Mixed Integer Linear Sets

从实践的角度来看，混合整数优化代表了一种非常强大的建模范例。然而，它的建模能力是有代价的。这两种类型变量的存在导致几何、代数、组合和算法性质的复杂性显著增加。具体地说，混合整数线性优化的割平面理论还没有处于与纯整数情况类似的发展水平。该研究方案的目的是研究一种新的基于无格点多面体的几何方法，并将其用于发展混合整数集的割平面理论。我们预计，这些新的发展将自然地扩展已为纯整数情况开发的理论，并阐明混合离散变量和连续变量所带来的额外复杂性。

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.