Taiwanese Journal of Mathematics

THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

Jia-wei Chen, Yeol Je Cho, and Zhongping Wan

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Abstract

In this paper, a new class of bilevel mixed equilibrium problems (for short, (BMEP)) is introduced and investigated in reflexive Banach space and some topological properties of solution sets for the lower level mixed equilibrium problem and the problem (BMEP) are established without coercivity. Subsequently, we construct a new iterative algorithm which can directly compute some solutions of the problem (BMEP). Some strong convergence theorems of the sequence generated by the proposed algorithm are also presented. Finally, the well-posedness and generalized well-posedness for the problem (BMEP) are introduced by an $\epsilon$-bilevel mixed equilibrium problem. Also, we explore the sufficient and necessary conditions for (generalized) well-posedness of the problem (BMEP) and show that, under some suitable conditions, the well-posedness and generalized well-posedness of (BMEP) are equivalent to the uniqueness and existence of its solutions, respectively. These results are new and improve some recent results in this field.

Article information

Source
Taiwanese J. Math., Volume 17, Number 2 (2013), 725-748.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705962

Digital Object Identifier
doi:10.11650/tjm.17.2013.2337

Mathematical Reviews number (MathSciNet)
MR3044531

Zentralblatt MATH identifier
1280.49016

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions)

Keywords
bilevel mixed equilibrium problem $\epsilon$-bilevel mixed equilibrium problem mixed equilibrium problem well-posedness Hausdorff metric

Citation

Chen, Jia-wei; Cho, Yeol Je; Wan, Zhongping. THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES. Taiwanese J. Math. 17 (2013), no. 2, 725--748. doi:10.11650/tjm.17.2013.2337. https://projecteuclid.org/euclid.twjm/1499705962


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References

  • P. N. Anh, J. K. Kim and L. D. Muu, An extragradient algorithm for solving bilevel pseudomonotone variational inequalities, J. Glob. Optim., doi:10.1007/s10898-012-9870-y.
  • L. Q. Anh, P. Q. Khanh and D. T. M. Van, Well-posedness under relaxed semicontinuity for bilevel equilibrium and optimization problems with equilibrium constraints, J. Optim. Theory Appl., doi:10.1007/s10957-011-9963-7.
  • A. S. Antipin, Iterative gradient prediction-type methods for computing fixed-point of extremal mappings. in: Parametric Optimization and Related Topics IV[C], (J. Guddat, H. Th. Jonden, F. Nizicka, G. Still and F. Twitt, eds.), Peter Lang, Frankfurt Main, 1997, pp. 11-24.
  • J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.
  • S. I. Birbil, G. Bouza, J. B. G. Frenk and G. Still, Equilibrium constrained optimization problems, Europ. J. Oper. Research, 169 (2006), 1108-1127.
  • E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Mathematics Stud., 63 (1994), 123-145.
  • O. Chadli, H. Mahdioui and J. C. Yao, Bilevel mixed equilibrium problems in Banach spaces: existence and algorithmic aspects, Numer. Algebra Cont. Optim., 1(3) (2011), 549-561.
  • J. W. Chen, Y. J. Cho and Z. P. Wan, Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space, Fixed Point Theory Appl., 2011, 2011:91.
  • J. W. Chen and Z. Wan, Existence of solutions and convergence analysis for a system of quasi-variational inclusions in Banach spaces, J. Inequal. Appl., 2011:49, doi:10.1186/ 1029-242X-2011-49.
  • J. W. Chen, Z. Wan and Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems, Math. Meth. Oper. Res., Doi 10.1007/s00186-012-0414-5.
  • J. W. Chen, Z. P. Wan and Y. Zou, Strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach spaces, Optim., (2011), doi:10.1080/ 02331934.2011.626779.
  • Y. J. Cho, X. Qin and J. I. Kang, Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems, Nonlinear Anal., 71 (2009), 4203-4214.
  • 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.
  • X. P. Ding, Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces, Acta Math. Sin. English. Ser., 28 (2011), 503-514.
  • X. P. Ding, Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces, Appl. Math. Mech.-Engl. Ed., \bf32(11) (2011), 1457-1474.
  • X. P. Ding, Existence and iterative algorithm of solutions for a class of bilevel generalized mixed equilibrium problems in Banach spaces, J. Glob. Optim., 2011, doi:10.1007/ s10898-011-9724-z.
  • X. P. Ding, Y. C. Liou and J. C. Yao, Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces, J. Glob. Optim., doi:10.1007/s10898-011-9712-3.
  • B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., Vol. 2011, Article ID 646452, 14 pp.
  • Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89-100.
  • S. M. Guu and J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Anal., 71(7-8) (2009), 2847-2855.
  • N. J. Huang, J. Li and B. H. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274.
  • M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Global Optim., 16(1) (2000), 57-67.
  • L. J. Lin, Mathematical programming with system of equilibrium constraints, J. Glob. Optim., 37 (2007), 275-286.
  • L. J. Lin, Existence theorems for bilevel problem with applications to mathematical program with equilibrium constraint and semi-infinite problem, J. Optim. Theory Appl., 137 (2008), 27-40.
  • Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, 1996.
  • Y. Lv, Z. Chen and Z. Wan, A neural network approach for solving mathematical programs with equilibrium constraints, Expert Sys. Appl., 38 (2011), 231-234.
  • A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Glob. Optim., 47 (2010), 287-292.
  • J. Salamon, Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems, J. Glob. Optim., 47 (2010), 173-183.
  • Z. Wan, J. W. Chen, H. Sun and L. Yuan, A new system of generalized mixed quasivariational inclusions with relaxed cocoercive operators and applications, J. Appl. Math., 2011 (2011), 1-22.
  • H. F. Xu and J. J. Ye, Necessary optimality conditions for two-stage stochastic programs with equilibrium constraints, SIAM J. Optim., 20(4) (2010), 1685-1715.
  • J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 305-369.
  • Y. Yao, Y. J. Cho and Y. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, Europ. J. Operat. Research, 212 (2011), 242-250.
  • R. Y. Zhong, N. J. Huang and Y. J. Cho, Boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems with an application, J. Inequal. Appl., Vol. 2011, Article ID 936428, 15 pp.