I have a set of numbers $x_i$ and I know sums of certain subsets $y_i=\sum x_{\sigma_k}$. All $x_i>0$ and I'm looking for a simple solution.
With some internet research I found that this might be related to problems in signal processing. So basically I have given a vector $\mathbf{y}$ and a matrix $\mathbf{A}$ with $y_i>0$ and $A_{ij}\in\{0,1\}$. I'm looking for a solution to the vector $\mathbf{x}$ ($x_i\geq 0$) with
$\mathbf{A}\mathbf{x}=\mathbf{y}$
where this linear equation is underdetermined.
Apparently to complete this problem several norms to minimize on $\mathbf{x}$ are possible. For my particular task it's not clear whether I need L0, L1 or L2 norm, so any solution will do - as long as it's simple. Approximate solution like iterative approaches are also fine.
Can you suggest a way to solve this problem? I'm looking for a reference to an algorithm which I can understand as a non-mathematician. Even better would be an open source implementation that I can download. And it would be perfect if it were a Python solution.