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

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Abstract

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.

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Data Availability

The datasets analysed during the current study are available at the following link: https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/regression.html

Notes

  1. The assumption \(\mathcal {A}\bar{x}-b=\bar{y}\ne 0\) is reasonable, since this is equivalent to the requirement that no overfitting occurs.

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Acknowledgements

We would like to thank the Editor-in-Chief Professor Chi-Wang Shu and the anonymous referees for their helpful suggestions which greatly improves the quality of the manuscript.

Funding

Chengjing Wang’s work was supported in part by the National Natural Science Foundation of China (No. U21A20169), Zhejiang Provincial Natural Science Foundation of China (Grant No. LTGY23H240002). Peipei Tang’s work was supported in part by the Scientific Research Foundation of Zhejiang University City College (No. X-202112).

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Wang, C., Tang, P. A Dual Semismooth Newton Based Augmented Lagrangian Method for Large-Scale Linearly Constrained Sparse Group Square-Root Lasso Problems. J Sci Comput 96, 45 (2023). https://doi.org/10.1007/s10915-023-02271-w

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