American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalites: Applications to Free Boundary Problems," John Wiley and Sons, New York, 1984.  Google Scholar

[3]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145.  Google Scholar

[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.  Google Scholar

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G. Cohen, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory. Appl., 32 (1980), 277-305. doi: 10.1007/BF00934554.  Google Scholar

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G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory. Appl., 59 (1988), 325-333.  Google Scholar

[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.  Google Scholar

[8]

K. Goebel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Patriksson, "Nonlinear Programming and Variational Inequality Problems: a unified approach," Kluwer, Dordrecht, 1999.  Google Scholar

[12]

E. Zeidler, "Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990.  Google Scholar

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.  Google Scholar

[2]

C. Baiocchi and A. Capelo, "Variational and Quasivariational Inequalites: Applications to Free Boundary Problems," John Wiley and Sons, New York, 1984.  Google Scholar

[3]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145.  Google Scholar

[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.  Google Scholar

[5]

G. Cohen, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory. Appl., 32 (1980), 277-305. doi: 10.1007/BF00934554.  Google Scholar

[6]

G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory. Appl., 59 (1988), 325-333.  Google Scholar

[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.  Google Scholar

[8]

K. Goebel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Patriksson, "Nonlinear Programming and Variational Inequality Problems: a unified approach," Kluwer, Dordrecht, 1999.  Google Scholar

[12]

E. Zeidler, "Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990.  Google Scholar

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