  A regularization algorithm (AR1pGN) for unconstrained nonlinear minimization
is considered, which uses a model consisting of a Taylor expansion of arbitrary
degree and regularization term involving a possibly non-smooth norm. It is
shown that the non-smoothness of the norm does not affect the
$O(\epsilon_1^{-(p+1)/p})$ upper bound on evaluation complexity for finding
first-order $\epsilon_1$-approximate minimizers using $p$ derivatives, and that
this result does not hinge on the equivalence of norms in $\Re^n$. It is also
shown that, if $p=2$, the bound of $O(\epsilon_2^{-3})$ evaluations for finding
second-order $\epsilon_2$-approximate minimizers still holds for a variant of
AR1pGN named AR2GN, despite the possibly non-smooth nature of the
regularization term. Moreover, the adaptation of the existing theory for
handling the non-smoothness results in an interesting modification of the
subproblem termination rules, leading to an even more compact complexity
analysis. In particular, it is shown when the Newton's step is acceptable for
an adaptive regularization method. The approximate minimization of quadratic
polynomials regularized with non-smooth norms is then discussed, and a new
approximate second-order necessary optimality condition is derived for this
case. An specialized algorithm is then proposed to enforce the first- and
second-order conditions that are strong enough to ensure the existence of a
suitable step in AR1pGN (when $p=2$) and in AR2GN, and its iteration complexity
is analyzed.
