American Institute of Mathematical Sciences

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July  2011, 7(3): 559-571. doi: 10.3934/jimo.2011.7.559

Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems

Abdelmalek Aboussoror 1, , Samir Adly 2, and  Vincent Jalby 2,

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

Received  July 2010 Revised  March 2011 Published  June 2011

In this paper, for a class of weak bilevel programming problems we provide sufficient conditions guaranteeing the existence of global solutions. These conditions are based on the use of reverse convex and convex maximization problems.
Keywords: convex analysis, reverse convex programs, d.c. programs., Two-level optimization.
Mathematics Subject Classification: 91A65, 90C26, 52A4.
Citation: Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Aboussoror, Reverse convex programs: Stability and global optimality, Pacific Journal of Optimization, 5 (2009), 143-153.  Google Scholar

[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.  Google Scholar

[7]

D. Azé, "Eléments d'Analyse Convexe et Variationnelle," Ellipses, Paris, 1997. Google Scholar

[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.  Google Scholar

[9]

S. Dempe, "Foundations of Bilevel Programming," Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-B. Hiriart-Urruty, Conditions for global optimality 2, Journal of Global Optimization, 13 (1998), 349-367. doi: 10.1023/A:1008365206132.  Google Scholar

[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.  Google Scholar

[15]

R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition, Springer-Verlag, Berlin, 1993.  Google Scholar

[16]

M. Laghdir, Optimality conditions in reverse convex optimization, Acta Mathematica Vietnamica, 28 (2003), 215-223.  Google Scholar

[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.  Google Scholar

[18]

C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.," Technical Note, University of Dijon, France, 1987. Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[20]

K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, 1997.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

A. Strekalovsky, Extremal problems on complements of convex sets, Cybernetics and System Analysis, 1 (1994), 88-100. Google Scholar

[24]

A. Strekalovsky, Global optimality conditions for nonconvex optimization, Journal of Global Optimization, 12 (1998), 415-434. doi: 10.1023/A:1008277314050.  Google Scholar

[25]

A. Strekalovsky, On convergence of a global search strategy for reverse convex problems,, Journal of Applied Mathematics and Decision Sciences, 2005 (): 149.  doi: 10.1155/JAMDS.2005.149.  Google Scholar

[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.  Google Scholar

[27]

I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions,, Journal of Applied Mathematics and Decision Sciences, 2006 (): 1.  doi: 10.1155/JAMDS/2006/29023.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H. Tuy, "Convex Analysis and Global Optimization," Nonconvex Optimization and its Applications, 22, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Aboussoror, Reverse convex programs: Stability and global optimality, Pacific Journal of Optimization, 5 (2009), 143-153.  Google Scholar

[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.  Google Scholar

[7]

D. Azé, "Eléments d'Analyse Convexe et Variationnelle," Ellipses, Paris, 1997. Google Scholar

[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.  Google Scholar

[9]

S. Dempe, "Foundations of Bilevel Programming," Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-B. Hiriart-Urruty, Conditions for global optimality 2, Journal of Global Optimization, 13 (1998), 349-367. doi: 10.1023/A:1008365206132.  Google Scholar

[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.  Google Scholar

[15]

R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition, Springer-Verlag, Berlin, 1993.  Google Scholar

[16]

M. Laghdir, Optimality conditions in reverse convex optimization, Acta Mathematica Vietnamica, 28 (2003), 215-223.  Google Scholar

[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.  Google Scholar

[18]

C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.," Technical Note, University of Dijon, France, 1987. Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[20]

K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, 1997.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

A. Strekalovsky, Extremal problems on complements of convex sets, Cybernetics and System Analysis, 1 (1994), 88-100. Google Scholar

[24]

A. Strekalovsky, Global optimality conditions for nonconvex optimization, Journal of Global Optimization, 12 (1998), 415-434. doi: 10.1023/A:1008277314050.  Google Scholar

[25]

A. Strekalovsky, On convergence of a global search strategy for reverse convex problems,, Journal of Applied Mathematics and Decision Sciences, 2005 (): 149.  doi: 10.1155/JAMDS.2005.149.  Google Scholar

[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.  Google Scholar

[27]

I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions,, Journal of Applied Mathematics and Decision Sciences, 2006 (): 1.  doi: 10.1155/JAMDS/2006/29023.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H. Tuy, "Convex Analysis and Global Optimization," Nonconvex Optimization and its Applications, 22, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

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