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

arXiv: https://arxiv.org/abs/1706.08732 · HTML: https://mlsanity.com/paper/1706.08732/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2017.

## Built on by (most cited first, 4 of 4)

- [Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates](https://mlsanity.com/paper/2205.10672/) (2022)
- [A proximal-proximal majorization-minimization algorithm for nonconvex tuning-free robust regression problems](https://mlsanity.com/paper/2106.13683/) (2021)
- [Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction](https://mlsanity.com/paper/2405.07965/) (2024)
- [Efficient Group Lasso Regularized Rank Regression with Simulation-Based Tuning](https://mlsanity.com/paper/2510.11546/) (2025)

## Abstract (arXiv)

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.