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G. Forsythe (1968)
On the asymptotic directions of thes-dimensional optimum gradient methodNumerische Mathematik, 11
H. Akaike (1959)
On a successive transformation of probability distribution and its application to the analysis of the optimum gradient methodAnn. Inst. Statist. Math. Tokyo, 11
H. Akaike (1959)
On a successive transformation of probability distribution and its application to the analysis of the optimum gradient methodAnnals of the Institute of Statistical Mathematics, 11
J. Hale, H. Koçak (1991)
Dynamics and Bifurcations
A. N. Shiryaev (1996)
Probability
E. Polak (1973)
Introduction to linear and nonlinear programming
L. V. Kantorovich, G. P. Akilov (1982)
Functional Analysis
J. Lasalle (1976)
The stability of dynamical systems
L. Pronzato, H. P. Wynn, A. A. Zhigljavsky (2000)
Dynamical Search
The result that for quadratic functions the classical steepest descent algorithm in R d converges locally to a two-point attractor was proved by Akaike. In this paper this result is proved for bounded quadratic operators in Hilbert space. The asymptotic rate of convergence is shown to depend on the starting point while, as expected, confirming the Kantorovich bounds. The introduction of a relaxation coefficient in the steepest-descent algorithm completely changes its behaviour, which may become chaotic. Different attractors are presented. We show that relaxation allows a significantly improved rate of convergence.
Acta Applicandae Mathematicae – Springer Journals
Published: Oct 19, 2004
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