Robust optimality and duality for bilevel optimization problems under uncertain data

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Document Type : Original paper

Authors

1 Department of Mathematics, National Institute of Technology Mizoram, Aizawl, 796012, Mizoram, India

2 Department of Mathematics, Satish Chandra College, Ballia, 277001, Uttar Pradesh, India

Abstract

The exploration of robust bilevel programming problems is a relatively new development in optimization theory. In this study, we examine a bilevel optimization problem in which both the upper-level and lower-level constraints involve uncertainty. By reducing the problem to a single-level, nonlinear and non-smooth program, we explore sufficient optimality conditions and duality theorems for robust optimal solutions of the considered non-smooth uncertain bilevel optimization problem, using Clarke subdifferentials. Leveraging the characteristics of Clarke subdifferentials, we propose Wolfe-type robust dual models. Additionally, we establish various duality theorems, including weak and strong robust duality, in terms of Clarke subdifferentials. Several illustrative examples are presented to confirm the applicability of the results developed.

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References

[1] J.F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Springer Science & Business Media, New York, 2013.
[2] Y. Beck, I. Ljubić, and M. Schmidt, A survey on bilevel optimization under uncertainty, European J. Oper. Res. 311 (2023), no. 2, 401–426.  https://doi.org/10.1016/j.ejor.2023.01.008
[3] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009. https://doi.org/10.1515/9781400831050
[4] J. Chen, E. Köbis, and J.C. Yao, Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints, J. Optim. Theory Appl. 181 (2019), no. 2, 411–436. https://doi.org/10.1007/s10957-018-1437-8
[5] T.D. Chuong, Optimality and duality for robust multiobjective optimization problems, Nonlinear Anal. 134 (2016), 127–143.  https://doi.org/10.1016/j.na.2016.01.002
[6] T.D. Chuong, Optimality conditions for nonsmooth multiobjective bilevel optimization problems, Ann. Oper. Res. 287 (2020), no. 2, 617–642.  https://doi.org/10.1007/s10479-017-2734-6
[7] T.D. Chuong, Robust optimality and duality in multiobjective optimization problems under data uncertainty, SIAM J. Optim. 30 (2020), no. 2, 1501–1526.  https://doi.org/10.1137/19M1251461
[8] F.H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics, 1990.
[9] S. Dempe, Foundations of Bilevel Programming, Springer Science & Business Media, New York, 2002.
[10] S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization 52 (2003), no. 3, 333–359.  https://doi.org/10.1080/0233193031000149894
[11] S. Dempe, Bilevel optimization: Theory, algorithms, applications and a bibliography, Bilevel Optimization: Advances and Next Challenges (S. Dempe and A. Zemkoho, eds.), vol. 161, Springer International Publishing, Cham, 2020, pp. 581–672.
[12] S. Dempe and P. Mehlitz, Semivectorial bilevel programming versus scalar bilevel programming, Optimization 69 (2020), no. 4, 657–679. https://doi.org/10.1080/02331934.2019.1625900
[13] N.A. Gadhi, K. Hamdaoui, and M. El idrissi, Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a Ψ-reformulation, Optimization 69 (2020), no. 4, 681–702. https://doi.org/10.1080/02331934.2019.1625901
[14] N.A. Gadhi and M. Ohda, Sufficient optimality conditions for a robust multiobjective problem, Asia-Pac. J. Oper. Res. 40 (2023), no. 3, 2250027.  https://doi.org/10.1142/S0217595922500270
[15] N.A. Gadhi and M. Ohda, Applying tangential subdifferentials in bilevel optimization, Optimization 73 (2024), no. 9, 2919–2932.  https://doi.org/10.1080/02331934.2023.2231501
[16] N.A. Gadhi and M. Ohda, Necessary optimality conditions for strictly robust bilevel optimization problems, Optimization (2024), 1–23.  https://doi.org/10.1080/02331934.2024.2370428
[17] J. Goseling and M. Lopuhaä-Zwakenberg, Robust optimization for local differential privacy, 2022 IEEE Int. Sym. Inf. Theory (ISIT) (Espoo, Finland), 2022, pp. 1629–1634.
[18] N.C. Hung, T.D. Chuong, and N.L.H. Anh, Optimality and duality for robust optimization problems involving inters- ection of closed sets, J. Optim. Theory Appl. 202 (2024), no. 2, 771–794.  https://doi.org/10.1007/s10957-024-02447-w
[19] V. Jeyakumar, G. Li, and G.M. Lee, Robust duality for generalized convex programming problems under data uncertainty, Nonlinear Anal. Theory Methods Appl. 75 (2012), no. 3, 1362–1373. https://doi.org/10.1016/j.na.2011.04.006
[20] G.M. Lee and P.T. Son, On nonsmooth optimality theorems for robust optimization problems, Bull. Korean Math. Soc. 51 (2014), no. 1, 287–301.  https://doi.org/10.4134/BKMS.2014.51.1.287
[21] F. Lin, X. Fang, and Z. Gao, Distributionally robust optimization: A review on theory and applications, Numer. Algebra Control Optim. 12 (2022), no. 1, 159–212.  http://dx.doi.org/10.3934/naco.2021057
[22] P. Mehlitz, L.I. Minchenko, and A.B. Zemkoho, A note on partial calmness for bilevel optimization problems with linearly structured lower level, Optim. Lett. 15 (2021), no. 4, 1277–1291. https://doi.org/10.1007/s11590-020-01636-6
[23] J. V. Outrata, On the numerical solution of a class of Stackelberg problems, Zeitschrift f¨ur Operations Research 34 (1990), no. 4, 255–277.  https://doi.org/10.1007/BF01416737
[24] A. Ren, Solving the general fuzzy random bilevel programming problem through me measure-based approach, IEEE Access 6 (2018), 25610–25620.  https://doi.org/10.1109/ACCESS.2018.2828706
[25] S. Saini, N. Kailey, and I. Ahmad, Optimality conditions and duality results for a robust bi-level programming problem, RAIRO Oper. Res. 57 (2023), no. 2, 525–539.  https://doi.org/10.1051/ro/2023026
[26] S. Wu and S. Chen, A bi-level algorithm for product line design and pricing, 2014 IEEE International Conference on Industrial Engineering and Engineering Management, 2014, pp. 14–18.
[27] J. Xiao, Y. Cai, Y. He, Y. Xie, and Z. Yang, A dual-randomness bi-level interval multi-objective programming model for regional water resources management, J. Contam. Hydrol. 241 (2021), 103816. https://doi.org/10.1016/j.jconhyd.2021.103816.
[28] J.J. Ye and D.L. Zhu, Optimality conditions for bilevel programming problems, Optimization 33 (1995), no. 1, 9–27. https://doi.org/10.1080/02331939508844060
[29] D. Zhang, H.H. Turan, R. Sarker, and D. Essam, Robust optimization approaches in inventory management: Part A—the survey, IISE Trans. 0 (2024), no. 0, In press.  https://doi.org/10.1080/24725854.2024.2381713
[30] T. Zhang, Z. Chen, Y. Zheng, and J. Chen, An improved simulated annealing algorithm for bilevel multiobjective programming problems with application, J. Nonlinear Sci. Appl. 9 (2016), no. 6, 3672–3685.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 20 March 2025

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