# An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

arXiv: https://arxiv.org/abs/2209.09119 · HTML: https://mlsanity.com/paper/2209.09119/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2022.

## Built on by (most cited first, 2 of 2)

- [An inexact LPA for DC composite optimization and application to matrix completions with outliers](https://mlsanity.com/paper/2303.16822/) (2023)
- [A Proximal Modified Quasi-Newton Method for Nonsmooth Regularized Optimization](https://mlsanity.com/paper/2409.19428/) (2024)

## Abstract (arXiv)

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 varrhoth 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∈(0,1), by assuming that cluster points satisfy a locally H"olderian 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 ℓ1-regularized Student's t-regressions, group penalized Student's t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method.