A new method for strong-weak linear bilevel programming problem

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April 2015, 11(2): 529-547. doi: 10.3934/jimo.2015.11.529

A new method for strong-weak linear bilevel programming problem

Yue Zheng ^1, , Zhongping Wan ^2, , Shihui Jia ^3, and Guangmin Wang ^4,

1.	College of Mathematics and Physics, Huanggang Normal University, Huanggang 438000, China
2.	School of Mathematics and Statistics, Wuhan University, Wuhan, 430072
3.	School of Science, Wuhan University of Science and Technology, Wuhan 430081, China
4.	School of Economics and Management, China University of Geosciences, Wuhan 430074, China

Received October 2012 Revised April 2014 Published September 2014

Abstract
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We first propose an exact penalty method to solve strong-weak linear bilevel programming problem (for short, SWLBP) for every fixed cooperation degree from the follower. Then, we prove that the solution of penalized problem is also that of the original problem under some conditions. Furthermore, we give some properties of the optimal value function (as a function of the follower's cooperation degree) of SWLBP. Finally, we develop a method to acquire the critical points of the optimal value function without enumerating all values of the cooperation degree from the follower, and thus this function is also achieved. Numerical results show that the proposed methods are feasible.

Keywords: partial cooperation, pessimistic formulation, penalty method, critical point., optimistic formulation, Bilevel programming.

Mathematics Subject Classification: Primary: 90C26, 90C3.

Citation: 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

References:

