Abstract
In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.
Article PDF
Similar content being viewed by others
References
E. Blum, and W. Oettli,From optimization and variational inequalities to equilibrium problems, Math. Student,63(1994), 123–145.
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski,Nonsmooth analysis and control theory, Springer Verlag, New York, NY, 1998.
F. Giannessi and A. Maugeri,Variational inequalities and network equilibrium problems, Plenum Press, New York, NY, 1995.
F. Giannessi, A. Maugeri and P. M. Pardalos,Equilibrium problems: nonsmooth optimization and variational inequality models, Kluwer Academics Publishers, Dordrecht, Holland, 2001.
R. Glowinski, J. L. Lions and R. Tremolieres,Numerical analysis of variational inequalities, North-Holland, Amsterdam, Holland, 1981.
M. Aslam Noor,Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl.,122(2004), 131–146.
M. Aslam Noor,Generalized mixed quasi equilibrium problems with trifunctions, Appl. Math. Letters,18(2005), 695–700.
M. Aslam Noor,Mixed quasivariational inequalities, Appl. Math. Comput.,146(2004), 553–578.
M. Aslam Noor,Iterative schemes for nonconvex variational inequalities, J. Optim. Theory Appl.,121(2004), 163–173.
M. Aslam Noor,Fundamentals of mixed quasivariational inequalities, Inter. J. Pure Appl. Math.,15(2004), 137–258.
M. Aslam Noor,Some developments in general variational inequalities, Appl. Math. Comput.,152(2004), 199–277.
M. Aslam Noor,Multivalued regularized equilibrium problems, J. Global Optim. In Press.
M. Aslam Noor and W. Oettli,On general nonlinear complementarity problems and quasiequilibria, Le Mathemat.,49(1994), 313–331.
M. Patriksson,Nonlinear programming and variational inequalities: A unified approach, Kluwer Academic Publishers, Dordrecht, Holland, 1999.
R. A. Poliquin, R. T. Rockafellar and L. Thibault,Local differentiability of distance functions, Trans. Amer. Math. Soc.,352(2000), 5231–5249.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Noor, M.A. Regularized mixed quasi equilibrium problems. J. Appl. Math. Comput. 23, 183–191 (2007). https://doi.org/10.1007/BF02831967
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02831967