# 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
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##### 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 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
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