[1]	A. Aboussoror and P. Loridan, Strong-weak Stackelberg problems in finite dimentional spaces,, Serdica Mathematical Journal, 21 (1995), 151. Google Scholar
[2]	A. Aboussoror and A. Mansouri, Weak linear bilevel programming problems: Existence of solutions via a penalty method,, Journal of Mathematical Analysis and Applications, 304* (2005), 399. doi: 10.1016/j.jmaa.2004.09.033. Google Scholar*
[3]	A. Aboussoror, S. Adly and V. Jalby, Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems,, Journal of Industrial and Management Optimization, 7 (2011), 559. doi: 10.3934/jimo.2011.7.559. Google Scholar
[4]	G. Anandalingam and D. J. White, A solution for the linear static Stackelberg problem using penalty function,, IEEE Transactions Automatic Control, 35 (1990), 1170. doi: 10.1109/9.58565. Google Scholar
[5]	J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications,, Kluwer Academic, (1998). doi: 10.1007/978-1-4757-2836-1. Google Scholar
[6]	M. Campelo, S. Dantas and S. Scheimberg, A note on a penalty function approach for solving bi-level linear programs,, Journal of Global Optimization, 16 (2000), 245. doi: 10.1023/A:1008308218364. Google Scholar
[7]	D. Cao and L. C. Leung, A partial cooperation model for non-unique linear two-level decision problems,, European Journal of Operational Research, 140* (2002), 134. doi: 10.1016/S0377-2217(01)00225-9. Google Scholar*
[8]	B. Colson, P. Marcotte and G. Savard, Bilevel programming: A survey,, 4OR: Q. J. Oper. Res., 3 (2005), 87. doi: 10.1007/s10288-005-0071-0. Google Scholar
[9]	B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization,, Ann. Oper. Res., 153* (2007), 235. doi: 10.1007/s10479-007-0176-2. Google Scholar*
[10]	S. Dassanayaka, Methods of Variational Analysis in Pessimistic Bilevel Programming,, PhD Thesis, (2010). Google Scholar
[11]	S. Dempe, Foundations of Bilevel Programming,, Nonconvex Optimization and its Applications Series, (2002). Google Scholar
[12]	S. Dempe, Annottated bibliography on bilevel programming and mathematical problems with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894. Google Scholar
[13]	S. Dempe and A. B. Zemkoho, The bilevel programming problem: Reformulations, constraint qualifications and optimality conditions,, Mathematical Programming, 138* (2013), 447. doi: 10.1007/s10107-011-0508-5. Google Scholar*
[14]	S. Dempe, B. S. Mordukhovich and A. B. Zemkoho, Necessary optimality conditions in pessimistic bilevel programming,, Optimization, 63 (2014), 505. doi: 10.1080/02331934.2012.696641. Google Scholar
[15]	J. L. Goffin, On convergence rates of subgradient optimization methods,, Mathematical Programming, 13 (1977), 329. doi: 10.1007/BF01584346. Google Scholar
[16]	P. Hansen, B. Jaumard and G. Savard, New branch-and-bound rules for linear bilevel programming,, SIAM J. on Scientific and Statistical Computing, 13 (1992), 1194. doi: 10.1137/0913069. Google Scholar
[17]	P. Loridan and J. Morgan, Weak via strong Stackelberg problem: New results,, Journal of Global Optimization, 8 (1996), 263. doi: 10.1007/BF00121269. Google Scholar
[18]	L. Mallozzi and J. Morgan, Hierarchical systems with weighted reaction set,, Nonlinear Optimization and Applications, (1996), 271. Google Scholar
[19]	R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970). Google Scholar
[20]	K. Shimizu, Y. Ishizuka and J. F. Bard, Nondifferentiable and Two-Level Mathematical Programming,, Kluwer Academic, (1997). doi: 10.1007/978-1-4615-6305-1. Google Scholar
[21]	M. Tawarmalani and N. V. Sahinidis, A polyhedral branch-and-cut approach to global optimization,, Mathematical Programming, 103 (2005), 225. doi: 10.1007/s10107-005-0581-8. Google Scholar
[22]	L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review,, Journal of Global Optimization, 5 (1994), 291. doi: 10.1007/BF01096458. Google Scholar
[23]	G. Wang, Z. Wan and X. Wang, Bibliography on bilevel programming,, Advances in Mathematics(in Chinese), 36 (2007), 513. Google Scholar
[24]	U. P. Wen and S. T. Hsu, Linear bilevel programming problems-a review,, Journal of Operational Research Society, 42 (1991), 125. Google Scholar
[25]	D. J. White and G. Anandalingam, A penalty function approach for solving bi-level linear programs,, Journal of Global Optimization, 3 (1993), 397. doi: 10.1007/BF01096412. Google Scholar
[26]	W. Wiesemann, A. Tsoukalas, P. Kleniati and B. Rustem, Pessimistic bilevel optimisation,, SIAM Journal on Optimization, 23 (2013), 353. doi: 10.1137/120864015. Google Scholar
[27]	J. Ye and D. Zhu, New necessary optimality conditions for bilevel programs by combining the mpec and value function approaches,, SIAM Journal on Optimization, 20 (2010), 1885. doi: 10.1137/080725088. Google Scholar

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References:

