I've run into an optimization problem that's giving me a bit of trouble. Essentially, for some matrix $\mathbf{A}$, I want to find a sparse vector $\mathbf{x}$ that fulfills $\mathbf{Ax}=\mathbf{0}$ as well as $|\mathbf{A}||\mathbf{x}|=\mathbf{b}$, for some known vector $\mathbf{b}$ (in this context, $|.|$ refers to an elementwise absolute value).
So far, I've come up with the idea of finding the solutions to $\mathbf{|A|x}=\mathbf{b}$, then restricting possible solutions to vectors with either all positive or negative entries. Given those vectors, I could construct a set of vectors whose elementwise absolute values fall in the determined space, then projecting it onto the kernel of $\mathbf{A}$. This feels wrong though, and I'm wondering if there's a cleaner way to go about it.
Edit: $\mathbf{A}$ is an underdetermined matrix