  This paper concerns the composite problem of minimizing the sum of a twice
continuously differentiable function $f$ and a nonsmooth convex function. For
this class of nonconvex and nonsmooth problems, by leveraging a practical
inexactness criterion and a novel selection strategy for iterates, we propose
an inexact $q$-order regularized proximal Newton method for $q\in[2,3]$, which
becomes an inexact cubic regularization (CR) method for $q=3$. We prove that
the whole iterate sequence converges to a stationary point for the KL objective
function; and when the objective function has the KL property of exponent
$\theta\in(0,\frac{q-1}{q})$, the convergence has a local $Q$-superlinear rate
of order $\frac{q-1}{\theta q}$. In particular, under a local H\"{o}lderian
error bound of order $\gamma\in(\frac{1}{q-1},1]$ on a second-order stationary
point set, we show that the iterate and objective value sequences converge to a
second-order stationary point and a second-order stationary value,
respectively, with a local $Q$-superlinear rate of order $\gamma(q\!-\!1)$,
specified as the $Q$-quadratic rate for $q=3$ and $\gamma=1$. This is the first
practical inexact CR method with $Q$-quadratic convergence rate for nonconvex
composite optimization. We validate the efficiency of the CR method with
ZeroFPR as the inner solver by applying it to composite optimization problems
with highly nonlinear $f$.
