zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 η-strongly monotone and κ-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 F(u * ),v-u * 0,vC·
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities
65J15Equations with nonlinear operators (numerical methods)
90C47Minimax problems
References:
[1]Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.
[2]Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984.
[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.
[5]Zeidler, E., Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, NY, 1985.
[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.
[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.
[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.
[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.
[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.
[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.