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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. onOptimization, 21 (2011), pp.~1201--1229] to solve the fused lasso problems, oneneeds to solve a sequence of regularized least squares subproblems. In order tomake the level-set method practical, we develop a highly efficient inexactsemismooth Newton based augmented Lagrangian method for solving thesesubproblems. The efficiency of our approach is based on several ingredientsthat constitute the main contributions of this paper. Firstly, an explicitformula for constructing the generalized Jacobian of the proximal mapping ofthe fused lasso regularizer is derived. Secondly, the special structure of thegeneralized Jacobian is carefully extracted and analyzed for the efficientimplementation of the semismooth Newton method. Finally, numerical results,including the comparison between our approach and several state-of-the-artsolvers, on real data sets are presented to demonstrate the high efficiency androbustness of our proposed algorithm in solving challenging large-scale fusedlasso problems.

On efficiently solving the subproblems of a level-set method for fused lasso problems