We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Non-linear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex quadratic sets and for intersection cuts for a wide variety of convex quadratic sets.
We study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to the Conic Mixed Integer Rounding (CMIR) cuts. We show that the CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and that it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that by exploiting the power of their extended formulation, families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.