On the complexity of finding first-order critical points in constrained nonlinear optimization

Coralia Cartis; Nicholas I M Gould; Philippe L. Toint

doi:10.1007/s10107-012-0617-9

On the complexity of finding first-order critical points in constrained nonlinear optimization

Coralia Cartis, Nicholas I M Gould, Philippe L. Toint

Namur Institute for Complex Systems

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Résumé

The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

langue originale	Anglais
Pages (de - à)	93-106
Nombre de pages	14
journal	Mathematical Programming
Volume	144
Numéro de publication	1-2
Les DOIs	https://doi.org/10.1007/s10107-012-0617-9
Etat de la publication	Publié - 2014

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10.1007/s10107-012-0617-9

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35 Citations
13 Article
3 Article de travail
1 Livre
1 Chapitre

Evaluation complexity of algorithms for nonconvex optimization
Cartis, C., Gould, N. I. M. & TOINT, P., juil. 2022, SIAM. 600 p. (SIAM-MOS Series on Optimization)
Résultats de recherche: Livre/Rapport/Revue › Livre
Adaptive regularization algorithms with inexact evaluations for nonconvex optimization
Bellavia, S., Gurioli, G., Morini, B. & Toint, P., 2 janv. 2020, Dans: SIAM Journal on Optimization. 29, 4, p. 2881-2915 35 p.
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Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities
Chen, X., Toint, P. & Wang, H., 15 avr. 2019, Dans: SIAM Journal on Optimization. 29, 1, p. 874-903 30 p.
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2 Actif

Complexity in nonlinear optimization
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Projet: Recherche
ADALGOPT: ADALGOPT - Algorithmes avancés en optimisation non-linéaire
Sartenaer, A. & TOINT, P.
1/01/87 → …
Projet: Axe de recherche

4 Présentation orale
3 Recherche/Enseignement dans une institution externe
1 Participation à un atelier/workshop, un séminaire, un cours
1 Visite à une institution académique externe

How much patience do you have? Issues in complexity for nonlinear optimization
Philippe Toint (Orateur invité)
5 févr. 2016
Activité: Discours ou présentation › Présentation orale
How much patience do you have? Issues in complexity for nonlinear optimization
Philippe Toint (Orateur)
31 janv. 2016
Activité: Discours ou présentation › Présentation orale
Hong Kong Polytechnic University
Philippe Toint (Chercheur visiteur)
31 janv. 2016 → 14 févr. 2016
Activité: Visite d'une organisation externe › Recherche/Enseignement dans une institution externe

Leverhulme Fellow
TOINT, Philippe (Bénéficiaire), sept. 2015
Prix: Bourse attribuée par concours

Contient cette citation

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title = "On the complexity of finding first-order critical points in constrained nonlinear optimization",

abstract = "The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. {\textcopyright} 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.",

keywords = "Constrained nonlinear optimization, Evaluation complexity, Worst-case analysis",

author = "Coralia Cartis and Gould, {Nicholas I M} and Toint, {Philippe L.}",

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doi = "10.1007/s10107-012-0617-9",

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AB - The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

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On the complexity of finding first-order critical points in constrained nonlinear optimization

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