Given a real matrix $A$ find a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to $0$).
A has size about $1000 \times 20$ and can be written as $[ A_P | A_N ]$ ($A_P$ contains only positive and $A_N$ only negative numbers)
Is there any deterministic algorithm which can reach the solution (global minimum) in polynomial time?