones, generalized gradients and mathematical programming in Banach spaces. Mathematics of operations research, 4(1), 79-97. https://doi.org/10.1287/moor.4.1.79
Huerga, L., Jiménez, B., Novo, V., & Vílchez, A. (2021). Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization. Mathematical Methods of Operations Research, 93(2), 413-436. https://doi.org/10.1007/s00186-020-00736-4
Jiménez, B., Novo, V., & Vílchez, A. (2018). A set scalarization function based on the oriented distance and relations with other set scalarizations. Optimization, 67(12), 2091-2116. https://doi.org/10.1080/02331934.2018.1533554
Jiménez, B., Novo, V., & Vílchez, A. (2020). Characterization of set relations through extensions of the oriented distance. Mathematical Methods of Operations Research, 91(1), 89-115. https://doi.org/10.1007/s00186-019-00661-1
Karuna, & Lalitha, C. S. (2019). External and internal stability in set optimization. Optimization, 68(4), 833-852. https://doi.org/10.1080/02331934.2018.1556663
Kassay, G., & Radulescu, V. (2018). Equilibrium problems and applications. Academic Press. https://doi.org/10.1016/C2015-0-06685-0
Khan, A. A., Tammer, C., & Zalinescu, C. (2016). Set-valued optimization. Springer-Verlag Berlin An.
https://doi.org/10.1007/978-3-642-54265-7
Khoshkhabar-amiranloo, S. (2019). Characterizations of generalized Levitin–Polyak well-posed set optimization problems. Optimization Letters, 13(1), 147-161. https://doi.org/10.1007/s11590-018-1258-6
Kuroiwa, D. (1998). The Natural Criteria in Set-Valued Optimization (NONLINEAR ANALYSIS AND CONVEX ANALYSIS). 数理解析研究所講究, 1031, 85-90.
Kuroiwa, D. (2003). Existence theorems of set optimization with set-valued maps. Journal of Information and Optimization sciences, 24(1), 73-84. https://doi.org/10.1080/02522667.2003.10699556
Luc, D. T. (1989). Theory of vector optimization. Springer.
https://doi.org/10.1007/978-3-642-50280-4
Qiu, Q. S., & Yang, X. M. (2012). Connectedness of Henig weakly efficient solution set for set-valued optimization problems. Journal of Optimization Theory and Applications, 152(2), 439-449. https://doi.org/10.1007/s10957-011-9906-3
Rockafellar, R. T. (1970). Convex analysis (Vol. 18). Princeton university press.
Warburton, A. R. (1983). Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives. Journal of optimization theory and applications, 40(4), 537-557. https://doi.org/10.1007/BF00933970
Xu, Y. D., & Li, S. J. (2014). Continuity of the solution set mappings to a parametric set optimization problem. Optimization Letters, 8(8), 2315-2327.
https://doi.org/10.1007/s11590-014-0738-6