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An Iterative Approach to Quadratic Optimization

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Abstract

Assume that C1, . . . , 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 x0H. The sequence (xn) is shown to converge in norm to the unique solution of the quadratic minimization problem minxC(1/2)〈Ax, x〉−〈x, u〉, 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.

Iterative algorithms quadratic optimization nonexpansive mappings convex feasibility problems Hilbert spaces 

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References

  1. 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.Google Scholar
  2. 2.
    Goebel, K., and Kirk, W.A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.Google Scholar
  3. 3.
    Goebel, K., and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, NY, 1984.Google Scholar
  4. 4.
    Halpern, B., Fixed Points of Nonexpanding Maps, Bulletin of the American Mathematical Society, Vol. 73, pp. 957-961, 1967.Google Scholar
  5. 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.Google Scholar
  6. 6.
    Wittmann, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol. 58, pp. 486-491, 1992.Google Scholar
  7. 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.Google Scholar
  8. 8.
    Reich, S., Approximating Fixed Points of Nonexpansive Mappings, Panamerican Mathematical Journal, Vol. 4, pp. 23-28, 1994.Google Scholar
  9. 9.
    ATtouch, H., Viscosity Solutions of Minimization Problems, SIAM Journal on Optimization, Vol. 6, pp. 769-806, 1996.Google Scholar
  10. 10.
    BAuschke, H. H., and Borwein, J.M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Reviews, Vol. 38, pp. 367-426, 1996.Google Scholar
  11. 11.
    Combettes, P.L., Hilbertian Convex Feasibility Problem: Convergence of Projection Methods, Applied Mathematics and Optimization, Vol. 35, pp. 311-330, 1997.Google Scholar
  12. 12.
    Oden, J.T., Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.Google Scholar
  13. 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).Google Scholar
  14. 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.Google Scholar
  15. 15.
    Xu, H. K., and Kim, T.W., Convergence of Hybrid Steepest Descent Methods for Variational Inequalities, Preprint, 2003.Google Scholar
  16. 16.
    Hiriart-Urruty, J. B., Conditions for Global Optimality 2, Journal of Global Optimization, Vol. 13, pp. 349-367, 1998.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • H.K. Xu
    • 1
  1. 1.Department of MathematicsUniversity of Durban-WestvilleDurbanSouth Africa

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