# A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization

arXiv: https://arxiv.org/abs/2106.09340 · HTML: https://mlsanity.com/paper/2106.09340/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2021.

## Built on by (most cited first, 4 of 4)

- [A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties](https://mlsanity.com/paper/2305.05828/) (2023)
- [A New Random Reshuffling Method for Nonsmooth Nonconvex Finite-sum Optimization](https://mlsanity.com/paper/2312.01047/) (2023)
- [Achieving Linear Speedup for Composite Federated Learning](https://mlsanity.com/paper/2602.03357/) (2026)
- [Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Framework](https://mlsanity.com/paper/2608.03001/) (2026)

## Abstract (arXiv)

We propose a novel trust region method for solving a class of nonsmooth, nonconvex composite-type optimization problems. The approach embeds inexact semismooth Newton steps for finding zeros of a normal map-based stationarity measure for the problem in a trust region framework. Based on a new merit function and acceptance mechanism, global convergence and transition to fast local q-superlinear convergence are established under standard conditions. In addition, we verify that the proposed trust region globalization is compatible with the Kurdyka-Lojasiewicz inequality yielding finer convergence results. We further derive new normal map-based representations of the associated second-order optimality conditions that have direct connections to the local assumptions required for fast convergence. Finally, we study the behavior of our algorithm when the Hessian matrix of the smooth part of the objective function is approximated by BFGS updates. We successfully link the KL theory, properties of the BFGS approximations, and a Dennis-Mor'e-type condition to show superlinear convergence of the quasi-Newton version of our method. Numerical experiments on sparse logistic regression, image compression, and a constrained log-determinant problem illustrate the efficiency of the proposed algorithm.