Asymptotic behaviour of a family of gradient algorithms in R^d and Hilbert spaces

Details Asymptotic behaviour of a family of gradient algorithms in R^d and Hilbert spaces Asymptotic behaviour of a family of gradient algorithms in R^d and Hilbert spaces

2008-02-29

arxiv.org/abs/0802.4382v1

Mathematical Programming, Series A 107 (2006) 409-438

Mathematics

Optimization and Control

The original publication is available at http://www.springerlink.com

Scientific paper

10.1007/s10107-005-0602-7

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 R^d 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.

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