A dual semismooth Newton based augmented Lagrangian method for largescale linearly constrained sparse group squareroot Lasso problems
Abstract
Squareroot Lasso problems are proven robust regression problems. Furthermore, squareroot 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 largescale linearly constrained sparse group squareroot 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.13878
 Bibcode:
 2021arXiv211113878W
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Statistics  Computation;
 Statistics  Machine Learning;
 65K05;
 90C06;
 90C25;
 90C90
 EPrint:
 31 pages, 6 tables