Projects per year
Abstract
Given a sufficiently smooth vectorvalued function $r(x)$,
a local minimizer of $\r(x)\_2$ within a closed, nonempty, convex set
$\calF$ is sought by modelling $\r(x)\^q_2 / q$
with a $p$th order Taylorseries 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 firstorder 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, nonempty, convex set
$\calF$ is sought by modelling $\r(x)\^q_2 / q$
with a $p$th order Taylorseries 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 firstorder 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 $p$.
Original language  English 

Number of pages  18 
Volume  122015 
Publication status  Published  18 Nov 2015 
Publication series
Name  naXys technical report 

Fingerprint Dive into the research topics of 'Improved worstcase evaluation complexity for potentially rankdeficient nonlinear leastEuclideannorm problems using higherorder regularized models'. Together they form a unique fingerprint.
Projects
 2 Active

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

Activities

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

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

International Conference on Numerical Analysis and Optimization
Philippe Toint (Contributor)
3 Aug 2016 → 7 Aug 2016Activity: Participating in or organising an event types › Participation in conference