  Support vector machine (SVM) has proved to be a successful approach for
machine learning. Two typical SVM models are the L1-loss model for support
vector classification (SVC) and $\epsilon$-L1-loss model for support vector
regression (SVR). Due to the nonsmoothness of the L1-loss function in the two
models, most of the traditional approaches focus on solving the dual problem.
In this paper, we propose an augmented Lagrangian method for the L1-loss model,
which is designed to solve the primal problem. By tackling the nonsmooth term
in the model with Moreau-Yosida regularization and the proximal operator, the
subproblem in augmented Lagrangian method reduces to a nonsmooth linear system,
which can be solved via the quadratically convergent semismooth Newton's
method. Moreover, the high computational cost in semismooth Newton's method can
be significantly reduced by exploring the sparse structure in the generalized
Jacobian. Numerical results on various datasets in LIBLINEAR show that the
proposed method is competitive with the most popular solvers in both speed and
accuracy.
