  The sparse group Lasso is a widely used statistical model which encourages
the sparsity both on a group and within the group level. In this paper, we
develop an efficient augmented Lagrangian method for large-scale
non-overlapping sparse group Lasso problems with each subproblem being solved
by a superlinearly convergent inexact semismooth Newton method. Theoretically,
we prove that, if the penalty parameter is chosen sufficiently large, the
augmented Lagrangian method converges globally at an arbitrarily fast linear
rate for the primal iterative sequence, the dual infeasibility, and the duality
gap of the primal and dual objective functions. Computationally, we derive
explicitly the generalized Jacobian of the proximal mapping associated with the
sparse group Lasso regularizer and exploit fully the underlying second order
sparsity through the semismooth Newton method. The efficiency and robustness of
our proposed algorithm are demonstrated by numerical experiments on both the
synthetic and real data sets.
