### 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 $p$.

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 |
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Number of pages | 18 |

Volume | 12-2015 |

Publication status | Published - 18 Nov 2015 |

### Publication series

Name | naXys technical report |
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### Cite this

Cartis, C., Gould, N. I. M., & Toint, P. (2015).

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