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### Abstract

The adaptive cubic regularization algorithms described in Cartis, Gould and Toint [Adaptive cubic regularisation methods for unconstrained optimization Part II: Worst-case function- and derivative-evaluation complexity, Math. Program. (2010), doi:10.1007/s10107-009-0337-y (online)]; [Part I: Motivation, convergence and numerical results, Math. Program. 127(2) (2011), pp. 245-295] for unconstrained (nonconvex) optimization are shown to have improved worst-case efficiency in terms of the function- and gradient-evaluation count when applied to convex and strongly convex objectives. In particular, our complexity upper bounds match in order (as a function of the accuracy of approximation), and sometimes even improve, those obtained by Nesterov [Introductory Lectures on Convex Optimization, Kluwer Academic Publishers, Dordrecht, 2004; Accelerating the cubic regularization of Newton's method on convex problems, Math. Program. 112(1) (2008), pp. 159-181] and Nesterov and Polyak [Cubic regularization of Newton's method and its global performance, Math. Program. 108(1) (2006), pp. 177-205] for these same problem classes, without requiring exact Hessians or exact or global solution of the subproblem. An additional outcome of our approximate approach is that our complexity results can naturally capture the advantages of both first- and second-order methods. © 2012 Taylor & Francis.

Original language | English |
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Pages (from-to) | 197-219 |

Number of pages | 23 |

Journal | Optimization Methods and Software |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2012 |

### Keywords

- unconstrained optimization
- worst-case analysis
- convexity
- cubic regularization
- ARC methods

## Fingerprint Dive into the research topics of 'Evaluation complexity of adaptive cubic regularization methods for convex unconstrained 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

## 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

## Polytechnic University of Hong Kong

Philippe Toint (Visiting researcher)

31 Jan 2016 → 14 Feb 2016

Activity: Visiting an external institution types › Research/Teaching in a external institution

## 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

## Cite this

Toint, P., Cartis, C., & Gould, N. (2012). Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.

*Optimization Methods and Software*,*27*(2), 197-219. https://doi.org/10.1080/10556788.2011.602076