×

zbMATH — the first resource for mathematics

Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. (English) Zbl 1137.47059
Summary: Assume that \(F\) is a nonlinear operator on a real Hilbert space \(H\) which is \(\eta\)-strongly monotone and \(\kappa\)-Lipschitzian on a nonempty closed convex subset \(C\) of \(H\). Assume also that \(C\) is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on \(H\). We construct an iterative algorithm with variable parameters which generates a sequence \({x_{n}}\) from an arbitrary initial point \(x_{0} H\). The sequence \({x_{n}}\) is shown to converge in norm to the unique solution \(u^{}\) of the variational inequality \(\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.\)

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
90C47 Minimax problems in mathematical programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980. · Zbl 0457.35001
[2] Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984. · Zbl 0536.65054
[3] Jaillet, P., Lamberton, D., and Lapeyre, B., Variational Inequalities and the Princing of American Options, Acta Applicandae Mathematicae, Vol. 21, pp. 263–289, 1990. · Zbl 0714.90004 · doi:10.1007/BF00047211
[4] Oden, J. T., Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1986. · Zbl 0578.70001
[5] Zeidler, E., Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, NY, 1985. · Zbl 0583.47051
[6] Yao, J. C., Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994. · Zbl 0813.49010 · doi:10.1287/moor.19.3.691
[7] Konnov, I., Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Germany, 2001. · Zbl 0982.49009
[8] Zeng, L. C., Iterative Algorithm for Finding Approximate Solutions to Completely Generalized Strongly Nonlinear Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 201, pp. 180–194, 1996. · Zbl 0853.65073 · doi:10.1006/jmaa.1996.0249
[9] Zeng, L. C., Completely Generalized Strongly Nonlinear Quasicomplementarity Problems in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 193, pp. 706–714, 1995. · Zbl 0832.47053 · doi:10.1006/jmaa.1995.1262
[10] Zeng, L. C., On a General Projection Algorithm for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 97, pp. 229–235, 1998. · Zbl 0907.90265 · doi:10.1023/A:1022687403403
[11] Yamada, I., The Hybrid Steepest-Descent Method for Variational Inequality Problems over the Intersection of the Fixed-Point Sets of Nonexpansive Mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Edited by D. Butnariu, Y. Censor, and S. Reich, North-Holland, Amsterdam, Holland, pp. 473–504, 2001. · Zbl 1013.49005
[12] Deutsch, F., and Yamada. I., Minimizing Certain Convex Functions over the Intersection of the Fixed-Point Sets of Nonexpansive Mappings, Numerical Functional Analysis and Optimization, Vol. 19, pp. 33–56, 1998. · Zbl 0913.47048 · doi:10.1080/01630569808816813
[13] Lions, P.L., Approximation de Points Fixes de Contractions, Comptes Rendus de L’Academie des Sciences de Paris, Vol. 284, pp. 1357–1359, 1977. · Zbl 0349.47046
[14] Bauschke, H. H., The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 150–159, 1996. · Zbl 0956.47024 · doi:10.1006/jmaa.1996.0308
[15] Wittmann, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol. 58, pp. 486–491, 1992. · Zbl 0797.47036 · doi:10.1007/BF01190119
[16] Xu, H. K., and Kim, T. H., Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 119, pp. 185–201, 2003. · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[17] Xu, H. K., An Iterative Approach to Quadratic Optimization, Journal of Optimization Theory and Applications, Vol. 116, pp. 659–678, 2003. · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[18] Goebel, K., and Kirk, W. A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990. · Zbl 0708.47031
[19] Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, Vol. 38, pp. 376–426, 1996. · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[20] Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Kluwer, Dordrecht, Holland, 2000. · Zbl 0859.65054
[21] Yamada, I., Ogura, N., and Shirakawa, N., A Numerically Robust Hybrid Steepest Descent Method for Convexly Constrained Generalized Inverse Problems, Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics, Edited by Z. Nashed and O. Scherzer, Vol. 313, pp. 269–305, 2002. · Zbl 1039.47051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.