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

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 $p$.