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

Etienne de Klerk (E.deKlerk***at***uvt.nl)
François Glineur (Francois.Glineur***at***uclouvain.be)
Adrien B. Taylor (Adrien.Taylor***at***uclouvain.be)

Abstract: 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 extend the result to 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 proof is computer-assisted, and relies on the resolution of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].

Keywords: gradient method, steepest descent, semidefinite programming, performance estimation problem,

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation:

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Entry Submitted: 06/30/2016
Entry Accepted: 06/30/2016
Entry Last Modified: 09/15/2016

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