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Abstract
An adaptive regularization algorithm using highorder models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains nonLipschitzian 'qnorm regularization terms for q ϵ (0; 1). It is shown that the algorithm using a pthorder Taylor model for p odd needs in general at most O(ϵ ^{(p+1)=p}) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ϵapproximate firstorder critical point. This result is obtained either with Taylor models, at the price of requiring the feasible set to be kernel centered" (which includes bound constraints and many other cases of interest), or with nonLipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [C. Cartis, N. I. M. Gould, and Ph. L. Toint, IMA J. Numer. Anal., 32 (2012), pp. 16621695], the new result shows that introducing nonLipschitzian singularities in the objective function may not affect the worstcase evaluation complexity order. The result also shows that using the problem's partially separable structure (if present) does not affect the complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.
Original language  English 

Pages (fromto)  874903 
Number of pages  30 
Journal  SIAM Journal on Optimization 
Volume  29 
Issue number  1 
DOIs  
Publication status  Published  15 Apr 2019 
Keywords
 Non_Lipschitz optimization
 Complexity theory
 partial separability
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Projects
 2 Active

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

Activities

Recent progress in evaluation complexity for nonlinear optimization
Philippe Toint (Speaker)
4 Oct 2019Activity: Talk or presentation types › Invited talk

Departement of Applied Mathematics, Polytechnic University of Hong Kong
Philippe Toint (Visiting researcher)
15 Nov 2018 → 15 Dec 2018Activity: Visiting an external institution types › Visiting an external academic institution

A path and some adventures in the jungle of highorder nonlinear optimization
Philippe Toint (Speaker)
23 Oct 2017Activity: Talk or presentation types › Invited talk