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

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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).
Original languageEnglish
Pages (from-to)1185–1199
Number of pages15
JournalOptimization Letters
Volume11
Issue number7
DOIs
StatePublished - Oct 2017

    Research areas

  • gradient method, steepest descent, semidefinite programming, performance estimation problem

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