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

TL;DR

This paper introduces a trust region method using a normal map-based semismooth Newton approach for nonsmooth, nonconvex composite optimization, achieving global and superlinear local convergence with practical applications demonstrated.

Contribution

It develops a novel trust region framework with inexact semismooth Newton steps, new normal map representations, and analyzes BFGS approximations for superlinear convergence.

Findings

Global convergence under standard conditions

Superlinear convergence with BFGS updates

Efficient performance on practical problems

Abstract

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-{\L}ojasiewicz 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.