In this paper we present GSSN, a globalized SCD semismooth \(^{*}\) Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth \(^{*}\) Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method.
