Mixed Integer Linear Sets

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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 types of variables results in a significant increase in complexity 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 research proposal is to examine a new geometric approach based on lattice point free polyhedra and use it to develop a cutting plane theory for mixed integer sets. We expect that these novel developments will naturally extend the theory that has been developed for the pure integer case and shed some light on the additional complexity that goes along with mixing discrete and continuous variables.