The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs

We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with k = 1 and k = 2, every optimal solution is integral. In contrast to this, for every k ≥ 3 there exist instances where every optimal solution takes non-integral values.