I would like to least-squares-"solve" a set of linear equations ($\underset{\mathbf{x}}{\mathrm{argmin}}\; \|\mathbf{Ax-b}\|_2$).
In my case, $\mathbf{b=0}$, e.g. the system is homogeneous. I also happen to know that all parameters must be positive, $\mathbf{x} \geq 0$. (The math works out because half of the coefficients are negative.)
I am assuming that enforcing $\|\mathbf{x}\|_2=1$, e.g. constraining the solution to a hypersphere of constant radius, will take care of the homogeneneity (lest we simply get the trivial solution).
I have found iterative methods (based on quadratic programming) to deal with the non-negativity constraint.
I have, however, not been able to find a way to combine the two constraints. When adding the fixed-norm constraint to the QP, the solver complains about non-convexity of the problem. Intuitively, I had assumed this was a convex problem. Am I wrong, and if yes, why? If not, how can I solve this problem in software?