LASSO with weighted 1-norm



I would like to use TFOCS to solve the following problem:
minimize norm(Wx,1)
s.t. norm(A
x-b,2) <= epsilon

Solving (1) is numerically difficult. So as I see it I can either solve the constrained problem,
minimize norm(Wx,1) + 0.5mu*(x-x0).^2
s.t. norm(A*x-b,2) <= epsilon

Or the unconstrained problem,
minimize norm(Wx,1)+ lambdanorm(A*x-b,2)

I can solve (2) with code straight from solver_BPDN_W.m. Problem (3), however, is similar to but different from the code in solver_L1RLS.m. How can I modify the code in solver_L1RLS.m to include the weight term W in the 1-norm? Code from solver_L1RLS.m:

[x,odata,opts] = tfocs( smooth_quad, { A, -b }, prox_l1( lambda ), x0, opts );

Thanks for your help!