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

TL;DR

This paper analyzes the local convergence of the semismooth Newton method for composite optimization, relaxing traditional assumptions and establishing superlinear convergence under broader conditions.

Contribution

It provides new characterizations of nonsingularity, verifies local smoothness for residual mappings, and introduces an algorithm with superlinear convergence on active manifolds.

Findings

Equivalent characterizations of nonsingularity for broad classes.

Local smoothness of residual mappings in composite optimization.

Superlinear convergence of the proposed algorithm under relaxed conditions.

Abstract

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.