Lâm Quốc Anh , Phạm Thanh Dược , Võ Thị Mộng Thuý * Đặng Thị Mỹ Vân

* Tác giả liên hệ (vtmthuy@tdu.edu.vn)

Abstract

This paper focuses on studying parametric vector optimization problems via improvement sets and investigating the Hausdorff continuity of weakly efficient solution mappings of these problems. Firstly, properties of improvement sets are discussed. Then, models of parametric vector optimization problems via improvement sets and their weakly efficient solutions are introduced. Finally, by using the properties of improvement sets and convexity conditions of a vector-valued mapping, sufficient conditions for the Hausdorff continuity of these weak efficient solution mappings are investigated.

Keywords: Hausdorff continuity, improvement set, vector optimization problem, weakly efficient solution

Tóm tắt

Trong bài báo này, mô hình bài toán tối ưu vector phụ thuộc tham số được tập trung nghiên cứu thông qua tập cải tiến và khảo sát tính liên tục Hausdorff của ánh xạ nghiệm hữu hiệu yếu cho các bài toán này. Trước tiên, một số tính chất của tập cải tiến được xây dựng. Sau đó, mô hình bài toán tối ưu vector thông qua tập cải tiến và nghiệm hữu hiệu yếu của chúng được đề xuất. Cuối cùng, bằng cách sử dụng các tính chất của tập cải tiến và tính lồi của hàm có giá trị vector, các điều kiện đủ cho tính liên tục Hausdorff của các ánh xạ nghiệm hữu hiệu yếu này được khảo sát.

Từ khóa: Bài toán tối ưu vector, nghiệm hữu hiệu yếu, tập cải tiến, tính liên tục Hausdorff

Article Details

Tài liệu tham khảo

Anh, L. Q., & Khanh, P. Q. (2007). On the stability of the solution sets of general multivalued vector quasiequilibrium problems. Journal of Optimization Theory and Applications, 135(2), 271-284.https://doi.org/10.1007/s10957-007-9250-9

Anh, L. Q., Duoc, P. T., & Tam, T. N. (2017). Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 13(4), 1685-1699.https://doi.org/10.3934/jimo.2017013

Anh, L. Q., Duoc, P. T., & Tam, T. N. (2020). On the stability of approximate solutions to set-valued equilibrium problems. Optimization, 69(7-8), 1583-1599.https://doi.org/10.1080/02331934.2019.1646744

Berge, C. (1963). Topological spaces. Oliver and Boyd, London, 213 pages.

Chicco, M., Mignanego, F., Pusillo, L., & Tijs, S. (2011). Vector optimization problems via improvement sets. Journal of Optimization Theory and Applications, 150(3), 516-529.https://doi.org/10.1007/s10957-011-9851-1

Chicco, M., & Rossi, A. (2015). Existence of optimal points via improvement sets. Journal of Optimization Theory and Applications, 167(2), 487-501.https://doi.org/10.1007/s10957-015-0744-6

Dhingra, M., & Lalitha, C. S. (2017). Set optimization using improvement sets. Yugoslav Journal of Operations Research, 27(2), 153-167.https://doi.org/10.2298/YJOR170115011D

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

Flores-Bazán, F., & Jiménez, B. (2009). Strict efficiency in set-valued optimization. Siam Journal on Control and Optimization, 48(2), 881-908. https://doi.org/10.1137/07070139X

Fu, J. Y., & Wang, Y. H. (2003). Arcwise connected cone-convex functions and mathematical programming. Journal of Optimization Theory and Applications, 118(2), 339-352.https://doi.org/10.1023/A:1025451422581

Göpfert, A., Riahi, H., Tammer, C., &Zalinescu, C. (2003). Variational methods in partially ordered spaces, Springer, Berlin.

Gutiérrez, C., Jiménez, B., & Novo, V. (2012). Improvement sets and vector optimization, European J. Oper. Res, 223(2), 304-311.https://doi.org/10.1016/j.ejor.2012.05.050

Hernández, E., Rodríguez-Marín, L., & Sama, M. (2010). On solutions of set-valued optimization problems. Computers and Mathematics with Applications, 60(5), 1401-1408. https://doi.org/10.1016/j.camwa.2010.06.022

Hu, S., & Papageorgiou, N. (1997). Handbook of multivalued analysis, Volume I: Theory, Kluwer, Boston.https://doi.org/10.1007/978-1-4615-6359-4

Jahn, J. (2009). Vector optimization, Springer, Berlin, 470 pages.

Jahn, J., & Ha, T. X. D. (2011). New order relations in set optimization. Journal of Optimization Theory and Applications, 148(2), 209-236.https://doi.org/10.1007/s10957-010-9752-8

Khan, A. A., Tammer, C., & Zalinescu, C. (2016). Set-valued optimization, Springer, Berlin, 781 pages.https://doi.org/10.1007/978-3-642-54265-7

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

Lalitha, C. S., & Chatterjee, P. (2015). Stability and scalarization in vector optimization using improvement sets. Journal of Optimization Theory and Applications, 166(3), 825-843.https://doi.org/10.1007/s10957-014-0686-4

Liang, H., Wan, Z., & Zhang, L. (2020). The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets. Journal of Inequalities and Applications, 2020(1), 1-14. https://doi.org/10.1186/s13660-020-02397-7

Luc, D.T. (2005). Generalized convexity in vector optimization. In Luc, D.T. (Eds.), Handbook of generalized convexity and generalized monotonicity, 195-236.

Maeda, T. (2012). On optimization problems with set-valued objective maps: existence and optimality. Journal of Optimization Theory and Applications, 153(2), 263-279. https://doi.org/10.1007/s10957-011-9952-x

Mao, J., Wang, S., & Han, Y. (2019). The stability of the solution sets for set optimization problems via improvement sets. Optimization, 68(11), 2171-2193. https://doi.org/10.1080/02331934.2019.1579813

Oppezzi, P., & Rossi, A. (2015). Improvement sets and convergence of optimal points. Journal of Optimization Theory and Applications, 165(2), 405-419. https://doi.org/10.1007/s10957-014-0669-5

Qiu, Q., & Yang, X. (2010). Some properties of approximate solutions for vector optimization problem with set-valued functions. Journal of Global Optimization, 47(1), 1-12. https://doi.org/10.1007/s10898-009-9452-9

Sach, P. H. (2005). New generalized convexity notion for set-valued maps and application to vector optimization. Journal of Optimization Theory and Applications, 125(1), 157-179. https://doi.org/10.1007/s10957-004-1716-4

Wei, H. Z., Zuo, X., & Chen, C. R. (2020). Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis. Numerical Algebra, Control & Optimization, 10(1), 107. https://doi.org/10.3934/naco.2019036

Zhao, K. Q., & Yang, X. M. (2013). A unified stability result with perturbations in vector optimization. Optimization Letters, 7(8), 1913-1919. https://doi.org/10.1007/s11590-012-0533-1

Zhao, K. Q., Yang, X. M., & Peng, J. W. (2013). Weak E-optimal solution in vector optimization. Taiwanese Journal ofMathematics, 17(4), 1287-1302. https://doi.org/10.1007/s11590-012-0533-1

Zhao, K. Q., & Yang, X. M. (2014). E-proper saddle points and E-proper duality in vector optimization with set-valued maps. Taiwanese Journal of Mathematics, 18(2), 483-495.https://doi.org/10.11650/tjm.18.2014.3473

Zhao, K. Q., & Yang, X. M. (2015). E-Benson proper efficiency in vector optimization. Optimization, 64(4), 739-752. https://doi.org/10.1080/02331934.2013.798321