On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadrat...

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Bibliographic Details
Published in:Optimization letters Vol. 11; no. 7; pp. 1185 - 1199
Main Authors: de Klerk, Etienne, Glineur, François, Taylor, Adrien B.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2017
Subjects:
Computational Intelligence
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Simulation
Semidefinite programming
Performance estimation problem
Steepest descent
Gradient method
ISSN:1862-4472, 1862-4480
Online Access:Get full text
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Summary:We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also give the tight worst-case complexity bound for a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proofs are computer-assisted, and rely on the resolutions of semidefinite programming performance estimation problems as introduced in the paper (Drori and Teboulle, Math Progr 145(1–2):451–482, 2014 ).
ISSN:1862-4472
1862-4480
DOI:10.1007/s11590-016-1087-4