# Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models

Coralia Cartis, Nicholas I M Gould, Philippe Toint

Research output: Working paper

### Abstract

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 $p$.
Original language English 18 12-2015 Published - 18 Nov 2015

### Publication series

Name naXys technical report

### Fingerprint

Euclidean norm
Local Minimizer
Vector-valued Functions
Taylor series
Criticality
Smooth function
Convex Sets
High Power
Divides
Higher Order
Lower bound
First-order
Derivative
Closed
Evaluation
Term
Approximation
Modeling
Derivatives
Model

### Cite this

@techreport{28146d5316bb485a9216fe68d0ea0abb,
title = "Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models",
abstract = "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 orderregularization 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 $p$.",
author = "Coralia Cartis and Gould, {Nicholas I M} and Philippe Toint",
year = "2015",
month = "11",
day = "18",
language = "English",
volume = "12-2015",
series = "naXys technical report",
type = "WorkingPaper",

}

2015. (naXys technical report).

Research output: Working paper

TY - UNPB

T1 - Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models

AU - Cartis, Coralia

AU - Gould, Nicholas I M

AU - Toint, Philippe

PY - 2015/11/18

Y1 - 2015/11/18

N2 - 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 orderregularization 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 $p$.

AB - 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 orderregularization 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 $p$.

M3 - Working paper

VL - 12-2015

T3 - naXys technical report

BT - Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models

ER -