Let $z_1,z_2,\dots,z_N$ be vectors from $\mathbb Z^m$ for some $m$.  The problem is:

Given a positive real number $p$, find the subset $A_p \subset \{ 1,2,\dots,N \}$ of size $|A_p| = p$ such that
$$
\|\sum_{i \in A_p} z_i \|
$$
is maximized, where $\|\cdot\|$ is the Euclidean norm.

I am most interested in cases where $m$ is small, not much more than $2$, and $N$ is large, potentially $1000$s.

I would be surprised if this problem had not previously been studied.  Has anybody seen it before?  Does it have a name?