From a practical perspective, mixed integer optimization represents a very powerful modeling paradigm. Its modeling power, however, comes with a price. The presence of both integer and continuous variables results in a significant increase in complexity over the pure integer case with respect to geometric, algebraic, combinatorial and algorithmic properties. Specifically, the theory of cutting planes for mixed integer linear optimization is not yet at a similar level of development as in the pure integer case. The goal of this paper is to discuss four research directions that are expected to contribute to the development of this field of optimization. In particular, we examine a new geometric approach based on lattice point free polyhedra and use it for developing a cutting plane theory for mixed integer sets. We expect that these novel developments will shed some light on the additional complexity that goes along with mixing discrete and continuous variables.
