3 deleted 4 characters in body

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 integer $p$, 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?

2 changes R^m to Z^m

Let $z_1,z_2,\dots,z_N$ be vectors from $\mathbb R^m$ 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? 1 # Maximum magnitude subset sum Let$z_1,z_2,\dots,z_N$be vectors from$\mathbb R^m$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?