Sự tồn tại nghiệm cho bài toán tối ưu vector dựa trên nón thứ tự kết hợp
Abstract
In this paper, a new cone is introduced and the convexity of strongly efficient solution sets to vector optimization problems via this cone is also discussed. Firstly, a mixed-ordered cone based on the positive Orthant cone, the Lorentz cone, and the Lexicographic cone is proposed. Then, the properties of the above cone and the relationships between it and others cones are observed. Finally, existence conditions and the convexity of the solution set to the vector optimization problem are formulated.
Tóm tắt
Trong bài báo này, một nón mới được giới thiệu và tính lồi của tập nghiệm hữu hiệu mạnh của một bài toán tối ưu vector thông qua nón mới này cũng được thảo luận. Đầu tiên, một nón thứ tự kết hợp dựa trên nón Orthant dương, nón Lorentz và nón từ điển được giới thiệu. Sau đó, các tính chất của nón này và mối quan hệ giữa nó với các nón khác được khảo sát. Cuối cùng, điều kiện tồn tại và tính lồi của tập nghiệm của bài toán tối ưu vector dựa trên nón mới được thiết lập.
Article Details
Tài liệu tham khảo
Anh, L. Q., Anh, N. T., Duoc, P. T, Khanh, L. T. V., & Thu, P. T. A. (2022). The connectedness of weakly and strongly efficient solution sets of nonconvex vector equilibrium problems, Applied Set-Valued Analysis Optimization, 4, 109-127.
Anh, L. Q., & Duy, T. Q. (2018). On penalty method for equilibrium problems in lexicographic order. Positivity, 22, 39-57.
Anh, L. Q., Duy, T. Q., & Hien, D. V. (2019). Stability for parametric vector quasi-equilibrium problems with variable cones. Numerical Functional Analysis and Optimization, 40, 461-483.
Anh, L. Q., & Danh, N. H. (2016). 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.
Anh, L. Q., Duy, T. Q., & Khanh, P. Q. (2016). Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity, 20, 61-80.
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.
Ansari, Q. H, Köbis, E., & Yao, J. C. (2018). Vector Variational inequalities and vector optimization, Springer, Berlin.
Aubin, J. P., & Frankowska, H. (1990). Set-valued
analysis, Birkhäuser Boston Inc., Boston.
Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). Nonlinear programming: theory and algorithms. John Wiley & Sons.
Bao, T. Q., & Tammer, C. (2019). Scalarization functionals with uniform level sets in set optimization. Journal of Optimization Theory and Applications, 182(1), 310-335.
Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2013). Nonlinear programming: theory and algorithms. John Wiley & Sons.
Bednarczuk, E. (1987). Well posedness of vector optimization problems. In recent advances and historical development of vector optimization (pp. 51-61). Springer, Berlin, Heidelberg.
Bianchi, M., Konnov, I. V., & Pini, R. (2010). Lexicographic and sequential equilibrium problems. Journal of Global Optimization, 46, 551-560.
Bourbaki, N. (2013). General topology: Chapters 1–4, Springer, Berlin.
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.
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.
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.
Ehrgott, M. (2005). Multicriteria optimization (Vol. 491). Springer Science & Business Media.
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.
Gajardo, P., & Seeger, A. (2014). Equilibrium problems involving the Lorentz cone. Journal of Global Optimization, 58(2), 321-340.
Gutiérrez, C., Huerga, L., Köbis, E., Tammer, C. (2021). A scalarization scheme for binary relations with applications to set-valued and robust optimization. Journal of Global Optimization, 79(1), 233-256.
Helbig, S. (1990). On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces. Journal of Optimization Theory and Applications, 65(2), 257-270.
Hirschberger, M. (2005). Connectedness of efficient points in convex and convex transformable vector optimization. Optimization, 54(3), 283-304.
Huang, X. X., & Yang, X. Q. (2006). Generalized Levitin--Polyak well-posedness in constrained optimization. SIAM Journal on Optimization, 17(1), 243-258.
Jahn, J. (2009). Vector optimization, Springer, Berlin.
Khoshkhabar-Amiranloo, S., Khorram, E. (2016). Scalar characterizations of cone-continuous set-valued maps. Applicable Analysis, 95(12), 2750-2765.
Kim, D. S., Phạm, T. S., & Tuyen, N. V. (2019). On the existence of Pareto solutions for polynomial vector optimization problems. Mathematical Programming, 177(1), 321-341.
Konnov, I. V. (2003). On lexicographic vector equilibrium problems. Journal of Optimization Theory and Applications, 118, 681-688.
Lalitha, C. S., & Chatterjee, P. (2014). Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. Journal of Global Optimization, 59(1), 191-205.
Luc, D. T (1989). Theory of vector optimization: lecture notes in economics and mathematical systems, vol. 319, Springer, Berlin.
Lucchetti, R. E., & Miglierina, E. (2004). Stability for convex vector optimization problems. Optimization, 53(5-6), 517-528.
Peng, Z. Y., Peng, J. W., Long, X. J., & Yao, J. C. (2018). On the stability of solutions for semi-infinite vector optimization problems. Journal of Global Optimization, 70(1), 55-69.
Peng, Z. Y., Wang, X., & Yang, X. M. (2019). Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems. Set-Valued and Variational Analysis, 27(1), 103-118.
Pervin, W. J. (2014). Foundations of general topology. Academic Press, London.
Sawaragi, Y., Nakayama, H., & Tanino, T. (Eds.). (1985). Theory of multiobjective optimization. Elsevier.
Sergienko, I. V., Lebedeva, T. T., & Semenova, N. V. (2000). Existence of solutions in vector optimization problems. Cybernetics and Systems Analysis, 36(6), 823-828.
Tanino, T. (1988). Stability and sensitivity analysis in convex vector optimization. SIAM Journal on Control and Optimization, 26(3), 521-536.
Tuyen, N. V. (2016). Convergence of the relative Pareto efficient sets. Taiwanese Journal of Mathematics, 20(5), 1149-1173.
Ustun, D., Carbas, S., & Toktas, A. (2021). Multi-objective optimization of engineering design problems through pareto-based bat algorithm. In Applications of Bat Algorithm and its Variants (pp. 19-43). Springer, Singapore.
Wu, J., & Chen, J. S. (2012). A proximal point algorithm for the monotone second-order cone complementarity problem. Computational Optimization and Applications, 51(3), 1037-1063.