Bilevel mixed equilibrium problems in Banach spaces : existence and algorithmic aspects

2011, 1(3): 549-561. doi: 10.3934/naco.2011.1.549

Bilevel mixed equilibrium problems in Banach spaces : existence and algorithmic aspects

Ouayl Chadli ^1, , Hicham Mahdioui ^2, and Jen-Chih Yao ^3,

1.	Department of Economics, Ibn Zohr University, B.P. 8658 Poste Dakhla, Agadir, Morocco
2.	Department of Mathematics, Ibn Zohr University, Agadir, Morocco
3.	Kaohsiung Medical University, Kaohsiung Medical University, Kaohsiung 80708

Received May 2011 Revised August 2011 Published September 2011

Abstract
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In this paper, we study the existence and algorithmic aspect of a class of bilevel mixed equilibrium problems (BMEP) in a Banach space setting. We introduce a suitable regularization of the bilevel mixed equilibrium problems by means of an auxiliary equilibrium problem. We show that it is possible to define from the auxiliary equilibrium problems a sequence strongly convergent to a solution of the bilevel mixed equilibrium problem. The results obtained are interesting in the sense that they improve some new results on bilevel mixed equilibrium problems and give answer to some open questions in literature related to the convergence of algorithms for (BMEP).

Keywords: Bilevel programming, variational inequalities, monotonicity., auxiliary principle, equilibrium problems.

Mathematics Subject Classification: Primary: 47J20, 49J40, 49J45; Secondary: 49M20, 47H05, 90C3.

Citation: Ouayl Chadli, Hicham Mahdioui, Jen-Chih Yao. Bilevel mixed equilibrium problems in Banach spaces : existence and algorithmic aspects. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 549-561. doi: 10.3934/naco.2011.1.549

References:

[1]	A. S. Antipin, The fixed points of extremal maps: computation by gradient methods, Zh. Vychisl. Mat. Mat. Fiz., 37 (1997), 42-53.
[2]	C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalites: Applications to Free Boundary Problems," John Wiley and Sons, New York, 1984.
[3]	E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145.
[4]	O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory. Appl., 105 (2000), 299-323. doi: 10.1023/A:1004657817758.
[5]	G. Cohen, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory. Appl., 32 (1980), 277-305. doi: 10.1007/BF00934554.
[6]	G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory. Appl., 59 (1988), 325-333.
[7]	X. P. Ding, Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces, J. Optim. Theory. Appl., 146 (2010), 347-357. doi: 10.1007/s10957-010-9651-z.
[8]	K. Goebel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
[9]	I. V. Konnov, Generalized monotone equilibrium problems and variational inequalities, in "Handbook of Generalized Convexity and Generalized Monotonicity " (eds. N. Hadjisavvas, S. Koml\'osi and S. Schaible), Springer-Verlag, (2005), 559-618.
[10]	A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Glob. Optim., 47 (2010), 287-292. doi: 10.1007/s10898-009-9476-1.
[11]	M. Patriksson, "Nonlinear Programming and Variational Inequality Problems: a unified approach," Kluwer, Dordrecht, 1999.
[12]	E. Zeidler, "Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990.

show all references

References:

[1]	A. S. Antipin, The fixed points of extremal maps: computation by gradient methods, Zh. Vychisl. Mat. Mat. Fiz., 37 (1997), 42-53.
[2]	C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalites: Applications to Free Boundary Problems," John Wiley and Sons, New York, 1984.
[3]	E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145.
[4]	O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory. Appl., 105 (2000), 299-323. doi: 10.1023/A:1004657817758.
[5]	G. Cohen, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory. Appl., 32 (1980), 277-305. doi: 10.1007/BF00934554.
[6]	G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory. Appl., 59 (1988), 325-333.
[7]	X. P. Ding, Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces, J. Optim. Theory. Appl., 146 (2010), 347-357. doi: 10.1007/s10957-010-9651-z.
[8]	K. Goebel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
[9]	I. V. Konnov, Generalized monotone equilibrium problems and variational inequalities, in "Handbook of Generalized Convexity and Generalized Monotonicity " (eds. N. Hadjisavvas, S. Koml\'osi and S. Schaible), Springer-Verlag, (2005), 559-618.
[10]	A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Glob. Optim., 47 (2010), 287-292. doi: 10.1007/s10898-009-9476-1.
[11]	M. Patriksson, "Nonlinear Programming and Variational Inequality Problems: a unified approach," Kluwer, Dordrecht, 1999.
[12]	E. Zeidler, "Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990.

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