On the worst case performance of the steepest descent algorithm for quadratic functions

The existing choices for the step lengths used by the classical steepest descent algorithm for minimizing a convex quadratic function require in the worst case O ( C log ( 1 / ε ) ) iterations to achieve a precision ε , where C is the Hessian condition number. We show how to construct a sequence of...

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Bibliographic Details
Published in:Mathematical programming 2016-11, Vol.160 (1-2), p.307-320
Main Author: Gonzaga, Clóvis C.
Format: Article
Language:English
Subjects:
Algorithms
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Conjugate gradient method
Descent
Formulas (mathematics)
Full Length Paper
Iterative methods
Linear programming
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Plugs
Polynomials
Quadratic equations
Quadratic programming
Studies
Texts
Theoretical
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Summary:The existing choices for the step lengths used by the classical steepest descent algorithm for minimizing a convex quadratic function require in the worst case O ( C log ( 1 / ε ) ) iterations to achieve a precision ε , where C is the Hessian condition number. We show how to construct a sequence of step lengths with which the algorithm stops in O ( C log ( 1 / ε ) ) iterations, with a bound almost exactly equal to that of the Conjugate Gradient method.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-0984-8