# American Institute of Mathematical Sciences

July  2019, 15(3): 1213-1223. doi: 10.3934/jimo.2018092

## Linear bilevel multiobjective optimization problem: Penalty approach

 1 School of Information and Mathematics, Yangtze University, Jingzhou 434023, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Yibing Lv

Received  May 2017 Revised  October 2017 Published  July 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 11771058, 11201039, 71471140, 91647204

In this paper, we are interested by the linear bilevel multiobjective programming problem, where both the upper level and the lower level have multiple objectives. We approach this problem via an exact penalty method. Then, we propose an exact penalty function algorithm. Numerical results showing viability of the algorithm proposed are presented.

Citation: 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
##### References:

show all references

##### References:
The Pareto optimal solution obtained in this paper
 Exam. No. The Pareto optimal solution $(x^{*}, y^{*})$ obtained in this paper Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
 Exam. No. The Pareto optimal solution $(x^{*}, y^{*})$ obtained in this paper Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
Results in this paper and that in [15]
 Exam. No. Results in this paper Results by the algorithm in [15] Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ $(x^{*}, y^{*})=(0.6, 2.4, 0, 0)$ $F(x^{*}, y^{*})=(3.521, 7.958)$ $F(x^{*}, y^{*})=(5.4, 4.2)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
 Exam. No. Results in this paper Results by the algorithm in [15] Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ $(x^{*}, y^{*})=(0.6, 2.4, 0, 0)$ $F(x^{*}, y^{*})=(3.521, 7.958)$ $F(x^{*}, y^{*})=(5.4, 4.2)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
 [1] 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, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020102 [2] 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 [3] Yanqin Bai, Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 543-556. doi: 10.3934/jimo.2014.10.543 [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] Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333 [8] 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 [9] 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 [10] 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 [11] 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 [12] Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477 [13] Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177 [14] Le Thi Hoai An, Tran Duc Quynh, Pham Dinh Tao. A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 167-185. doi: 10.3934/naco.2012.2.167 [15] Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323 [16] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [17] Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385 [18] Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497 [19] Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033 [20] Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019102

2018 Impact Factor: 1.025

## Tools

Article outline

Figures and Tables