# SPGL1¶

SPGL1 is a solver for large-scale one-norm regularized least squares.

It is designed to solve any of the following three problems:

1. Basis pursuit denoise (BPDN):
$min \quad ||\mathbf{x}||_1 \quad subj. to \quad ||\mathbf{A}\mathbf{x} - \mathbf{b}||_2 <= \sigma$
1. Basis pursuit (BP):
$min \quad ||\mathbf{x}||_1 \quad subj. to \quad \mathbf{A}\mathbf{x} = \mathbf{b}$
1. Lasso:
$min \quad ||\mathbf{A}\mathbf{x} - \mathbf{b}||_2 \quad subj. to \quad ||\mathbf{x}||_1 <= \tau$

The matrix $$\mathbf{A}$$ can be defined explicitly, or as a scipy.sparse.linalg.LinearOperator that returns both both $$\mathbf{Ax}$$ and $$\mathbf{A}^H\mathbf{b}$$.

SPGL1 can solve these three problems in both the real and complex domains.

## References¶

The algorithm implemented by SPGL1 is described in these two papers:

• E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM J. on Scientific Computing, 31(2):890-912, November 2008
• E. van den Berg and M. P. Friedlander, Sparse optimization with least-squares constraints, Tech. Rep. TR-2010-02, Dept of Computer Science, Univ of British Columbia, January 2010

## History¶

SPGL1 has been initially implemented in MATLAB by E. van den Berg and M. P. Friedlander. This project is aimed at porting of their algorithm in Python. Small modifications are implemented in some areas of the code where more appropriate implementation choices were identified for the Python programming language.