  This paper focuses on the minimization of a sum of a twice continuously
differentiable function $f$ and a nonsmooth convex function. An inexact
regularized proximal Newton method is proposed by an approximation to the
Hessian of $f$ involving the $\varrho$th power of the KKT residual. For
$\varrho=0$, we justify the global convergence of the iterate sequence for the
KL objective function and its R-linear convergence rate for the KL objective
function of exponent $1/2$. For $\varrho\in(0,1)$, by assuming that cluster
points satisfy a locally H\"{o}lderian error bound of order $q$ on a
second-order stationary point set and a local error bound of order
$q>1\!+\!\varrho$ on the common stationary point set, respectively, we
establish the global convergence of the iterate sequence and its superlinear
convergence rate with order depending on $q$ and $\varrho$. A dual semismooth
Newton augmented Lagrangian method is also developed for seeking an inexact
minimizer of subproblems. Numerical comparisons with two state-of-the-art
methods on $\ell_1$-regularized Student's $t$-regressions, group penalized
Student's $t$-regressions, and nonconvex image restoration confirm the
efficiency of the proposed method.
