  An adaptive regularization algorithm using high-order models is proposed for
partially separable convexly constrained nonlinear optimization problems whose
objective function contains non-Lipschitzian $\ell_q$-norm regularization terms
for $q\in (0,1)$. It is shown that the algorithm using an $p$-th order Taylor
model for $p$ odd needs in general at most $O(\epsilon^{-(p+1)/p})$ evaluations
of the objective function and its derivatives (at points where they are
defined) to produce an $\epsilon$-approximate first-order critical point. This
result is obtained either with Taylor models at the price of requiring the
feasible set to be 'kernel-centered' (which includes bound constraints and many
other cases of interest), or for non-Lipschitz models, at the price of passing
the difficulty to the computation of the step. Since this complexity bound is
identical in order to that already known for purely Lipschitzian minimization
subject to convex constraints [CartGoulToin2016], the new result shows that
introducing non-Lipschitzian singularities in the objective function may not
affect the worst-case evaluation complexity order. The result also shows that
using the problem's partially separable structure (if present) does not affect
complexity order either. A final (worse) complexity bound is derived for the
case where Taylor models are used with a general convex feasible set.
