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 and Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102
References:
[1]

Z. Ankhili and A. Mansouri, An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research, 197 (2009), 36-41.  doi: 10.1016/j.ejor.2008.06.026.

[2]

J. F. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.

[3]

H. I. Calvete and C. Gale, Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics, 234 (2010), 950-959.  doi: 10.1016/j.cam.2008.12.010.

[4]

M. Campelo and S. Scheimberg, A study of local solutions in linear bilevel programming, Journal of Optimization Theory and Applications, 125 (2005), 63-84.  doi: 10.1007/s10957-004-1711-9.

[5]

X. ChiZ. P. Wan and Z. Hao, Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem, Journal of Industrial and Management Optimization, 11 (2015), 1111-1125.  doi: 10.3934/jimo.2015.11.1111.

[6]

B. ColsonP. Marcotte and G. Savard, An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256.  doi: 10.1007/s10479-007-0176-2.

[7]

S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002.

[8]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical Programming, Ser.A, 131 (2012), 37-48.  doi: 10.1007/s10107-010-0342-1.

[9]

S. DempeN. Gadhi and A. B. Zemkoho, New optimality conditions for the semivectorial bilevel optimization problem, Journal of Optimization Theory and Applications, 157 (2013), 54-74.  doi: 10.1007/s10957-012-0161-z.

[10]

G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449.  doi: 10.1007/s10107-008-0259-0.

[11]

Y. B. Lv and Z. P. Wan, Linear bilevel multiobjective optimization problem: Penalty approach, Journal of Industrial and Management Optimization, 2018. doi: 10.3934/jimo.2018092.

[12]

Y. B. Lv, An exact penalty function approach for solving the linear bilevel multiobjective programming problem, Filomat, 29 (2015), 773-779.  doi: 10.2298/FIL1504773L.

[13]

Y. B. Lv and Z. P. Wan, Solving linear bilevel multiobjective programming problem via exact penalty function approach, Journal of Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0780-7.

[14]

I. Nishizaki and M. Sakawa, Stackelberg solutions to multiobjective two-level linear programming problem, Journal of Optimization Theory and Applications, 103 (1999), 161-182.  doi: 10.1023/A:1021729618112.

[15]

J. S. Pang, Three modeling paradiams in mathematical programming, Mathematical Programming, 125 (2010), 297-323.  doi: 10.1007/s10107-010-0395-1.

[16]

C. O. PieumeP. Marcotte and L. P. Fotso, Solving bilevel linear multiobjective programming problems, American Journal of Operations Research, 1 (2011), 214-219. 

[17]

L. Vicente and P. Calamai, Bilevel and multilevel programming: A bibligraphy review, Journal of Global Optimization, 5 (1994), 291-306.  doi: 10.1007/BF01096458.

[18]

D. J. White, Multiobjective programming and penalty function, Journal of Optimization Theory and Applications, 43 (1984), 583-599.  doi: 10.1007/BF00935007.

[19]

J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184.  doi: 10.1287/moor.1100.0480.

[20]

G. ZhangJ. Lu and T. Dillon, Decentralized multi-objective bilevel decision making with fuzzy demands, Knowledge-Based Systems, 20 (2007), 495-507. 

show all references

References:
[1]

Z. Ankhili and A. Mansouri, An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research, 197 (2009), 36-41.  doi: 10.1016/j.ejor.2008.06.026.

[2]

J. F. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.

[3]

H. I. Calvete and C. Gale, Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics, 234 (2010), 950-959.  doi: 10.1016/j.cam.2008.12.010.

[4]

M. Campelo and S. Scheimberg, A study of local solutions in linear bilevel programming, Journal of Optimization Theory and Applications, 125 (2005), 63-84.  doi: 10.1007/s10957-004-1711-9.

[5]

X. ChiZ. P. Wan and Z. Hao, Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem, Journal of Industrial and Management Optimization, 11 (2015), 1111-1125.  doi: 10.3934/jimo.2015.11.1111.

[6]

B. ColsonP. Marcotte and G. Savard, An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256.  doi: 10.1007/s10479-007-0176-2.

[7]

S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002.

[8]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical Programming, Ser.A, 131 (2012), 37-48.  doi: 10.1007/s10107-010-0342-1.

[9]

S. DempeN. Gadhi and A. B. Zemkoho, New optimality conditions for the semivectorial bilevel optimization problem, Journal of Optimization Theory and Applications, 157 (2013), 54-74.  doi: 10.1007/s10957-012-0161-z.

[10]

G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449.  doi: 10.1007/s10107-008-0259-0.

[11]

Y. B. Lv and Z. P. Wan, Linear bilevel multiobjective optimization problem: Penalty approach, Journal of Industrial and Management Optimization, 2018. doi: 10.3934/jimo.2018092.

[12]

Y. B. Lv, An exact penalty function approach for solving the linear bilevel multiobjective programming problem, Filomat, 29 (2015), 773-779.  doi: 10.2298/FIL1504773L.

[13]

Y. B. Lv and Z. P. Wan, Solving linear bilevel multiobjective programming problem via exact penalty function approach, Journal of Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0780-7.

[14]

I. Nishizaki and M. Sakawa, Stackelberg solutions to multiobjective two-level linear programming problem, Journal of Optimization Theory and Applications, 103 (1999), 161-182.  doi: 10.1023/A:1021729618112.

[15]

J. S. Pang, Three modeling paradiams in mathematical programming, Mathematical Programming, 125 (2010), 297-323.  doi: 10.1007/s10107-010-0395-1.

[16]

C. O. PieumeP. Marcotte and L. P. Fotso, Solving bilevel linear multiobjective programming problems, American Journal of Operations Research, 1 (2011), 214-219. 

[17]

L. Vicente and P. Calamai, Bilevel and multilevel programming: A bibligraphy review, Journal of Global Optimization, 5 (1994), 291-306.  doi: 10.1007/BF01096458.

[18]

D. J. White, Multiobjective programming and penalty function, Journal of Optimization Theory and Applications, 43 (1984), 583-599.  doi: 10.1007/BF00935007.

[19]

J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184.  doi: 10.1287/moor.1100.0480.

[20]

G. ZhangJ. Lu and T. Dillon, Decentralized multi-objective bilevel decision making with fuzzy demands, Knowledge-Based Systems, 20 (2007), 495-507. 

Table 1.  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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