A linesearch-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization

TL;DR

This paper introduces a new linesearch-based semismooth Newton method with adaptive parameters for nonsmooth, nonconvex composite optimization, offering improved convergence analysis and practical efficiency over existing trust region methods.

Contribution

It develops a simplified linesearch variant of a semismooth Newton method with adaptive parameter estimation, avoiding explicit Lipschitz constants and weaker convergence assumptions.

Findings

Global convergence and Kurdyka-Łojasiewicz convergence established.

Achieves transition to fast local q-superlinear convergence.

Numerical experiments show improved efficiency in applications like sparse logistic regression.

Abstract

We propose a novel linesearch variant of the trust region normal map-based semismooth Newton method developed in [Ouyang and Milzarek, Math. Program. 212(1-2), 389--435 (2025)] for solving a class of nonsmooth, nonconvex composite-type optimization problems. Our approach uses adaptive parameter estimation techniques, which allow us to avoid explicit and potentially expensive Lipschitz constant computations. We provide extensive convergence results including global convergence, convergence of the iterates under the Kurdyka-{\L}ojasiewicz inequality, and transition to fast local q-superlinear convergence. Compared to the original trust region framework, the linesearch-based algorithm is simpler and the overall convergence analysis can be conducted under weaker assumptions -- in particular, without requiring explicit boundedness conditions on the Hessian approximations and iterates. Numerical experiments on sparse logistic regression, image compression, and nonlinear least squares with group penalty terms demonstrate the efficiency of the proposed approach.