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We develop a fast and robust algorithm for solving large scale convexcomposite optimization models with an emphasis on the $\ell_1$-regularizedleast squares regression (Lasso) problems. Despite the fact that there exist alarge number of solvers in the literature for the Lasso problems, we found thatno solver can efficiently handle difficult large scale regression problems withreal data. By leveraging on available error bound results to realize theasymptotic superlinear convergence property of the augmented Lagrangianalgorithm, and by exploiting the second order sparsity of the problem throughthe semismooth Newton method, we are able to propose an algorithm, called {\scSsnal}, to efficiently solve the aforementioned difficult problems. Under verymild conditions, which hold automatically for Lasso problems, both the primaland the dual iteration sequences generated by {\sc Ssnal} possess a fast linearconvergence rate, which can even be superlinear asymptotically. Numericalcomparisons between our approach and a number of state-of-the-art solvers, onreal data sets, are presented to demonstrate the high efficiency and robustnessof our proposed algorithm in solving difficult large scale Lasso problems.

A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems