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Optimal selection of cleaner products in a green supply chain with risk aversion
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 |
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. Google Scholar |
[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. 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. |
[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. |
show all references
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. Google Scholar |
[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. 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. |
[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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