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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

 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

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.
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. [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. [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. [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. [5] J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications,, Kluwer Academic, (1998). doi: 10.1007/978-1-4757-2836-1. [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. [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. [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. [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. [10] S. Dassanayaka, Methods of Variational Analysis in Pessimistic Bilevel Programming,, PhD Thesis, (2010). [11] S. Dempe, Foundations of Bilevel Programming,, Nonconvex Optimization and its Applications Series, (2002). [12] S. Dempe, Annottated bibliography on bilevel programming and mathematical problems with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894. [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. [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. [15] J. L. Goffin, On convergence rates of subgradient optimization methods,, Mathematical Programming, 13 (1977), 329. doi: 10.1007/BF01584346. [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. [17] P. Loridan and J. Morgan, Weak via strong Stackelberg problem: New results,, Journal of Global Optimization, 8 (1996), 263. doi: 10.1007/BF00121269. [18] L. Mallozzi and J. Morgan, Hierarchical systems with weighted reaction set,, Nonlinear Optimization and Applications, (1996), 271. [19] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970). [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. [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. [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. [23] G. Wang, Z. Wan and X. Wang, Bibliography on bilevel programming,, Advances in Mathematics(in Chinese), 36 (2007), 513. [24] U. P. Wen and S. T. Hsu, Linear bilevel programming problems-a review,, Journal of Operational Research Society, 42 (1991), 125. [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. [26] W. Wiesemann, A. Tsoukalas, P. Kleniati and B. Rustem, Pessimistic bilevel optimisation,, SIAM Journal on Optimization, 23 (2013), 353. doi: 10.1137/120864015. [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.

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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. [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. [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. [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. [5] J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications,, Kluwer Academic, (1998). doi: 10.1007/978-1-4757-2836-1. [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. [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. [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. [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. [10] S. Dassanayaka, Methods of Variational Analysis in Pessimistic Bilevel Programming,, PhD Thesis, (2010). [11] S. Dempe, Foundations of Bilevel Programming,, Nonconvex Optimization and its Applications Series, (2002). [12] S. Dempe, Annottated bibliography on bilevel programming and mathematical problems with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894. [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. [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. [15] J. L. Goffin, On convergence rates of subgradient optimization methods,, Mathematical Programming, 13 (1977), 329. doi: 10.1007/BF01584346. [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. [17] P. Loridan and J. Morgan, Weak via strong Stackelberg problem: New results,, Journal of Global Optimization, 8 (1996), 263. doi: 10.1007/BF00121269. [18] L. Mallozzi and J. Morgan, Hierarchical systems with weighted reaction set,, Nonlinear Optimization and Applications, (1996), 271. [19] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970). [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. [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. [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. [23] G. Wang, Z. Wan and X. Wang, Bibliography on bilevel programming,, Advances in Mathematics(in Chinese), 36 (2007), 513. [24] U. P. Wen and S. T. Hsu, Linear bilevel programming problems-a review,, Journal of Operational Research Society, 42 (1991), 125. [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. [26] W. Wiesemann, A. Tsoukalas, P. Kleniati and B. Rustem, Pessimistic bilevel optimisation,, SIAM Journal on Optimization, 23 (2013), 353. doi: 10.1137/120864015. [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.
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