Suppose I have a family of $n$ vectors in $\mathbb{R}^d$: $v_1,\dots,v_n.$ Is there a poly-time algorithm that computes a subset $S\subset [n]:=\{1,\dots,n\}$ of size $1\leq k\leq n$ for which the (Euclidean) norm of the sum $\|\sum_{i\in S}v_i\|_2$ is as large as possible? That is, I am looking for an algorithm that computes: $$\operatorname*{arg max}_{S\subset [n],\# S=k} \left\|\sum_{i\in S}v_i\right\|_2.$$
References for exact methods, approximate solutions and/or hardness results would be most welcome.
