  We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the
elastic-net penalized Huber loss regression and quantile regression in high
dimensional settings. Unlike existing coordinate descent type algorithms, the
SNCD updates each regression coefficient and its corresponding subgradient
simultaneously in each iteration. It combines the strengths of the coordinate
descent and the semismooth Newton algorithm, and effectively solves the
computational challenges posed by dimensionality and nonsmoothness. We
establish the convergence properties of the algorithm. In addition, we present
an adaptive version of the "strong rule" for screening predictors to gain extra
efficiency. Through numerical experiments, we demonstrate that the proposed
algorithm is very efficient and scalable to ultra-high dimensions. We
illustrate the application via a real data example.

Computation;Machine Learning