Xu, H. K.; Kim, T. H. Convergence of hybrid steepest-descent methods for variational inequalities. (English) Zbl 1045.49018 J. Optimization Theory Appl. 119, No. 1, 185-201 (2003). 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 devise an iterative algorithm which generates a sequence (\(x_n\)) from an arbitrary initial point \(x_0 \in H\). The sequence (\(x_n\)) is shown to converge in norm to the unique solution \(u^{\ast}\) of the variational inequality \[ \langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0,\qquad \text{for } v \in C. \] Applications to constrained pseudoinverses are included. Cited in 2 ReviewsCited in 186 Documents MSC: 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 90C30 Nonlinear programming Keywords:iterative algorithms; hybrid steepest-descent methods; convergence; nonexpansive mappings; Hilbert space; constrained pseudoinverses PDFBibTeX XMLCite \textit{H. K. Xu} and \textit{T. H. Kim}, J. Optim. Theory Appl. 119, No. 1, 185--201 (2003; Zbl 1045.49018) 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 Pricing of American Options, Acta Applicandae Mathematicae, Vol. 21, pp. 263-289, 1990. · Zbl 0714.90004 [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] Konnov, I., Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Germany, 2001. · Zbl 0982.49009 [7] 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 [8] 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 [9] 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 [10] Halpern, B., Fixed Points of Nonexpanding Maps, American Mathematical Society Bulletin, Vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 [11] 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 [12] Wittmann, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol. 58, pp. 486-491, 1992. · Zbl 0797.47036 [13] Xu, H. K., An Iterative Approach to Quadratic Optimization, Journal of Optimization Theory and Applications, Vol. 116, pp. 659-678, 2003. · Zbl 1043.90063 [14] Geobel, K., and Kirk, W. A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990. [15] Iusem, A. N., An Iterative Algorithm for the Variational Inequality Problem, Computational and Applied Mathematics, Vol. 13, pp. 103-114, 1994. · Zbl 0811.65049 [16] Censor, Y., Iusem, A. N., and Zenios, S. A., An Interior-Point Method with Bregman Functions for the Variational Inequality Problem with Paramonotone Operators, Mathematical Programming, Vol. 81, pp. 373-400, 1998. · Zbl 0919.90123 [17] Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, Vol. 38, pp. 367-426, 1996. · Zbl 0865.47039 [18] Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Kluwer, Dordrecht, Holland, 2000. · Zbl 0859.65054 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.