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A variable fixing version of the two-block nonlinear constrained Gauss–Seidel algorithm for \(\ell _1\)-regularized least-squares

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Abstract

The problem of finding sparse solutions to underdetermined systems of linear equations is very common in many fields as e.g. signal/image processing and statistics. A standard tool for dealing with sparse recovery is the \(\ell _1\)-regularized least-squares approach that has recently attracted the attention of many researchers. In this paper, we describe a new version of the two-block nonlinear constrained Gauss–Seidel algorithm for solving \(\ell _1\)-regularized least-squares that at each step of the iteration process fixes some variables to zero according to a simple active-set strategy. We prove the global convergence of the new algorithm and we show its efficiency reporting the results of some preliminary numerical experiments.

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Acknowledgments

The authors thank anonymous referees for useful comments that helped to improve the manuscript. The second author would also like to thank Professor Luigi Grippo for kindly sharing with him many ideas and insights that led to this paper. The work of the first author was supported by “National Group of Computing Science (GNCS-INDAM)”.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy

    Margherita Porcelli

  2. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Francesco Rinaldi

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  1. Margherita Porcelli
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  2. Francesco Rinaldi
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Correspondence to Margherita Porcelli.

Appendix

Appendix

In Tables 1, 2 and 3, we report the average CPU time (Av-CPU), the average relative error (Av-rel.err.) and the average number on nonzero components of the computed solution (Av-nnz), corresponding to runs summarized in the performance profiles in Figs. 2 and 3. Problems where the false termination of FPC-AS with the criterion (40) occurs 1 over 10 runs is denoted with the symbol ‘*’ and the average values reported in the tables do not take into account these cases.

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Porcelli, M., Rinaldi, F. A variable fixing version of the two-block nonlinear constrained Gauss–Seidel algorithm for \(\ell _1\)-regularized least-squares. Comput Optim Appl 59, 565–589 (2014). https://doi.org/10.1007/s10589-014-9653-0

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