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Existence of solutions to weak nonlinear bilevel problems via MinSup and d.c. problems

Published online by Cambridge University Press:  17 May 2008

Abdelmalek Aboussoror  and
Abdelatif Mansouri
Abdelmalek Aboussoror
Affiliation:
Université Cadi Ayyade, Faculté Polydisciplinaire de Safi, B.P. 4162, Safi, Morocco; aboussororabdel@hotmail.com
Abdelatif Mansouri
Affiliation:
Université Cadi Ayyad, Faculté des sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakech, Morocco; mansouri@ucam.ac.ma
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Abstract

In this paper, which is an extension of [4], we first show the existence of solutions to a class of Min Sup problems with linked constraints, which satisfy a certain property. Then, we apply our result to a class of weak nonlinear bilevel problems. Furthermore, for such a class of bilevel problems, we give a relationship with appropriate d.c. problems concerning the existence of solutions.

Keywords

Min Sup problemsvariational convergencebilevel programmingd.c. programming.
Type
Research Article
Information
RAIRO - Operations Research , Volume 42 , Issue 2: CIRO 05 , April 2008 , pp. 87 - 103
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

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References

Aboussoror, A. and Loridan, P., Existence of Solutions to Two-Level Optimization Problems with Nonunique Lower-Level Solutions. J. Math. Anal. Appl. 254 (2001) 348–357. CrossRef
Aboussoror, A., Weak Bilevel Programming Problems: Existence of Solutions. Adv. Math. Res. 1 (2002) 83–92.
Aboussoror, A. and Mansouri, A., Weak linear bilevel programming problems: existence of solutions via a penalty method. J. Mat. Anal. Appl. 304 (2005) 399–408. CrossRef
A. Aboussoror and A. Mansouri, Sufficient conditions for Min Sup problems and application to bilevel programs, in Proc CIRO'05, IV Conférence Internationale en Recherche Opérationnelle, Théorie et Applications 1 (2005) 99–107.
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis. Pure and Applied Mathematics (New York), John Wiley & Sons, New York (1984).
H. Attouch, Variational Convergences for Functions and Operators. Pitman, Boston (1984).
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-Linear Parametric Optimization. Akademie-Verlag, Berlin (1982).
De Giorgi, E. and Franzoni, T., Su un Tipo di Convergenza Variazionale. Atti Accad. Naz. Lincei Sci. Fi. Mat. Natur. 58 (1975) 842–850.
Dempe, S., Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52 (2003) 333–359. CrossRef
A.L. Dontchev and Zolezzi, Well-Posed Optimization Problems. Lect. Notes in Mathematics. Springer-Verlag, Berlin. 1543 (1993).
Falk, J.E., A linear Max-Min problem. Math. Program. 5 (1973) 169–188. CrossRef
Lignola, M.B. and Morgan, J., Topological existence and stability for Min-Sup problems. J. Math. Anal. Appl. 151 (1990) 164–180. CrossRef
Lignola, M.B. and Morgan, J., Semicontinuities of marginal functions in a sequential setting. Optimization 24 (1994) 241–252. CrossRef
P. Loridan and J. Morgan, Approximate Solutions for Two-Level Optimization Problems, in Trends in Mathematical Optimization, International Series of Numerical Mathematics, edited by K. Hoffman, J.-B. Hiriart-Urruty, C. Lemarechal and J. Zowe, Birkhäuser Verlag, Basel 84 (1988) 181–196.
P. Loridan and J. Morgan, On Strict ε-Solutions for Two-Level Optimization Problem, in Operations Research Proceedings of the International Conference on Operations Research 90, Vienna, edited by W. Buhler, G. Feichtinger, F. Hartl, F.J. Radermacher and P. Stahly, Springer-Verlag, Berlin (1992) 165–172.
Lucchetti, R., Mignanego, F. and Pieri, G., Existence theorem of equilibrium points in Stackelberg games with constraints. Optimization 18 (1987) 857–866. CrossRef
C. Michelot, Caractérisation des minima locaux des fonctions de la classe D.C., Technical Note, University of Dijon (1987).
Pham Dinh Tao, Le Thi Hoai, An, Convex analysis approach to d.c. programming: theory, algorithms and applications. Acta Mathematica Vietnamica 22 (1997) 289–355.
R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970).
K. Shimizu and E. Aiyoshi, Necessary Conditions for Min-Max Problems and algorithms by a relaxation procedure. IEEE Transactions on Automatic Control: AC- 25(1) (1980) 62–66.