### Abstract

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 |
---|---|

Number of pages | 18 |

Volume | 12-2015 |

Publication status | Published - 18 Nov 2015 |

### Publication series

Name | naXys technical report |
---|

### Fingerprint

### Cite this

*Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models*. (naXys technical report).

}

**Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models.** / Cartis, Coralia; Gould, Nicholas I M; Toint, Philippe.

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 -