### Abstract

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

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 |

### Fingerprint

### Keywords

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

### Cite this

*Optimization Methods and Software*,

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

}

*Optimization Methods and Software*, vol. 27, no. 2, pp. 197-219. https://doi.org/10.1080/10556788.2011.602076

**Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization.** / Toint, Philippe; Cartis, Coralia; Gould, Nick.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization

AU - Toint, Philippe

AU - Cartis, Coralia

AU - Gould, Nick

N1 - Publication code : FP SB092/2010/05 ; SB04977/2010/05

PY - 2012/4/1

Y1 - 2012/4/1

N2 - 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.

AB - 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.

KW - unconstrained optimization

KW - worst-case analysis

KW - convexity

KW - cubic regularization

KW - ARC methods

UR - http://www.scopus.com/inward/record.url?scp=84859180670&partnerID=8YFLogxK

U2 - 10.1080/10556788.2011.602076

DO - 10.1080/10556788.2011.602076

M3 - Article

VL - 27

SP - 197

EP - 219

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 2

ER -