# American Institute of Mathematical Sciences

## Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem

 1 School of Information and Mathematics, Yangtze University, Jingzhou 434023, China 2 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China 3 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

* Corresponding author: Jianlin Jiang. Email: jiangjianlin@nuaa.edu.cn

Received  August 2018 Revised  October 2018 Published  September 2019

Fund Project: Supported by the National Natural Science Foundation of China grant 11971230, 11771058, 91647204, 11571169. Supported by the Outstanding Youth Foundation of Hubei Province of China(2019CFA088)

In this paper, a linear bilevel multiobjective programming problem is concerned. Based on the method of replacing the lower level problem with its optimality conditions, and taking the complementary constraints as the penalty term of the upper level objectives, we obtain the exact penalized multiobjective programming problem $(P_{K})$. The concept of equilibrium point of problem $(P_{K})$ is introduced and its properties are analyzed. Thereafter, we propose a penalty method-based equilibrium point algorithm, which only needs to solve a series of linear programming problems, for the linear bilevel multiobjective programming problem. Numerical results showing viability of the equilibrium point approach are presented.

Citation: Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020102
##### References:

show all references

##### References:
Pareto optimal solution corresponding to the fixed weights
 Examples in this paper Pareto optimal solutions to the fixed weights of the upper level objectives $\eta=(0.4,0.6)$ Exam.2 $(x^{*},y^{*})=(0,0.5)$, $F(x^{*},y^{*})=(1,-2)$ Exam.3 $(x^{*},y^{*})=(0,1.0,1.0)$, $F(x^{*},y^{*})=(2,1)$
 Examples in this paper Pareto optimal solutions to the fixed weights of the upper level objectives $\eta=(0.4,0.6)$ Exam.2 $(x^{*},y^{*})=(0,0.5)$, $F(x^{*},y^{*})=(1,-2)$ Exam.3 $(x^{*},y^{*})=(0,1.0,1.0)$, $F(x^{*},y^{*})=(2,1)$
 [1] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [2] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [3] Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585 [4] Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial & Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529 [5] Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965 [6] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [7] Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33 [8] Yanqun Liu. An exterior point linear programming method based on inclusive normal cones. Journal of Industrial & Management Optimization, 2010, 6 (4) : 825-846. doi: 10.3934/jimo.2010.6.825 [9] Liming Sun, Li-Zhi Liao. An interior point continuous path-following trajectory for linear programming. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1517-1534. doi: 10.3934/jimo.2018107 [10] Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004 [11] Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020109 [12] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [13] Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697 [14] Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-9. doi: 10.3934/jimo.2018170 [15] Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049 [16] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [17] Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074 [18] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51 [19] Soodabeh Asadi, Hossein Mansouri. A Mehrotra type predictor-corrector interior-point algorithm for linear programming. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 147-156. doi: 10.3934/naco.2019011 [20] Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37

2018 Impact Factor: 0.545