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### Abstract

An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most O(|log(ϵ)|ϵ-2) evaluations of the problem’s functions and their derivatives for finding an ϵ-approximate first-order stationary point. This complexity bound therefore generalizes that provided by Bellavia et al. (Theoretical study of an adaptive cubic regularization method with dynamic inexact Hessian information. arXiv:1808.06239, 2018) for inexact methods for smooth nonconvex problems, and is within a factor | log (ϵ) | of the optimal bound known for smooth and nonsmooth nonconvex minimization with exact evaluations. A practically more restrictive variant of the algorithm with worst-case complexity O(| log (ϵ) | + ϵ
^{- 2}) is also presented.

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

Number of pages | 19 |

Journal | Mathematical Programming |

DOIs | |

Publication status | Accepted/In press - 1 Jan 2020 |

### Keywords

- evaluation complexity
- nonsmooth problems
- nonconvex optimization
- inexact evaluations
- composite functions
- Evaluation complexity
- Nonconvex optimization
- Composite functions
- Nonsmooth problems
- Inexact evaluations

## Fingerprint Dive into the research topics of 'An algorithm for the minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity'. Together they form a unique fingerprint.

## Projects

- 1 Active

## Complexity in nonlinear optimization

TOINT, P., Gould, N. I. M. & Cartis, C.

1/11/08 → …

Project: Research

## Activities

## Recent results in worst-case evaluation complexity for smooth and non-smooth, exact and inexact, nonconvex optimization

Philippe TOINT (Speaker)

Activity: Talk or presentation types › Invited talk

## 5th Conference on Numerical Analysis and Optimization

Philippe Toint (Contributor)

Activity: Participating in or organising an event types › Participation in workshop, seminar, course

## ENSEEIHT-IRIT

Philippe Toint (Visiting researcher)

Activity: Visiting an external institution types › Visiting an external academic institution