zbMATH — the first resource for mathematics

An iterative approach to quadratic optimization. (English) Zbl 1043.90063
Summary: Assume that \(C_{1}, \dots , C_{N}\) are \(N\) closed convex subsets of a real Hilbert space \(H\) having a nonempty intersection \(C\). Assume also that each \(C_i\) is the fixed point set of a nonexpansive mapping \(T_i\) of \(H\). We devise an iterative algorithm which generates a sequence (\(x_n\)) from an arbitrary initial \(x_{0}{\in}H\). The sequence (\(x_n\)) is shown to converge in norm to the unique solution of the quadratic minimization problem \(min_{x\in C}(1/2){\langle}Ax, x{\rangle}-{\langle}x, u{\rangle}\), where \(A\) is a bounded linear strongly positive operator on \(H\) and \(u\) is a given point in \(H\). Quadratic-quadratic minimization problems are also discussed.

90C20 Quadratic programming
90C48 Programming in abstract spaces
Full Text: DOI
[1] 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
[2] Goebel, K., and Kirk, W.A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990. · Zbl 0708.47031
[3] Goebel, K., and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, NY, 1984.
[4] Halpern, B., Fixed Points of Nonexpanding Maps, Bulletin of the American Mathematical Society, Vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[5] Lions, P. L., Approximation de Points Fixes de Contractions, Comptes Rendus de l’Academie des Sciences, Serie I-Mathematique, Vol. 284, pp. 1357-1359, 1997. · Zbl 0349.47046
[6] 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
[7] Bauschke, H.H., The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Spaces, Journal of Mathematical Analysisis and Applications, Vol. 202, pp. 150-159, 1996. · Zbl 0956.47024 · doi:10.1006/jmaa.1996.0308
[8] Reich, S., Approximating Fixed Points of Nonexpansive Mappings, Panamerican Mathematical Journal, Vol. 4, pp. 23-28, 1994. · Zbl 0856.47032
[9] ATtouch, H., Viscosity Solutions of Minimization Problems, SIAM Journal on Optimization, Vol. 6, pp. 769-806, 1996. · Zbl 0859.65065 · doi:10.1137/S1052623493259616
[10] BAuschke, H. H., and Borwein, J.M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Reviews, Vol. 38, pp. 367-426, 1996. · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[11] Combettes, P.L., Hilbertian Convex Feasibility Problem: Convergence of Projection Methods, Applied Mathematics and Optimization, Vol. 35, pp. 311-330, 1997. · Zbl 0872.90069 · doi:10.1007/BF02683333
[12] Oden, J.T., Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1986. · Zbl 0578.70001
[13] O’hara, J. G., Pillay, P., and Xu, H.K., Iterative Approaches to Finding Nearest Common Fixed Points in Hilbert Spaces, Nonlinear Analysis (to appear).
[14] 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
[15] Xu, H. K., and Kim, T.W., Convergence of Hybrid Steepest Descent Methods for Variational Inequalities, Preprint, 2003. · Zbl 1045.49018
[16] Hiriart-Urruty, J. B., Conditions for Global Optimality 2, Journal of Global Optimization, Vol. 13, pp. 349-367, 1998. · Zbl 0922.90109 · doi:10.1023/A:1008365206132
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.