Projects per year

### Abstract

The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order $p$ (for $p\geq 1$) and to assume Lipschitz continuity of the $p$-th derivative, then an $\epsilon$-approximate first-order critical point can be computed in at most $O(\epsilon^{-(p+1)/p})$ evaluations of the problem's objective function and its derivatives. This generalizes and subsumes results known for $p=1$ and $p=2$.

Original language | English |
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Place of Publication | 2015 |

Publisher | Namur center for complex systems |

Number of pages | 8 |

Volume | naXys-05-2015 |

Publication status | Published - Jun 2015 |

### Publication series

Name | naXys Technical Reports |
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Publisher | naXys |

Volume | 05-2015 |

### Keywords

- Nonlinear optimization
- Complexity theory
- High-order models

## Fingerprint Dive into the research topics of 'Worst-case evaluation complexity for unconstrained nonlinear 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

## A path and some adventures in the jungle of high-order nonlinear optimization

Philippe Toint (Speaker)

23 Oct 2017

Activity: Talk or presentation types › Invited talk

## A path and some adventures in the jungle of high-order nonlinear optimization

Philippe Toint (Speaker)

24 Oct 2017

Activity: Talk or presentation types › Invited talk

## High-order optimality conditions in nonlinear optimization: necessary conditions and a conceptual approach of evaluation complexity

Philippe Toint (Speaker)

10 Aug 2016

Activity: Talk or presentation types › Invited talk

## Cite this

Birgin, E., Gardenghi, J., Martinez, J-M., Santos, S. A., & Toint, P. (2015).

*Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models*. (naXys Technical Reports; Vol. 05-2015). Namur center for complex systems.