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### Abstract

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.

Original language | English |
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Title of host publication | Invited Lectures |

Editors | Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana |

Publisher | World Scientific Publishing Co Pte Ltd |

Pages | 3729-3768 |

Number of pages | 40 |

ISBN (Electronic) | 9789813272934 |

Publication status | Published - 1 Jan 2018 |

Event | 2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil Duration: 1 Aug 2018 → 9 Aug 2018 |

### Publication series

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

### Conference

Conference | 2018 International Congress of Mathematicians, ICM 2018 |
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Country | Brazil |

City | Rio de Janeiro |

Period | 1/08/18 → 9/08/18 |

### Keywords

- Nonlinear optimization
- Evaluation complexity
- Regularization methods

## Fingerprint Dive into the research topics of 'Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization'. Together they form a unique fingerprint.

## Projects

- 2 Active

## Complexity in nonlinear optimization

TOINT, P., Gould, N. I. M. & Cartis, C.

1/11/08 → …

Project: Research

## Cite this

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