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We propose an algorithm, semismooth Newton coordinate descent (SNCD), for theelastic-net penalized Huber loss regression and quantile regression in highdimensional settings. Unlike existing coordinate descent type algorithms, theSNCD updates each regression coefficient and its corresponding subgradientsimultaneously in each iteration. It combines the strengths of the coordinatedescent and the semismooth Newton algorithm, and effectively solves thecomputational challenges posed by dimensionality and nonsmoothness. Weestablish the convergence properties of the algorithm. In addition, we presentan adaptive version of the "strong rule" for screening predictors to gain extraefficiency. Through numerical experiments, we demonstrate that the proposedalgorithm is very efficient and scalable to ultra-high dimensions. Weillustrate the application via a real data example.

Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression