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.
Rinaldi, G., Voigt, U., & Woeginger, G. (2002). The mathematics of playing golf, or: A new class of difficult non-linear mixed integer programs. Mathematical programming, 93, 77-86. https://doi.org/10.1007/s101070200298