  In applying the level-set method developed in [Van den Berg and Friedlander,
SIAM J. on Scientific Computing, 31 (2008), pp.~890--912 and SIAM J. on
Optimization, 21 (2011), pp.~1201--1229] to solve the fused lasso problems, one
needs to solve a sequence of regularized least squares subproblems. In order to
make the level-set method practical, we develop a highly efficient inexact
semismooth Newton based augmented Lagrangian method for solving these
subproblems. The efficiency of our approach is based on several ingredients
that constitute the main contributions of this paper. Firstly, an explicit
formula for constructing the generalized Jacobian of the proximal mapping of
the fused lasso regularizer is derived. Secondly, the special structure of the
generalized Jacobian is carefully extracted and analyzed for the efficient
implementation of the semismooth Newton method. Finally, numerical results,
including the comparison between our approach and several state-of-the-art
solvers, on real data sets are presented to demonstrate the high efficiency and
robustness of our proposed algorithm in solving challenging large-scale fused
lasso problems.
