Problem
I have a linear system of the form $Ax=b$, where the matrix $A$ is not necessarily square and $\operatorname{nullity}(A)>0$. Each entry of the matrix $A$ and of the vector $b$ is non-negative. What is the particular solution where $\forall i. x_i \geq 0$ and $\langle x, x \rangle$ is minimal?
Details
We can get the set of solutions to the general form of $Ax=b$ by finding some particular solution $x_p$ and then expanding it using the null space: $x \in \{x_p + v, v \in \operatorname{null}(A)\}$. Then we could find the point in that set that minimizes $\langle x, x \rangle$ by subtracting the projection onto the null space: $x = x_p - \operatorname{proj}(x_p, \operatorname{null}(A))$. However, this solution does not necessarily satisfy $\forall i. x_i \geq 0$, and I can't see how to correct the solution to satisfy that constraint.
The nuclear option is to solve numerically using a quadratic constrained optimization solver, but I would prefer an exact solution and a solution that has well-known time bounds.
Does such a solution exist in closed form, or is the best possible approach numerical?