  This paper explores a specific type of nonconvex sparsity-promoting
regularization problems, namely those involving $\ell_p$-norm regularization,
in conjunction with a twice continuously differentiable loss function. We
propose a novel second-order algorithm designed to effectively address this
class of challenging nonconvex and nonsmooth problems, showcasing several
innovative features: (i) The use of an alternating strategy to solve a
reweighted $\ell_1$ regularized subproblem and the subspace approximate Newton
step. (ii) The reweighted $\ell_1$ regularized subproblem relies on a convex
approximation to the nonconvex regularization term, enabling a closed-form
solution characterized by the soft-thresholding operator. This feature allows
our method to be applied to various nonconvex regularization problems. (iii)
Our algorithm ensures that the iterates maintain their sign values and that
nonzero components are kept away from 0 for a sufficient number of iterations,
eventually transitioning to a perturbed Newton method. (iv) We provide
theoretical guarantees of global convergence, local superlinear convergence in
the presence of the Kurdyka-\L ojasiewicz (KL) property, and local quadratic
convergence when employing the exact Newton step in our algorithm. We also
showcase the effectiveness of our approach through experiments on a diverse set
of model prediction problems.
