# On the local convergence of the semismooth Newton method for composite optimization

arXiv: https://arxiv.org/abs/2211.01127 · HTML: https://mlsanity.com/paper/2211.01127/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2022.

## Built on by (most cited first, 1 of 1)

- [Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Framework](https://mlsanity.com/paper/2608.03001/) (2026)

## Abstract (arXiv)

In this paper, we consider a large class of nonlinear equations derived from first-order type methods for solving composite optimization problems. Traditional approaches to establishing superlinear convergence rates of semismooth Newton-type methods for solving nonlinear equations usually postulate either nonsingularity of the B-Jacobian or smoothness of the equation. We investigate the feasibility of both conditions. For the nonsingularity condition, we present equivalent characterizations in broad generality, and illustrate that they are easy-to-check criteria for some examples. For the smoothness condition, we show that it holds locally for a large class of residual mappings derived from composite optimization problems. Furthermore, we investigate a relaxed version of the smoothness condition - smoothness restricted to certain active manifolds. We present a conceptual algorithm utilizing such structures and prove that it has a superlinear convergence rate.