Biểu diễn nón Bishop-Phelps trong không gian hữu hạn chiều
Abstract
The aim of the article is to study representations of Bishop-Phelps cones in finite-dimensional spaces under various norms. First, the definitions of cones in finite-dimensional spaces are recalled, accompanied by examples illustrating Bishop-Phelps cones with both empty and non-empty interiors. Next, the article explores the properties of Bishop-Phelps cones. Finally, these cones are utilized to represent foundational cones in finite-dimensional spaces, including the non-negative Orthant cones, Lorentz cones, and other related cones.
Tóm tắt
Mục tiêu của bài báo là nghiên cứu sự biểu diễn của nón Bishop-Phelps trong không gian hữu hạn chiều dưới các chuẩn khác nhau. Đầu tiên, định nghĩa về các nón trong không gian hữu hạn chiều được nhắc lại, kèm theo các ví dụ minh họa về nón Bishop-Phelps có cả phần trong bằng rỗng và khác rỗng. Tiếp theo, bài báo xem xét các tính chất của nón Bishop-Phelps. Cuối cùng, những nón này được sử dụng để biểu diễn các nón cơ bản trong không gian hữu hạn chiều như nón Orthant không âm, nón Lorentz, và các nón có liên quan khác.
Article Details
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Tài liệu tham khảo
Anh, L. Q., & Danh, N. H. (2016a). Tính nửa liên tục trên của ánh xạ nghiệm bài toán cân bằng mạnh theo nón Lorentz. Tạp chí Khoa học Trường Đại học Cần Thơ, 43, 26-33. https://doi.org/10.22144/ctu.jvn.2016.161
Anh, L. Q., & Duy, T. Q. (2018). On penalty method for equilibrium problems in lexicographic order. Positivity, 22, 39-57. https://doi.org/10.1007/s11117-017-0496-7
Anh, L. Q., Duy, T. Q., & Khanh, P. Q. (2016b). Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity, 20, 61-80. https://doi.org/10.1007/s11117-015-0341-9
Anh, L. Q., Duy, T. Q., Kruger, A. Y., & Thao, N. H. (2014). Well-posedness for lexicographic vector equilibrium problems. In Constructive Nonsmooth Analysis and Related Topics (pp. 159-174), Springer, New York. https://doi.org/10.1007/978-1-4614-8615-2_10
Bednarczuk, E. M. (1996). Bishop-Phelps cones and convexity: applications to stability of vector optimization problems (Doctoral dissertation, INRIA).
Bianchi, M., Konnov, I. V., & Pini, R. (2010). Lexicographic and sequential equilibrium problems. Journal of Global Optimization, 46, 551-560.
https://doi.org/10.1007/s10898-009-9439-6
Bishop, E., & Phelps, R. R. (1962). The support functionals of a convex set, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R. I., 7, 27–35.
Bueno, M. I., Furtado, S., & Sivakumar, K. C. (2021). Linear maps preserving the Lorentz-cone spectrum in certain subspaces of . Banach Journal of Mathematical Analysis, 15(3), 1-20.
https://doi.org/10.1007/s43037-021-00140-y
Chang, Y. L., Huang, C. H., Chen, J. S., & Hu, C. C. (2018). Some inequalities for means defined on the Lorentz cone. Mathematical Inequalities and Applications, 21(4), 1015-1028. https://doi.org/10.7153/mia-2018-21-69
Dong, L., Tang, J., & Zhou, J. (2012). A smoothing Newton algorithm for solving the monotone second-order cone complementarity problems. Journal of Applied Mathematics and Computing, 40(1), 45-61. https://doi.org/10.1007/s12190-012-0550-3
Eichfelder, G., & Ha, T. X. D. (2013). Optimality conditions for vector optimization problems with variable ordering structures. Optimization, 62(5), 597-627. https://doi.org/10.1080/02331934.2011.575939
Eichfelder, G., & Pilecka, M. (2018). Ordering structures and their applications. Applications of Nonlinear Analysis, 265-304. https://doi.org/10.1007/978-3-319-89815-5_9
Fang, L., He, G., & Hu, Y. (2009). A new smoothing Newton-type method for second-order cone programming problems. Applied Mathematics and Computation, 215(3), 1020-1029. https://doi.org/10.1016/j.amc.2009.06.029
Ha, T. X. D., & Jahn, J. (2017). Properties of Bishop-Phelps cones. Journal of Nonlinear Convex Analysis, 18(3), 415-429.
Ha, T. X. D., & Jahn, J. (2023). Bishop–Phelps cones given by an equation in Banach spaces. Optimization, 72(5), 1309-1346. https://doi.org/10.1080/02331934.2021.2011870
James, R. C. (1972). Reflexivity and the sup of linear functionals. Israel Journal of Mathematics, 13(3-4), 289-300. https://doi.org/10.1007/BF02762803
Jahn, J. (2009a). Vector Optimization, Springer, Berlin, 470 pages.
Jahn, J. (2009b). Bishop-Phelps Cones in Optimization, International Journal of Optimization: Theory, Methods and Applications, 1, 123-139.
Luc, D. T. (1989). Theory of vector optimization, Springer, Berlin, 183 pages. https://doi.org/10.1007/978-3-642-50280-4
Konnov, I. V. (2003). On lexicographic vector equilibrium problems. Journal of Optimization Theory and Applications, 118, 681-688. https://doi.org/10.1023/B:JOTA.0000004877.39408.80
Petschke, M. (1990). On a theorem of Arrow, Barankin, and Blackwell. SIAM Journal on Control and Optimization, 28(2), 395-401. https://doi.org/10.1137/0328021
Phelps, R. R. (1974). Support cones in Banach spaces and their applications. Advances in Mathematics, 13(1), 1-19. https://doi.org/10.1016/0001-8708(74)90062-0