# American Institute of Mathematical Sciences

June  2020, 13(6): 1743-1755. doi: 10.3934/dcdss.2020102

## 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, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102
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##### 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 $(x^{*},y^{*})=(0,1.0,1.0)$, $F(x^{*},y^{*})=(2,1)$
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