When Lift-and-Project Cuts Are Different

Abstract

In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is regular, in which case the cut is equivalent to an intersection cut from some basis of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from non-split disjunctions. While it is known since Balas and Perregaard (2003) that such equivalence always holds for lift-and-project cuts from split disjunctions, Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016).Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut. Third, we describe a numerical procedure that verifies if a lift-and-project cut is not regular.Finally, we use this method to evaluate how often lift-and-project cuts are irregular, and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB benchmarks.