An Inexact Proximal Newton Method for Nonconvex Composite Minimization

Abstract: In this paper, we propose an inexact proximal Newton-type method for nonconvex composite problems. We establish the global convergence rate of the order $\mathcal{O}(k^{-1/2})$ in terms of the minimal norm of the KKT residual mapping and the local superlinear convergence rate in terms of the sequence generated by the proposed algorithm under the higher-order metric $q$-subregularity property. When the Lipschitz constant of the corresponding gradient is known, we show that the proposed algorithm is well-defined without line search. Extensive numerical experiments on the $\ell_1$-regularized Student's $t$-regression and the group penalized Student's $t$-regression show that the performance of the proposed method is comparable to the state-of-the-art proximal Newton-type methods.