Nguyễn Thái Anh , Phạm Thanh Dược , Lâm Thị Vân Khánh Phạm Trần Anh Thư *

* Tác giả liên hệ (thupta9@fe.edu.vn)

Abstract

In this paper, the non-convex vector optimization problem was considered and discussed the properties of its weakly efficient solution set. Firstly, some concepts about generalized convexities of vector valued mappings were provided and studied their relationships. Next, based on the Hiriart-Urruty oriented distance function, a new nonlinear scalar function for the underlying problem was presented and investigated its pseudo semicontinuous property. Finally, these concepts and properties of the Hiriart-Urruty oriented distance function were used to formulate sufficient conditions for the existence and the connectedness of the weakly efficient solution set of the reference problem.

Keywords: Optimization problem, Hiriart-Urruty oriented distance funtion, scalar method, connectedness

Tóm tắt

Trong bài báo này, bài toán tối ưu vector không lồi được xem xét và thảo luận các tính chất của tập nghiệm hữu hiệu yếu đối với bài toán này. Trước hết, một số khái niệm về tính lồi tổng quát của ánh xạ giá trị vector được đưa ra và nghiên cứu các mối quan hệ của chúng. Tiếp đó, dựa trên hàm khoảng cách định hướng theo nghĩa Hiriart-Urruty, một hàm vô hướng phi tuyến mới cho bài toán đang xét được giới thiệu và nghiên cứu tính giả nửa liên tục của nó. Cuối cùng, các khái niệm và tính chất của hàm khoảng cách định hướng Hiriart-Urruty được sử dụng để thiết lập các điều kiện đủ cho sự tồn tại và tính liên thông của tập nghiệm hữu hiệu yếu của bài toán trên.

Từ khóa: Bài toán tối ưu, hàm khoảng cách định hướng Hiriart-Urruty, phương pháp vô hướng hóa, tính liên thông

Article Details

Tài liệu tham khảo

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(4), 461-483. https://doi.org/10.1080/01630563.2018.1556688

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 and Optimization 4(1), 109-127. https://doi.org/10.23952/asvao.4.2022.1.08

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

Bialas, W., & Karwan, M. (1982). On two-level optimization. IEEE transactions on automatic control, 27(1), 211-214. https://doi.org/10.1109/TAC.1982.1102880

Bourbaki, N. (2013). General Topology: Chapters 1–4 (18). Springer Science & Business Media.

Cheng, Y. (2001). On the connectedness of the solution set for the weak vector variational inequality. Journal of Mathematical Analysis and Applications, 260(1), 1-5. https://doi.org/10.1006/jmaa.2000.7389

Farajzadeh, A. P. (2015). On the convexity of the solution set of symmetric vector equilibrium problems. Filomat, 29(9), 2097-2105. https://doi.org/10.2298/FIL1509097F

Flores-Bazán, F. (2004). Semistrictly quasiconvex mappings and non-convex vector optimization. Mathematical Methods of Operations Research, 59(1), 129-145. https://doi.org/10.1007/s001860300321

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

Göpfert, A., Riahi, H., Tammer, C., & Zalinescu, C. (2006). Variational methods in partially ordered spaces. Springer Science and Business Media.

Han, Y., & Huang, N. J. (2018). Existence and connectedness of solutions for generalized vector quasi-equilibrium problems. Journal of Optimization Theory and Applications, 179(1), 65-85. https://doi.org/10.1007/s10957-016-1032-9

Han, Y., Wang, S. H., & Huang, N. J. (2019). Arcwise connectedness of the solution sets for set optimization problems. Operations Research Letters, 47(3), 168-172. https://doi.org/10.1016/j.orl.2019.03.005

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

Huong, N. T. T., Yao, J. C., & Yen, N. D. (2017). Connectedness structure of the solution sets of vector variational inequalities. Optimization, 66(6), 889-901. https://doi.org/10.1080/02331934.2016.1172073

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

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

Luc, D, T. (1989). Theory of vector optimization. Springer. https://doi.org/10.1007/978-3-642-50280-4

Morgan, J., & Scalzo, V. (2004). Pseudocontinuity in optimization and nonzero-sum games. Journal of optimization Theory and Applications, 120(1), 181-197. https://doi.org/10.1023/B:JOTA.0000012738.90889.5b

Rockafellar, R. T. (1970). Convex Analysis, Princeton Univ. Press, Princeton, NJ.

Ruíz-Canales, P., & Rufián-Lizana, A. (1995). A characterization of weakly efficient points. Mathematical Programming, 68(1), 205-212. https://doi.org/10.1007/BF01585765

Sun, E. J. (1996). On the connectedness of the efficient set for strictly quasiconvex vector minimization problems. Journal of Optimization Theory and Applications, 89(2), 475-481. https://doi.org/10.1007/BF02192541

Tanaka, T. (1994). Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. Journal of Optimization Theory and Applications, 81(2), 355-377. https://doi.org/10.1007/BF02191669

Xu, Y., & Zhang, P. (2018). Connectedness of solution sets of strong vector equilibrium problems with an application. Journal of Optimization Theory and Applications, 178(1), 131-152. https://doi.org/10.1007/s10957-018-1244-2

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