Despite the success of branch-and-cut methods for solving mixed integer bilevel linear optimization problems (MIBLPs) in practice, there have remained some gaps in the theory surrounding these methods. In this paper, we take a first step towards laying out a theory of valid inequalities and cutting-plane methods for MIBLPs that parallels the existing theory for mixed integer linear optimization problems (MILPs). We provide a general scheme for classifying valid inequalities and illustrate how the known classes of valid inequalities fit into this categorization. We also introduce new classes of valid inequalities---one based on a generalization of Chvatal inequalities for MILPs and several more based on a notion of parametric inequality similar to subadditive inequalities for MILPs. Finally, we compare the performance of all classes discussed in the paper using the open source solver MibS.
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Laboratory for Computational Optimization Research at Lehigh (COR@L) Technical Report 20T-013.
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