We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular function to be single-valued and locally Lipschitz continuous. By virtue of this property, we propose a hybrid of proximal gradient and semismooth Newton methods for solving these composite optimization problems, which is a globalized semismooth Newton method. The whole sequence is shown to converge to an L-stationary point under a Kurdyka-Łojasiewicz exponent assumption. Under an additional error bound condition and some other mild conditions, we prove that the sequence converges to a nonisolated L-stationary point at a superlinear convergence rate. Numerical comparison with several existing second order methods reveal that our approach performs comparably well in solving both the q (0<q<1) quasi-norm regularized problems and the fused zero-norm regularization problems.
Wu et al. (Sat,) studied this question.