  We consider an important problem in signal processing, which consists in
finding the sparsest solution of a linear system $\Phi x=b$. This problem has
applications in several areas, but is NP-hard in general. Usually an
alternative convex problem is considered, based on minimizing the (weighted)
$\ell_{1}$ norm. For this alternative to be useful, weights should be chosen as
to obtain a solution of the original NP-hard problem. A well known algorithm
for this is the Re-Weighted $\ell_{1}$, proposed by Cand\`es, Wakin and Boyd.
In this article we introduce a new methodology for updating the weights of a
Re-Weighted $\ell_{1}$ algorithm, based on identifying these weights as
Lagrange multipliers. This is then translated into an algorithm with
performance comparable to the usual methodology, but allowing an interpretation
of the weights as Lagrange multipliers. The methodology may also be used for a
noisy linear system, obtaining in this case a Re-Weighted LASSO algorithm, with
a promising performance according to the experimental results.
