Granularity in Nonlinear Mixed-Integer Optimization

We study a new technique to check the existence of feasible points for mixed-integer nonlinear optimization problems that satisfy a structural requirement called granularity. For granular optimization problems, we show how rounding the optimal points of certain purely continuous optimization problems can lead to feasible points of the original mixed-integer nonlinear problem. To this end, we generalize results for the mixed-integer linear case from Neumann et al. (Comput Optim Appl 72:309-337, 2019). We study some additional issues caused by nonlinearity and show how to overcome them by extending the standard granularity concept to an advanced version, which we call pseudo-granularity. In a computational study on instances from a standard test library, we demonstrate that pseudo-granularity can be expected in many nonlinear applications from practice, and that its explicit use can be beneficial.