Will irrational parameters make a problem not well-defined on complexity
The appearance of square roots does not make this an ill-defined problem.

Note that $(\sqrt{2\sum_{i\in S}a_i}-1)^2=0$ if $\sum_{i\in S}a_i = 1/2$ and $(\sqrt{2\sum_{i\in S}a_i}-1)^2&gt;0$ otherwise. Therefore, the problem is easily seen to be NP-hard by reduction from subset sum.