I want to solve optimization problem of the type basis pursuit denoising (BPDN). What I want to minimize is the l_1 norm of vector x (length N), |x|_1, subject to |Ax - B|_2 <= sigma, where A is an MxN matrix, B is length M vector, and sigma is a positive scalar. In addition to that constraint, there is also another one, namely I want also, let's say, x(1:P) = 0 where P < N obviouisly.
That's what I want to do and now I have already found a MATLAB code designed specially to solve BPDN problem but it's the standard one I think, it can only assume the first constraint involving inequality with sigma. What I want to do now is that I want to add the second constraint while using the code I'm having without modifying it, that is I would like to add my own constraint directly in my MATLAB script in which the BPDN problem will be executed.
Could somebody give an idea?
EDIT: My idea is to set the first up to the P-th column of matrix A to have zero entries because this way the product Ax is the same as the original problem. In coming up with this idea I also assume that the BPDN code I found will automatically set the elements x(1:P) to zero because it will have the full freedom in deciding the values of these elements due to the multiplication with zero-entries columns and it must also minimize the |x|_1. Will this way equivalent with the original problem?
