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Résumé
Given a sufficiently smooth vector-valued function $r(x)$,
a local minimizer of $\|r(x)\|_2$ within a closed, non-empty, convex set
$\calF$ is sought by modelling $\|r(x)\|^q_2 / q$
with a $p$-th order Taylor-series approximation plus a $(p+1)$-st order
regularization term for given even $p$ and some appropriate associated $q$.
The resulting algorithm is guaranteed to find a value $\bar{x}$ for which
$\|r(\bar{x})\|_2 \leq \epsilon_p$ or $\chi(\bar{x}) \leq \epsilon_d$, for
some first-order criticality measure $\chi(x)$ of $\|r(x)\|_2$ within $\calF$,
using at most $O(\max\{\max(\epsilon_d,\chi_{\min})^{-(p+1)/p},
\max(\epsilon_p,r_{\min})^{-1/2^i}\})$
evaluations of $r(x)$ and its derivatives;
here $r_{\min}$ and $\chi_{\min} \geq 0$
are any lower bounds on $\|r(x)\|_2$ and $\chi(x)$, respectively,
and $2^i$ is the highest power of $2$ that divides $p$.
a local minimizer of $\|r(x)\|_2$ within a closed, non-empty, convex set
$\calF$ is sought by modelling $\|r(x)\|^q_2 / q$
with a $p$-th order Taylor-series approximation plus a $(p+1)$-st order
regularization term for given even $p$ and some appropriate associated $q$.
The resulting algorithm is guaranteed to find a value $\bar{x}$ for which
$\|r(\bar{x})\|_2 \leq \epsilon_p$ or $\chi(\bar{x}) \leq \epsilon_d$, for
some first-order criticality measure $\chi(x)$ of $\|r(x)\|_2$ within $\calF$,
using at most $O(\max\{\max(\epsilon_d,\chi_{\min})^{-(p+1)/p},
\max(\epsilon_p,r_{\min})^{-1/2^i}\})$
evaluations of $r(x)$ and its derivatives;
here $r_{\min}$ and $\chi_{\min} \geq 0$
are any lower bounds on $\|r(x)\|_2$ and $\chi(x)$, respectively,
and $2^i$ is the highest power of $2$ that divides $p$.
langue originale | Anglais |
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Nombre de pages | 18 |
Volume | 12-2015 |
Etat de la publication | Publié - 18 nov. 2015 |
Série de publications
Nom | naXys technical report |
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Empreinte digitale Examiner les sujets de recherche de « Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models ». Ensemble, ils forment une empreinte digitale unique.
Projets
- 2 Actif
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Complexity in nonlinear optimization
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Projet: Recherche
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ADALGOPT: ADALGOPT - Algorithmes avancés en optimisation non-linéaire
1/01/87 → …
Projet: Axe de recherche
Activités
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A path and some adventures in the jungle of high-order nonlinear optimization
Philippe Toint (Orateur)
24 oct. 2017Activité: Types de discours ou de présentation › Discours invité
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A path and some adventures in the jungle of high-order nonlinear optimization
Philippe Toint (Orateur)
23 oct. 2017Activité: Types de discours ou de présentation › Discours invité
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International Conference on Numerical Analysis and Optimization
Philippe Toint (Orateur)
3 août 2016 → 7 août 2016Activité: Types de Participation ou d'organisation d'un événement › Participation à une conférence, un congrès