  The octagonal shrinkage and clustering algorithm for regression (OSCAR),
equipped with the $\ell_1$-norm and a pair-wise $\ell_{\infty}$-norm
regularizer, is a useful tool for feature selection and grouping in
high-dimensional data analysis. The computational challenge posed by OSCAR, for
high dimensional and/or large sample size data, has not yet been well resolved
due to the non-smoothness and inseparability of the regularizer involved. In
this paper, we successfully resolve this numerical challenge by proposing a
sparse semismooth Newton-based augmented Lagrangian method to solve the more
general SLOPE (the sorted L-one penalized estimation) model. By appropriately
exploiting the inherent sparse and low-rank property of the generalized
Jacobian of the semismooth Newton system in the augmented Lagrangian
subproblem, we show how the computational complexity can be substantially
reduced. Our algorithm presents a notable advantage in the high-dimensional
statistical regression settings. Numerical experiments are conducted on real
data sets, and the results demonstrate that our algorithm is far superior, in
both speed and robustness, than the existing state-of-the-art algorithms based
on first-order iterative schemes, including the widely used accelerated
proximal gradient (APG) method and the alternating direction method of
multipliers (ADMM).
