Approximate bound/algorithm for "product of sums maximization" problem
Intuitively, you are looking for partitioning the given numbers into k sets that are all have the same sum. Indeed, otherwise, you can benefit by taking two of these sets, and make them as balanced as possible. 

As such, you can take an approximation algorithm for subset sum, and turn it into an approximation algorithm to partition. Then extend it to handle $k$-sides partitions, and you would get the required approximation. See for example here: http://valis.cs.uiuc.edu/~sariel/teach/courses/473/notes/08_approx_III.pdf

Then you need to modify it to care about your target function....

I assume that going through the details, you would get an $(1+\epsilon)$-approximation algorithm and its running time would be $(n/\epsilon)^{O(k)}$. But better running time might be possible if you are more careful.