Any ideas or references on how to approach this problem? Every element in a set has a parameter $p_i \geq 0$ and $c_i \geq 0$. The objective is to find a subset which maximizes

$\prod_{i \in S} p_i + \sum_{i \in S} c_i$

I tried to prove complexity results but was unsuccessful. My guess is that it is NP-Hard, but I cannot find a reduction from any known problem. A greedy heuristic which sequentially adds elements which give the highest increase works fine, but can give arbitrarily bad results.