  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.
