Minimization over the ℓ1-ball

Active-Set algorithm for minimization over the ℓ1-ball (AS-L1) is a solver for the following problem with ℓ1 constraint:

      min f(x)
s.t. ||x||_1 <= τ,

with given objective function f(x) and ℓ1-ball radius τ.

AS-L1 uses an active-set strategy and a projected spectral gradient direction with non-monotone line search.

Reference paper

A. Cristofari, M. De Santis, S. Lucidi, F. Rinaldi (2022). Minimization over the l1-ball using an active-set non-monotone projected gradient. arXiv preprint 2108.00237.

Authors

• Andrea Cristofari (e-mail: andrea.cristofari@unipd.it)
• Marianna De Santis (e-mail: mdesantis@diag.uniroma1.it)
• Stefano Lucidi (e-mail: lucidi@diag.uniroma1.it)
• Francesco Rinaldi (e-mail: rinaldi@math.unipd.it)

Licensing

AS-L1 is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. AS-L1 is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with AS-L1. If not, see http://www.gnu.org/licenses/.

Copyright 2022 Andrea Cristofari, Marianna De Santis, Stefano Lucidi, Francesco Rinaldi.

How to use AS-L1

This directory should contain the files COPYING.txt and README.md, plus the following subdirectories:

• lasso, with a Matlab implementation of AS-L1 to solve lasso problems. Namely, the objective function has the form

    f(x) = 0.5||Ax-b||^2,
  

  with given matrix A and vector b.

  This subdirectory should contain the following files:

  • as_l1_lasso.m, where the algorithm is implemented;
  • main.m, with an example of how to call AS-L1 in Matlab;
  • usage.txt, where inputs, outputs and options of the algorithm are explained in detail.

• logistic_regression, with a Matlab implementation of AS-L1 to solve logistic regression problems. Namely, the objective function has the form

    f(x) = sum_{i=1}^m log(1+exp(-y_i A_i^T x)),
  

  with given matrix A = [A_1 ... A_m]^T and vector y = [y_1 ... y_m].

  This subdirectory should contain the following files:

  • as_l1_log_reg.m, where the algorithm is implemented;
  • main.m, with an example of how to call AS-L1 in Matlab;
  • usage.txt, where inputs, outputs and options of the algorithm are explained in detail.