the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to relax the constraint by introducting a cost of violating it?
Just to be more clear: ideally, I'd like to be able to solve something like $x'Qx + f'x + \lambda max(Ax, 0) \rightarrow \min_x$, where $max(Ax, 0)$ is component-wise maximum of Ax and 0.