A Dual Semismooth Newton Based Augmented Lagrangian Method for Large-Scale Linearly Constrained Sparse Group Square-Root Lasso Problems

• Published in: Journal of scientific computing 2023-08, Vol.96 (2), p.45, Article 45
• Main Authors: Wang, Chengjing, Tang, Peipei
• Format: Article
• Language: English
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Constraints; Feature selection; Machine learning; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Numerical analysis; Robustness (mathematics); Special Issue on Machine Learning on Scientific Computing; Theoretical
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-023-02271-w

Square-root Lasso problems have already be shown to be 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.