Projets par an

### Résumé

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing our previous results. To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton’s method as well as of their linesearch variants. For each method in this class and arbitrary accuracy threshold ∊ 2 (0; 1), we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is α-Hölder continuous (for given α 2 [0; 1]), for which the method in question takes at least b∊-(2+α)/(1+α^{)}c function evaluations to generate a first iterate whose gradient is smaller than ∊ in norm. Moreover, we also construct another function on which Newton’s takes b∊-^{2}c evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for α = 1, this lower bound is of the same order in ∊ as the upper bound on the worst-case evaluation complexity of the cubic regularization method and other algorithms in a class of methods recently proposed by Curtis, Robinson and Samadi or by Royer and Wright, thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton’s method is suboptimal.

langue originale | Anglais |
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titre | Invited Lectures |

rédacteurs en chef | Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana |

Editeur | World Scientific Publishing Co Pte Ltd |

Pages | 3729-3768 |

Nombre de pages | 40 |

ISBN (Electronique) | 9789813272934 |

Etat de la publication | Publié - 1 janv. 2018 |

Evénement | 2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brésil Durée: 1 août 2018 → 9 août 2018 |

### Série de publications

Nom | Proceedings of the International Congress of Mathematicians, ICM 2018 |
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Volume | 4 |

### Une conférence

Une conférence | 2018 International Congress of Mathematicians, ICM 2018 |
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Pays | Brésil |

La ville | Rio de Janeiro |

période | 1/08/18 → 9/08/18 |

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## Projets

- 2 Actif

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Projet: Recherche

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Projet: Axe de recherche

## Contient cette citation

*Invited Lectures*(p. 3729-3768). (Proceedings of the International Congress of Mathematicians, ICM 2018; Vol 4). World Scientific Publishing Co Pte Ltd.