[1]	A. Aboussoror and P. Loridan, Strong-weak Stackelberg problems in finite dimentional spaces,, Serdica Mathematical Journal, 21 (1995), 151. Google Scholar
[2]	A. Aboussoror and A. Mansouri, Weak linear bilevel programming problems: Existence of solutions via a penalty method,, Journal of Mathematical Analysis and Applications, 304* (2005), 399. doi: 10.1016/j.jmaa.2004.09.033. Google Scholar*
[3]	A. Aboussoror, S. Adly and V. Jalby, Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems,, Journal of Industrial and Management Optimization, 7 (2011), 559. doi: 10.3934/jimo.2011.7.559. Google Scholar
[4]	G. Anandalingam and D. J. White, A solution for the linear static Stackelberg problem using penalty function,, IEEE Transactions Automatic Control, 35 (1990), 1170. doi: 10.1109/9.58565. Google Scholar
[5]	J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications,, Kluwer Academic, (1998). doi: 10.1007/978-1-4757-2836-1. Google Scholar
[6]	M. Campelo, S. Dantas and S. Scheimberg, A note on a penalty function approach for solving bi-level linear programs,, Journal of Global Optimization, 16 (2000), 245. doi: 10.1023/A:1008308218364. Google Scholar
[7]	D. Cao and L. C. Leung, A partial cooperation model for non-unique linear two-level decision problems,, European Journal of Operational Research, 140* (2002), 134. doi: 10.1016/S0377-2217(01)00225-9. Google Scholar*
[8]	B. Colson, P. Marcotte and G. Savard, Bilevel programming: A survey,, 4OR: Q. J. Oper. Res., 3 (2005), 87. doi: 10.1007/s10288-005-0071-0. Google Scholar
[9]	B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization,, Ann. Oper. Res., 153* (2007), 235. doi: 10.1007/s10479-007-0176-2. Google Scholar*
[10]	S. Dassanayaka, Methods of Variational Analysis in Pessimistic Bilevel Programming,, PhD Thesis, (2010). Google Scholar
[11]	S. Dempe, Foundations of Bilevel Programming,, Nonconvex Optimization and its Applications Series, (2002). Google Scholar
[12]	S. Dempe, Annottated bibliography on bilevel programming and mathematical problems with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894. Google Scholar
[13]	S. Dempe and A. B. Zemkoho, The bilevel programming problem: Reformulations, constraint qualifications and optimality conditions,, Mathematical Programming, 138* (2013), 447. doi: 10.1007/s10107-011-0508-5. Google Scholar*
[14]	S. Dempe, B. S. Mordukhovich and A. B. Zemkoho, Necessary optimality conditions in pessimistic bilevel programming,, Optimization, 63 (2014), 505. doi: 10.1080/02331934.2012.696641. Google Scholar
[15]	J. L. Goffin, On convergence rates of subgradient optimization methods,, Mathematical Programming, 13 (1977), 329. doi: 10.1007/BF01584346. Google Scholar
[16]	P. Hansen, B. Jaumard and G. Savard, New branch-and-bound rules for linear bilevel programming,, SIAM J. on Scientific and Statistical Computing, 13 (1992), 1194. doi: 10.1137/0913069. Google Scholar
[17]	P. Loridan and J. Morgan, Weak via strong Stackelberg problem: New results,, Journal of Global Optimization, 8 (1996), 263. doi: 10.1007/BF00121269. Google Scholar
[18]	L. Mallozzi and J. Morgan, Hierarchical systems with weighted reaction set,, Nonlinear Optimization and Applications, (1996), 271. Google Scholar
[19]	R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970). Google Scholar
[20]	K. Shimizu, Y. Ishizuka and J. F. Bard, Nondifferentiable and Two-Level Mathematical Programming,, Kluwer Academic, (1997). doi: 10.1007/978-1-4615-6305-1. Google Scholar
[21]	M. Tawarmalani and N. V. Sahinidis, A polyhedral branch-and-cut approach to global optimization,, Mathematical Programming, 103 (2005), 225. doi: 10.1007/s10107-005-0581-8. Google Scholar
[22]	L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review,, Journal of Global Optimization, 5 (1994), 291. doi: 10.1007/BF01096458. Google Scholar
[23]	G. Wang, Z. Wan and X. Wang, Bibliography on bilevel programming,, Advances in Mathematics(in Chinese), 36 (2007), 513. Google Scholar
[24]	U. P. Wen and S. T. Hsu, Linear bilevel programming problems-a review,, Journal of Operational Research Society, 42 (1991), 125. Google Scholar
[25]	D. J. White and G. Anandalingam, A penalty function approach for solving bi-level linear programs,, Journal of Global Optimization, 3 (1993), 397. doi: 10.1007/BF01096412. Google Scholar
[26]	W. Wiesemann, A. Tsoukalas, P. Kleniati and B. Rustem, Pessimistic bilevel optimisation,, SIAM Journal on Optimization, 23 (2013), 353. doi: 10.1137/120864015. Google Scholar
[27]	J. Ye and D. Zhu, New necessary optimality conditions for bilevel programs by combining the mpec and value function approaches,, SIAM Journal on Optimization, 20 (2010), 1885. doi: 10.1137/080725088. Google Scholar

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