  Compressed sensing is a new methodology for constructing sensors which allow
sparse signals to be efficiently recovered using only a small number of
observations. The recovery problem can often be stated as the one of finding
the solution of an underdetermined system of linear equations with the smallest
possible support. The most studied relaxation of this hard combinatorial
problem is the $l_1$-relaxation consisting of searching for solutions with
smallest $l_1$-norm. In this short note, based on the ideas of Lagrangian
duality, we introduce an alternating $l_1$ relaxation for the recovery problem
enjoying higher recovery rates in practice than the plain $l_1$ relaxation and
the recent reweighted $l_1$ method of Cand\`es, Wakin and Boyd.
