I've written a C# solver for linear least squares problems with inequality constraints. That is, given $A$, $b$, $G$, $h$
$$\min\|Ax-b\|^2\text{ s.t. }Gx\ge h$$
I have a few hand crafted test problems that my solver gives the correct answer for, and now I'd like to throw a gauntlet of randomly generated problems of various ranks at it to make sure there aren't any edge cases I'm missing.
So what I need is a way to determine that a given $b$ vector calculated satisfies the constraints $Gx \ge h$ (which is easy to check for) and that the solution vector can't be improved by perturbing it in a given non-constrained direction. The second part is what I'm at a loss for.