An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

TL;DR

This paper introduces an inexact regularized proximal Newton method for nonconvex, nonsmooth optimization, providing convergence guarantees and demonstrating efficiency through numerical experiments on statistical and image restoration problems.

Contribution

It proposes a novel inexact regularized proximal Newton method with convergence analysis and practical algorithms for nonconvex nonsmooth optimization.

Findings

Global convergence and R-linear rate for $ ho=0$ case.

Superlinear convergence under certain error bounds.

Numerical results outperforming state-of-the-art methods.

Abstract

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.