Abstract
In this paper, we suggest and analyze a new self-adaptive inexact implicit method with a variable parameter for general mixed quasi variational inequalities, where the skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of this method. We use a self-adaptive technique to adjust parameter ρ at each iteration. The global convergence of the proposed method is proved under some mild conditions. Preliminary numerical results indicate that the self-adaptive adjustment rule is necessary in practice.
Similar content being viewed by others
References
Bnouhachem A.: A self-adaptive method for solving general mixed variational inequalities. J. Math. Anal. Appl. 309, 136–150 (2005)
Brezis H.: Operateurs maximaux monotone et semigroupes de contractions dans les espace d’Hilbert. North-Holland, Amsterdam (1973)
Glowinski R., Lions J.L., Tremoliers R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)
He B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999)
He B.S., Liao L.Z.: Improvement of some projection methods for monotone variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)
Lions J.L., Stampacchia G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493–512 (1967)
Noor M.A.: General variational inequalities. Appl. Math. Lett. 1, 119–121 (1988)
Noor M.A.: An implicit method for mixed variational inequalities. Appl. Math. Lett. 11, 109–113 (1998)
Noor M.A.: Pseudomonotone general mixed variational inequalities. Appl. Math. Comput. 141, 529–540 (2003)
Noor M.A.: Fundamentals of mixed quasi variational inequalities. Inter. J. Pure Appl. Math. 15, 137–258 (2004)
Noor M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)
Noor M.A., Noor K.I.: Self-adaptive projection algorithms for general variational inequalities. Appl. Math. Comput. 151, 659–670 (2004)
Noor M.A., Noor K.I.: On general mixed quasi variational inequalities. J. Optim. Theory Appl. 120, 579–599 (2004)
Solodov M.V., Svaiter B.F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Math. Program. Ser. B 88, 371–389 (2000)
Solodov M.V., Svaiter B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001)
Stampacchia G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)
Wang S.L., Yang H., He B.S.: Implicit method with variable parameter for mixed monotone variational inequalities. J. Optim. Theory Appl. 111, 431–443 (2001)
Zeng L.C., Yao J.C.: Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities. Math. Meth. Oper. Res. 62, 211–224 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Muhammad Aslam Noor is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.
Rights and permissions
About this article
Cite this article
Bnouhachem, A., Noor, M.A. New inexact implicit method for general mixed quasi variational inequalities. J Glob Optim 44, 409–421 (2009). https://doi.org/10.1007/s10898-008-9339-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9339-1
Keywords
- General mixed quasi variational inequalities
- Implicit methods
- Resolvent operator
- Inexact methods
- Self-adaptive rules