Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

We study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to Conic Mixed Integer Rounding (CMIR) cuts. We show that CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.