Mathematical Programming

, Volume 107, Issue 3, pp 409–438 | Cite as

Asymptotic behaviour of a family of gradient algorithms in ℝd and Hilbert spaces

  • Luc Pronzato
  • Henry P. Wynn
  • Anatoly A. Zhigljavsky
Article

Abstract

The asymptotic behaviour of a family of gradient algorithms (including the methods of steepest descent and minimum residues) for the optimisation of bounded quadratic operators in ℝd and Hilbert spaces is analyzed. The results obtained generalize those of Akaike (1959) in several directions. First, all algorithms in the family are shown to have the same asymptotic behaviour (convergence to a two-point attractor), which implies in particular that they have similar asymptotic convergence rates. Second, the analysis also covers the Hilbert space case. A detailed analysis of the stability property of the attractor is provided.

Mathematics Subject Classification (2000)

90C25 68Q25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Luc Pronzato
    • 1
  • Henry P. Wynn
    • 2
  • Anatoly A. Zhigljavsky
    • 3
  1. 1.Laboratoire I3S, CNRS - UNSALes Algorithmes - Bât. Euclide BSophia Antipolis CedexFrance
  2. 2.Department of StatisticsLondon School of EconomicsLondonUK
  3. 3.School of MathematicsCardiff UniversityCardiffUK

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