Software for reweighted L1 minmization?
I am not sure that it will answer the question exactly, but it appears that the glmnet package (R : glmnet, Python : glmnet) do something similar : enforcing sparsity trough iteration.
A small exemple of what you can get : coefficients as a function of the L1 norm. You can see the number of non-zero coefficients (degrees of freedom) evolving .