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Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems
| 1. | Université Cadi Ayyad, Faculté Polydisciplinaire de Safi, B.P. 4162, Sidi Bouzid, Safi, Morocco |
| 2. | Laboratoire XLIM UMR-CNRS 6172, Université de Limoges Département de Mathématiques, 123 Avenue Albert Thomas 87060 Limoges Cedex, France, France |
References:
| [1] |
A. Aboussoror and P. Loridan, Existence of solutions to two-level optimization problems with nonunique lower-level solutions, Journal of Mathematical Analysis and Applications, 254 (2001), 348-357.
doi: 10.1006/jmaa.2000.7001. |
| [2] |
A. Aboussoror, Weak bilevel programming problems: Existence of solutions, in "Advances in Mathematics Research," 1, 83-92, Adv. Math. Res., Nova Sci. Publ., Hauppauge, NY, 2002. |
| [3] |
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), 399-408.
doi: 10.1016/j.jmaa.2004.09.033. |
| [4] |
A. Aboussoror and A. Mansouri, Existence of solutions to weak nonlinear bilevel problems via MinSup and D.C. problems, RAIRO Operations Research, 42 (2008), 87-102.
doi: 10.1051/ro:2008012. |
| [5] |
A. Aboussoror, Reverse convex programs: Stability and global optimality, Pacific Journal of Optimization, 5 (2009), 143-153. |
| [6] |
A. Aboussoror and S. Adly, A Fenchel-Lagrange duality approach for a bilevel programming problem with extremal-value function, Journal of Optimization Theory and Applications, 149 (2011), 254-268.
doi: 10.1007/s10957-011-9831-5. |
| [7] |
D. Azé, "Eléments d'Analyse Convexe et Variationnelle," Ellipses, Paris, 1997. |
| [8] |
M. Breton, A. Alj and A. Haurie, Sequential Stackelberg equilibria in two-person games, Journal of Optimization Theory and Applications, 59 (1988), 71-97.
doi: 10.1007/BF00939867. |
| [9] |
S. Dempe, "Foundations of Bilevel Programming," Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002. |
| [10] |
M. Dür, R. Horst and M. Locatelli, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217 (1998), 637-649.
doi: 10.1006/jmaa.1997.5745. |
| [11] |
J. Haberl, Maximization of generalized convex functionals in locally convex spaces, Journal of Optimization Theory and Applications, 121 (2004), 327-359.
doi: 10.1023/B:JOTA.0000037408.31141.e4. |
| [12] |
J.-B. Hiriart-Urruty, From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality, in "Nonsmooth Optimization and Related Topics," 219-239, Plenum Press, New York, 1989. |
| [13] |
J.-B. Hiriart-Urruty, Conditions for global optimality 2, Journal of Global Optimization, 13 (1998), 349-367.
doi: 10.1023/A:1008365206132. |
| [14] |
J.-B. Hiriart-Urruty and Y. S. Ledyaev, A note on the characterization of the global maxima of a (tangentially) convex function over a convex set, Journal of Convex Analysis, 3 (1996), 55-61. |
| [15] |
R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition, Springer-Verlag, Berlin, 1993. |
| [16] |
M. Laghdir, Optimality conditions in reverse convex optimization, Acta Mathematica Vietnamica, 28 (2003), 215-223. |
| [17] |
R. Lucchetti, F. Mignanego and G. Pieri, Existence theorem of equilibrium points in Stackelberg games with constraints, Optimization, 18 (1987), 857-866.
doi: 10.1080/02331938708843300. |
| [18] |
C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.," Technical Note, University of Dijon, France, 1987. |
| [19] |
R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. |
| [20] |
K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, 1997. |
| [21] |
K. Shimizu and Y. Ishizuka, Optimality conditions and algorithms for parameter design problems with two-level structure, IEEE Transactions on Automatic Control, 30 (1985), 986-993.
doi: 10.1109/TAC.1985.1103803. |
| [22] |
A. Strekalovsky, On global maximum search of convex functions over a feasible set, Journal of Numerical Mathematics and Mathematical Physics, 3 (1993), 349-363. |
| [23] |
A. Strekalovsky, Extremal problems on complements of convex sets, Cybernetics and System Analysis, 1 (1994), 88-100. |
| [24] |
A. Strekalovsky, Global optimality conditions for nonconvex optimization, Journal of Global Optimization, 12 (1998), 415-434.
doi: 10.1023/A:1008277314050. |
| [25] |
A. Strekalovsky, On convergence of a global search strategy for reverse convex problems, Journal of Applied Mathematics and Decision Sciences, 2005, 149-164.
doi: 10.1155/JAMDS.2005.149. |
| [26] |
T. Tanino and T. Ogawa, An algorithm for solving two-level convex optimization problems, Int. J. Systems Sci., 15 (1984), 163-174.
doi: 10.1080/00207728408926552. |
| [27] |
I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions, Journal of Applied Mathematics and Decision Sciences, 2006, 1-16.
doi: 10.1155/JAMDS/2006/29023. |
| [28] |
H. Tuy, Convex programs with an additional reverse convex constraint, Journal of Optimization Theory and Applications, 52 (1987), 463-486.
doi: 10.1007/BF00938217. |
| [29] |
H. Tuy and N. V. Thuong, On the global minimization of a convex function under general non convex constraints, Applied Mathematics and Optimization, 18 (1988), 119-142.
