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Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities
Journal of Inequalities and Applications volume 2007, Article number: 019270 (2007)
Abstract
We study the strong convergence of a hybrid steepest descent method with variable parameters for the general variational inequality (GVI). Consequently, as an application, we obtain some results concerning the constrained generalized pseudoinverse. Our results extend and improve the result of Yao and Noor (2007) and many others.
References
Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus de l'Académie des Sciences 1964, 258: 4413–4416.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313.
Noor MA: General variational inequalities. Applied Mathematics Letters 1988,1(2):119–122. 10.1016/0893-9659(88)90054-7
Noor MA: Some developments in general variational inequalities. Applied Mathematics and Computation 2004,152(1):199–277. 10.1016/S0096-3003(03)00558-7
Noor MA, Noor KI: Self-adaptive projection algorithms for general variational inequalities. Applied Mathematics and Computation 2004,151(3):659–670. 10.1016/S0096-3003(03)00368-0
Noor MA: Wiener-Hopf equations and variational inequalities. Journal of Optimization Theory and Applications 1993,79(1):197–206. 10.1007/BF00941894
Yao Y, Noor MA: On modified hybrid steepest-descent methods for general variational inequalities. Journal of Mathematical Analysis and Applications 2007,334(2):1276–1289. 10.1016/j.jmaa.2007.01.036
Glowinski R: Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer, New York, NY, USA; 1984:xv+493.
Jaillet P, Lamberton D, Lapeyre B: Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 1990,21(3):263–289. 10.1007/BF00047211
Zeidler E: Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. Springer, New York, NY, USA; 1985:xxii+662.
Yamada I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000). Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.
Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. Journal of Optimization Theory and Applications 2003,119(1):185–201.
Zeng LC, Wong NC, Yao JC: Convergence of hybrid steepest-descent methods for generalized variational inequalities. Acta Mathematica Sinica 2006,22(1):1–12.
Noor MA: Wiener-Hopf equations and variational inequalities. Journal of Optimization Theory and Applications 1993,79(1):197–206. 10.1007/BF00941894
Zeng LC, Wong NC, Yao JC: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. Journal of Optimization Theory and Applications 2007,132(1):51–69. 10.1007/s10957-006-9068-x
Song Y, Chen R: An approximation method for continuous pseudocontractive mappings. Journal of Inequalities and Applications 2006, 2006: 9 pages.
Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Applied Mathematics Letters 2007,20(7):751–757. 10.1016/j.aml.2006.09.003
Chen R, Zhu Z: Viscosity approximation fixed points for nonexpansive and-accretive operators. Fixed Point Theory and Applications 2006, 2006: 10 pages.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017
Geobel K, Kirk WA: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge, UK; 1990.
Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian Journal of Mathematics 1999,41(3):435–453.
Engl HW, Hanke M, Neubauer A: Regularization of Inverse Problems. Volume 13. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000.
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Yu, Y., Chen, R. Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities. J Inequal Appl 2007, 019270 (2007). https://doi.org/10.1155/2007/19270
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DOI: https://doi.org/10.1155/2007/19270