# Quadratic programming: maximizing the euclidean norm

The setting is the following, I have a full-dimensional polytope $$\mathcal{F}: A \cdot x \leq \vec{b}$$ that has been transformed such that it contains the origin: $$\vec{0} \in \mathcal{F}$$. I need to find the point with a maximum $$L_2$$ norm in this polytope, which equates solving the following quadratic program:

$$\arg \max_{\vec{x}\in \mathcal{F}} \ \vec{x}^T \cdot I \cdot \vec{x}$$

I tried doing so using cvxpy, but that tells me that this program is not a disciplined convex. I can kind of imagine that this program has a concave objective function and therefore is not solvable using convex optimization techniques, so I am wondering whether there is some other way of solving it.

Its clear that the solution to this problem is at a vertex of the polytope, and I therefore think that there must be some linear programming trick to find it without enumerating all vertices.

Whether or not your problem has a solution depends on whether or not the feasible region is unbounded. Let $$P:=\{x:Ax\leq b\}$$. You say that $$P$$ is a polytope (and hence is bounded). In this case, the Bolzano-Weierstrass theorem guarantees the existence of a global maximum (since the feasible region is compact).
For the LCP (linear complementarity problem) approach, notice that the KKT conditions of the original problem $$\max_x\{\|x\|_2^2 : Ax\leq b\}$$ can be written as $$x = A^\top\lambda,\;Ax\leq b,\;\lambda\geq0,\;\lambda^\top(Ax-b)=0,$$ which give the LCP$$(q,M)$$ $$\mbox{find } \lambda : 0 \leq b-Ax=-AA^\top\lambda + b \perp \lambda \geq0,$$ where $$q := b$$ and $$M := -AA^\top$$. Then you may use standard LCP techniques to obtain a (local) solution.