# A dual semismooth Newton based augmented Lagrangian method for large-scale linearly constrained sparse group square-root Lasso problems

arXiv: https://arxiv.org/abs/2111.13878 · HTML: https://mlsanity.com/paper/2111.13878/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 50 references that are in the map.
- First posted on arXiv 27 Nov 2021.

## Builds on (most cited first, 6 of 6)

- [Algorithms for Fitting the Constrained Lasso](https://mlsanity.com/paper/1611.01511/) (2016)
- [A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems](https://mlsanity.com/paper/1607.05428/) (2016)
- [An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems](https://mlsanity.com/paper/1712.05910/) (2017)
- [The flare Package for High Dimensional Linear Regression and Precision Matrix Estimation in R](https://mlsanity.com/paper/2006.15419/) (2020)
- [On Regularized Square-root Regression Problems: Distributionally Robust Interpretation and Fast Computations](https://mlsanity.com/paper/2109.03632/) (2021)
- [Variational Analysis of the Ky Fan k-norm](https://mlsanity.com/paper/1601.07430/) (2016)

## Built on by (most cited first, 1 of 1)

- [Learning the hub graphical Lasso model with the structured sparsity via an efficient algorithm](https://mlsanity.com/paper/2308.08852/) (2023)

## Abstract (arXiv)

Square-root Lasso problems are proven robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the SSN method. To apply the SSN method, the positive definiteness of the generalized Jacobian is very important. Hence we characterize the equivalence of its positive definiteness and the constraint nondegeneracy condition of the corresponding primal problem. In numerical implementation, we fully employ the second order sparsity so that the Newton direction can be efficiently obtained. Numerical experiments demonstrate the efficiency of the proposed algorithm.