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

TL;DR

This paper introduces a dual semismooth Newton augmented Lagrangian method to efficiently solve large-scale linearly constrained sparse group square-root Lasso problems, addressing nonsmoothness and leveraging second order sparsity.

Contribution

It develops a novel dual SSN-ALM algorithm that effectively handles nonsmooth terms and large-scale problems in structured sparse regression.

Findings

Algorithm demonstrates high efficiency in numerical experiments.

Effectively handles nonsmooth and large-scale problems.

Characterizes positive definiteness via primal constraint nondegeneracy.

Abstract

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.