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Square-root Lasso problems are proven robust regression problems.Furthermore, square-root regression problems with structured sparsity alsoplays an important role in statistics and machine learning. In this paper, wefocus on the numerical computation of large-scale linearly constrained sparsegroup square-root Lasso problems. In order to overcome the difficulty thatthere are two nonsmooth terms in the objective function, we propose a dualsemismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. Thatis, we apply the ALM to the dual problem with the subproblem solved by the SSNmethod. To apply the SSN method, the positive definiteness of the generalizedJacobian is very important. Hence we characterize the equivalence of itspositive definiteness and the constraint nondegeneracy condition of thecorresponding primal problem. In numerical implementation, we fully employ thesecond order sparsity so that the Newton direction can be efficiently obtained.Numerical experiments demonstrate the efficiency of the proposed algorithm.

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