doi: 10.1007/BF01443618. |
| [30] |
H. Tuy, "Convex Analysis and Global Optimization," Nonconvex Optimization and its Applications, 22, Kluwer Academic Publishers, Dordrecht, 1998. |
show all references
References:
| [1] |
A. Aboussoror and P. Loridan, Existence of solutions to two-level optimization problems with nonunique lower-level solutions, Journal of Mathematical Analysis and Applications, 254 (2001), 348-357.
doi: 10.1006/jmaa.2000.7001. |
| [2] |
A. Aboussoror, Weak bilevel programming problems: Existence of solutions, in "Advances in Mathematics Research," 1, 83-92, Adv. Math. Res., Nova Sci. Publ., Hauppauge, NY, 2002. |
| [3] |
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), 399-408.
doi: 10.1016/j.jmaa.2004.09.033. |
| [4] |
A. Aboussoror and A. Mansouri, Existence of solutions to weak nonlinear bilevel problems via MinSup and D.C. problems, RAIRO Operations Research, 42 (2008), 87-102.
doi: 10.1051/ro:2008012. |
| [5] |
A. Aboussoror, Reverse convex programs: Stability and global optimality, Pacific Journal of Optimization, 5 (2009), 143-153. |
| [6] |
A. Aboussoror and S. Adly, A Fenchel-Lagrange duality approach for a bilevel programming problem with extremal-value function, Journal of Optimization Theory and Applications, 149 (2011), 254-268.
doi: 10.1007/s10957-011-9831-5. |
| [7] |
D. Azé, "Eléments d'Analyse Convexe et Variationnelle," Ellipses, Paris, 1997. |
| [8] |
M. Breton, A. Alj and A. Haurie, Sequential Stackelberg equilibria in two-person games, Journal of Optimization Theory and Applications, 59 (1988), 71-97.
doi: 10.1007/BF00939867. |
| [9] |
S. Dempe, "Foundations of Bilevel Programming," Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002. |
| [10] |
M. Dür, R. Horst and M. Locatelli, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217 (1998), 637-649.
doi: 10.1006/jmaa.1997.5745. |
| [11] |
J. Haberl, Maximization of generalized convex functionals in locally convex spaces, Journal of Optimization Theory and Applications, 121 (2004), 327-359.
doi: 10.1023/B:JOTA.0000037408.31141.e4. |
| [12] |
J.-B. Hiriart-Urruty, From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality, in "Nonsmooth Optimization and Related Topics," 219-239, Plenum Press, New York, 1989. |
| [13] |
J.-B. Hiriart-Urruty, Conditions for global optimality 2, Journal of Global Optimization, 13 (1998), 349-367.
doi: 10.1023/A:1008365206132. |
| [14] |
J.-B. Hiriart-Urruty and Y. S. Ledyaev, A note on the characterization of the global maxima of a (tangentially) convex function over a convex set, Journal of Convex Analysis, 3 (1996), 55-61. |
| [15] |
R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition, Springer-Verlag, Berlin, 1993. |
| [16] |
M. Laghdir, Optimality conditions in reverse convex optimization, Acta Mathematica Vietnamica, 28 (2003), 215-223. |
| [17] |
R. Lucchetti, F. Mignanego and G. Pieri, Existence theorem of equilibrium points in Stackelberg games with constraints, Optimization, 18 (1987), 857-866.
doi: 10.1080/02331938708843300. |
| [18] |
C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.," Technical Note, University of Dijon, France, 1987. |
| [19] |
R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. |
| [20] |
K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, 1997. |
| [21] |
K. Shimizu and Y. Ishizuka, Optimality conditions and algorithms for parameter design problems with two-level structure, IEEE Transactions on Automatic Control, 30 (1985), 986-993.
doi: 10.1109/TAC.1985.1103803. |
| [22] |
A. Strekalovsky, On global maximum search of convex functions over a feasible set, Journal of Numerical Mathematics and Mathematical Physics, 3 (1993), 349-363. |
| [23] |
A. Strekalovsky, Extremal problems on complements of convex sets, Cybernetics and System Analysis, 1 (1994), 88-100. |
| [24] |
A. Strekalovsky, Global optimality conditions for nonconvex optimization, Journal of Global Optimization, 12 (1998), 415-434.
doi: 10.1023/A:1008277314050. |
| [25] |
A. Strekalovsky, On convergence of a global search strategy for reverse convex problems, Journal of Applied Mathematics and Decision Sciences, 2005, 149-164.
doi: 10.1155/JAMDS.2005.149. |
| [26] |
T. Tanino and T. Ogawa, An algorithm for solving two-level convex optimization problems, Int. J. Systems Sci., 15 (1984), 163-174.
doi: 10.1080/00207728408926552. |
| [27] |
I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions, Journal of Applied Mathematics and Decision Sciences, 2006, 1-16.
doi: 10.1155/JAMDS/2006/29023. |
| [28] |
H. Tuy, Convex programs with an additional reverse convex constraint, Journal of Optimization Theory and Applications, 52 (1987), 463-486.
doi: 10.1007/BF00938217. |
| [29] |
H. Tuy and N. V. Thuong, On the global minimization of a convex function under general non convex constraints, Applied Mathematics and Optimization, 18 (1988), 119-142.
doi: 10.1007/BF01443618. |
| [30] |
H. Tuy, "Convex Analysis and Global Optimization," Nonconvex Optimization and its Applications, 22, Kluwer Academic Publishers, Dordrecht, 1998. |
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