Projects per year

### Abstract

When solving the general smooth nonlinear and possibly nonconvex optimization problem involving equality and/or inequality constraints, an approximate first-order critical point of accuracy ∈ can be obtained by a second-order method using cubic regularization in at most O(∈<sup>-3/2</sup> ) evaluations of problem functions, the same order bound as in the unconstrained case. This result is obtained by first showing that the same result holds for inequality constrained nonlinear least-squares. As a consequence, the presence of (possibly nonconvex) equality/inequality constraints does not affect the complexity of finding approximate first-order critical points in nonconvex optimization. This result improves on the best known (O(∈<sup>-2</sup> )) evaluation-complexity bound for solving general nonconvexly constrained optimization problems.

Original language | English |
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Pages (from-to) | 836-851 |

Number of pages | 16 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 53 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 |

### Keywords

- Complexity theory
- Nonlinear optimization
- Constrained problems
- Least-squares problems

## Fingerprint Dive into the research topics of 'On the evaluation complexity of constrained nonlinear least-squares and general constrained nonlinear optimization using second-order methods'. 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)

Activity: Talk or presentation types › Invited talk

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

Philippe Toint (Invited speaker)

Activity: Talk or presentation types › Oral presentation

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

Philippe Toint (Speaker)

Activity: Talk or presentation types › Oral presentation

## Prizes

## Cite this

*SIAM Journal on Numerical Analysis*,

*53*(2), 836-851. https://doi.org/10.1137/130915546