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

Authors:
Luc Pronzato;Henry P. Wynn;Anatoly A. Zhigljavsky
Affiliations:
Laboratoire I3S, CNRS - UNSA, Les Algorithmes - Bât. Euclide B, B.P. 121, 2000 route des Lucioles, 06903, Sophia Antipolis Cedex, France;Department of Statistics, London School of Economics, B.P. 121, 2000 route des Lucioles, WC2A 2AE, London, UK;School of Mathematics, Cardiff University, B.P. 121, Senghennydd Road, CF24 4AG, Cardiff, UK
Venue:
Mathematical Programming: Series A and B
Year:
2006

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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.