Projects per year

### Abstract

It is shown that the steepest-descent and Newton's methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to O(ε^{-2}) to drive the norm of the gradient below ε. This shows that the upper bound of O(ε^{-2}) evaluations known for the steepest descent is tight and that Newton's method may be as slow as the steepest-descent method in the worst case. The improved evaluation complexity bound of O(ε^{-3/2} ) evaluations known for cubically regularized Newton's methods is also shown to be tight. © 2010 Society for Industrial and Applied Mathematics.

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

Pages (from-to) | 2833-2852 |

Number of pages | 20 |

Journal | SIAM Journal on Optimization |

Volume | 20 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 2010 |

## Fingerprint Dive into the research topics of 'On the complexity of steepest descent, Newton's and regularized Newton's methods for nonconvex unconstrained optimization problems'. 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

## How much patience do you have? Issues in complexity for nonlinear optimization

Philippe Toint (Invited speaker)

5 Feb 2016

Activity: Talk or presentation types › Oral presentation

## How much patience do you have? Issues in complexity for nonlinear optimization

Philippe Toint (Speaker)

31 Jan 2016

Activity: Talk or presentation types › Oral presentation

## How much patience do you have? Issues in complexity for nonlinear optimization

Philippe Toint (Invited speaker)

10 Dec 2015

Activity: Talk or presentation types › Oral presentation

## Cite this

Cartis, C., Gould, N., & Toint, P. (2010). On the complexity of steepest descent, Newton's and regularized Newton's methods for nonconvex unconstrained optimization problems.

*SIAM Journal on Optimization*,*20*(6), 2833-2852. https://doi.org/10.1137/090774100