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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)ϵ2) evaluations of the problem’s functions and their derivatives for finding an ϵapproximate firstorder 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 worstcase 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
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Projects
 1 Active

Complexity in nonlinear optimization
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Project: Research
Activities

Recent results in worstcase evaluation complexity for smooth and nonsmooth, exact and inexact, nonconvex optimization
Philippe TOINT (Speaker)
8 May 2020Activity: Talk or presentation types › Invited talk

5th Conference on Numerical Analysis and Optimization
Philippe Toint (Contributor)
6 Jan 2020 → 9 Jan 2020Activity: Participating in or organising an event types › Participation in workshop, seminar, course

ENSEEIHTIRIT
Philippe Toint (Visiting researcher)
4 Feb 2019 → 3 Apr 2019Activity: Visiting an external institution types › Visiting an external academic institution