Nguyễn Thị Ngọc Như * Phạm Thanh Dược

* Tác giả liên hệ (nhum0720009@gstudent.ctu.edu.vn)

Abstract

This paper investigates bilevel optimization problems and their well-posedness. First, many kinds of approximate solutions to such problems are defined, and then based on these approximate solutions, various kinds of well-posedness for the underlying problems are introduced. By using conditions related to the continuity properties of multivariable functions, sufficient conditions for the relationship between the mentioned well-posedness properties are formulated. Many examples are given to illustrate the obtained results.

Keywords: Approximate solutions, bilevel mathematical programming, bilevel optimization problem, well-posedness

Tóm tắt

Trong bài báo này, bài toán quy hoạch hai mức và tính chất đặt chỉnh của chúng được tập trung nghiên cứu. Trước hết, các dạng xấp xỉ nghiệm của bài toán đang xét được xây dựng và từ đó, các khái niệm đặt chỉnh theo nhiều nghĩa khác nhau của lớp bài toán này cũng được đề xuất. Bằng việc sử dụng các điều kiện liên quan đến tính liên tục của hàm nhiều biến, điều kiện đủ cho các mối quan hệ của các loại đặt chỉnh đã được đề xuất ở trên được thiết lập. Một số ví dụ minh họa cho kết quả nghiên cứu cũng được đưa ra.

Từ khóa: Bài toán quy hoạch hai mức, Nghiệm xấp xỉ, Sự đặt chỉnh

Article Details

Tài liệu tham khảo

Bednarczuck, E. (1994). An approach to well-posedness in vector optimization: consequences to stability. Control and cybernetics, 23, 107-122.

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

Chen, G. Y., Huang, X., & Yang, X. (2006). Vector optimization: set-valued and variational analysis. Springer Science & Business Media.

Camacho-Vallejo, J. F., González-Rodríguez, E., Almaguer, F. J., & González-Ramírez, R. G. (2015). A bilevel optimization model for aid distribution after the occurrence of a disaster. Journal of Cleaner Production, 105, 134-145. https://doi.org/10.1016/j.jclepro.2014.09.069

Dempe, S., Kalashnikov, V. V., & Kalashnykova, N. (2006). Optimality conditions for bilevel programming problems. In Optimization with multivalued mappings, 3-28, Springer, Boston, MA. https://doi.org/10.1007/0-387-34221-4_1

Dempe, S. (2018). Bilevel optimization: theory, algorithms and applications. TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik.

Göpfert, A., Riahi, H., Tammer, C., & Zalinescu, C. (2003). Variational methods in partially ordered spaces. CMS Books in Mathematics.

Grötschel, M., Lovász, L., & Schrijver, A. (2012). Geometric algorithms and combinatorial optimization. Springer Science & Business Media.

Hu, S., & Papageorgiou, N. S. (1997). Handbook of Multivalued Analysis. Vol. I. Theory, vol. 419 of. Mathematics and its Applications. https://doi.org/10.1007/978-1-4615-6359-4

Hansen, E., & Walster, G. W. (Eds.). (2003). Global optimization using interval analysis: revised and expanded. CRC Press. https://doi.org/10.1201/9780203026922

John, J. (2004). Vector Optimization, Theory, Application, and Extensions.

Jiang, C., Han, X., & Xie, H. (2021). Nonlinear Interval Optimization for Uncertain Problems. Springer Verlag, Singapro. https://doi.org/10.1007/978-981-15-8546-3

Khan, A. A., Tammer, C., & Zalinescu, C. (2016). Set-valued optimization. Springer-Verlag Berlin An. https://doi.org/10.1007/978-981-15-8546-3

Kassay, G., & Radulescu, V. (2018). Equilibrium problems and applications. Academic Press.

Kis, T., Kovács, A., & Mészáros, C. (2021). On optimistic and pessimistic bilevel optimization models for demand response management. Energies, 14, 2095. https://doi.org/10.3390/en14082095

Lignola, M. B., & Morgan, J. (1997). Stability of regularized bilevel programming problems. Journal of Optimization Theory and Applications, 93(3), 575-596. https://doi.org/10.1023/A:1022695113803

Li, G., Tang, L., Huang, Y., & Yang, X. (2022). Stability for semivectorial bilevel programs. Journal of industrial & management optimization, 18(1), 427. https://doi.org/10.3934/jimo.2020161

Marti, K. (2005). Stochastic optimization methods. Berlin: Springer.

Miglierina, E., Molho, E., & Rocca, M. (2005). Well-posedness and scalarization in vector optimization. Journal of Optimization Theory and Applications, 126(2), 391-409. https://doi.org/10.1007/s10957-005-4723-1

Mehlitz, P., & Zemkoho, A. B. (2021). Sufficient optimality conditions in bilevel programming. Mathematics of operations research, 46(4), 1573-1598. https://doi.org/10.1287/moor.2021.1122

Pardalos, P. M., Žilinskas, A., & Žilinskas, J. (2017). Non-convex multi-objective optimization. New York: Springer International Publishing. https://doi.org/10.1007/978-3-319-61007-8

Sinha, A., Malo, P., & Deb, K. (2017). A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE transactions on evolutionary computation, 22(2), 276-295. https://doi.org/10.1109/TEVC.2017.2712906

Ye, J. J., & Zhu, D. (2010). New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM journal on optimization, 20(4), 1885-1905. https://doi.org/10.1137/080725088