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### Abstract

An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p, (Formula presented.), of the unconstrained objective function, and that is guaranteed to find a first- and second-order critical point in at most (Formula presented.) function and derivatives evaluations, where ε
_{1} and ε
_{1} are prescribed first- and second-order optimality tolerances. This is a simple algorithm and associated analysis compared to the much more general approach in Cartis et al. [Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints, arXiv:1811.01220, 2018] that addresses the complexity of criticality higher-than two; here, we use standard optimality conditions and practical subproblem solves to show a same-order sharp complexity bound for second-order criticality. Our approach also extends the method in Birgin et al. [Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models, Math. Prog. A 163(1) (2017), pp. 359–368] to finding second-order critical points, under the same problem smoothness assumptions as were needed for first-order complexity.

Original language | English |
---|---|

Pages (from-to) | 243-256 |

Number of pages | 14 |

Journal | Optimization Methods and Software |

Volume | 35 |

Issue number | 2 |

DOIs | |

Publication status | Published - 3 Mar 2020 |

### Keywords

- complexity analysis
- Nonconvex optimization
- regularization methods

## Fingerprint Dive into the research topics of 'A concise second-order complexity analysis for unconstrained optimization using high-order regularized models'. 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

## Activities

- 1 Invited talk

## Recent results in worst-case evaluation complexity for smooth and non-smooth, exact and inexact, nonconvex optimization

Philippe TOINT (Speaker)

Activity: Talk or presentation types › Invited talk

## Cite this

*Optimization Methods and Software*,

*35*(2), 243-256. https://doi.org/10.1080/10556788.2019.1678033