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We propose a novel trust region method for solving a class of nonsmooth,nonconvex composite-type optimization problems. The approach embeds inexactsemismooth Newton steps for finding zeros of a normal map-based stationaritymeasure for the problem in a trust region framework. Based on a new meritfunction and acceptance mechanism, global convergence and transition to fastlocal q-superlinear convergence are established under standard conditions. Inaddition, we verify that the proposed trust region globalization is compatiblewith the Kurdyka-{\L}ojasiewicz inequality yielding finer convergence results.We further derive new normal map-based representations of the associatedsecond-order optimality conditions that have direct connections to the localassumptions required for fast convergence. Finally, we study the behavior ofour algorithm when the Hessian matrix of the smooth part of the objectivefunction is approximated by BFGS updates. We successfully link the KL theory,properties of the BFGS approximations, and a Dennis-Mor{\'e}-type condition toshow superlinear convergence of the quasi-Newton version of our method.Numerical experiments on sparse logistic regression, image compression, and aconstrained log-determinant problem illustrate the efficiency of the proposedalgorithm.

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