On the steepest descent algorithm for quadratic functions

The steepest descent algorithm with exact line searches (Cauchy algorithm) is inefficient, generating oscillating step lengths and a sequence of points converging to the span of the eigenvectors associated with the extreme eigenvalues. The performance becomes very good if a short step is taken at ev...

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Bibliographic Details
Published in:Computational optimization and applications 2016-03, Vol.63 (2), p.523-542
Main Authors: Gonzaga, Clóvis C., Schneider, Ruana M.
Format: Article
Language:English
Subjects:
Algorithms
Cauchy problems
Convex and Discrete Geometry
Descent
Eigenvalues
Eigenvectors
Linear programming
Management Science
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Operations Research
Operations Research/Decision Theory
Optimization
Optimization algorithms
Polynomials
Roots
Searching
Statistics
Studies
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Summary:The steepest descent algorithm with exact line searches (Cauchy algorithm) is inefficient, generating oscillating step lengths and a sequence of points converging to the span of the eigenvectors associated with the extreme eigenvalues. The performance becomes very good if a short step is taken at every (say) ten iterations. We show a new method for estimating short steps, and propose a method alternating Cauchy and short steps. Finally, we use the roots of a certain Chebyshev polynomial to further accelerate the methods.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-015-9775-z