I'm aware of several solution methods and have several solvers at my disposal, but I can't for the life of me find analysis on the complexity. In particular, I'm interested in the complexity of solving the following problem:
$ \begin{aligned} & \underset{ x }{ \text{ minimize} } && || x_q - x || \\ & \text{subject to} && r_i^\intercal x \le c_i & i=1 \ldots n \end{aligned} $
Where $x,x_q \in \mathbb{R}^d$. Which is, in words "find the nearest point in a polytope to some point $x_q$". I'm told by a colleague that it's $O(n^3)$, but he isn't certain about that. Also, how does complexity scale with dimension? Is it $O(d n^3)$?