### 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≥ 1 ) and to assume Lipschitz continuity of the p-th derivative, then an ϵ-approximate first-order critical point can be computed in at most O(ϵ
^{-}
^{(}
^{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 |
---|---|

Pages (from-to) | 359-368 |

Number of pages | 10 |

Journal | Mathematical Programming |

Volume | 163 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 15 Apr 2017 |

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### Keywords

- Evaluation complexity
- High-order models
- Nonlinear optimization
- Regularization
- Unconstrained optimization

### Cite this

*Mathematical Programming*,

*163*(1-2), 359-368. https://doi.org/10.1007/s10107-016-1065-8

}

*Mathematical Programming*, vol. 163, no. 1-2, pp. 359-368. https://doi.org/10.1007/s10107-016-1065-8

**Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models.** / Birgin, E. G.; Gardenghi, J. L.; Martínez, J. M.; Santos, S. A.; Toint, P. L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

AU - Birgin, E. G.

AU - Gardenghi, J. L.

AU - Martínez, J. M.

AU - Santos, S. A.

AU - Toint, P. L.

PY - 2017/4/15

Y1 - 2017/4/15

N2 - 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≥ 1 ) and to assume Lipschitz continuity of the p-th derivative, then an ϵ-approximate first-order critical point can be computed in at most O(ϵ - ( 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.

AB - 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≥ 1 ) and to assume Lipschitz continuity of the p-th derivative, then an ϵ-approximate first-order critical point can be computed in at most O(ϵ - ( 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.

KW - Evaluation complexity

KW - High-order models

KW - Nonlinear optimization

KW - Regularization

KW - Unconstrained optimization

UR - http://www.scopus.com/inward/record.url?scp=84984820379&partnerID=8YFLogxK

U2 - 10.1007/s10107-016-1065-8

DO - 10.1007/s10107-016-1065-8

M3 - Article

AN - SCOPUS:84984820379

VL - 163

SP - 359

EP - 368

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -