### 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].

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].

Original language | English |
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Place of Publication | Itacha |

Publisher | Cornell University Library |

Number of pages | 10 |

Publication status | Published - 30 Jun 2016 |

### Publication series

Name | arXiv |
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Volume | arXiv:1606.09365 |

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### Cite this

de Klerk, E., Glineur, F., & Taylor, A. (2016).

*On the Worst-Case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions*. (arXiv; Vol. arXiv:1606.09365). Cornell University Library. http://arxiv.org/abs/1606.09365