Generating Disjunctive Cuts for Mixed Integer Programs — Doctoral Dissertation

This report constitutes the Doctoral Dissertation for Michael Perregaard and is a collection of results on the efficient generation of disjunctive cuts for mixed integer programs. Disjunctive cuts is a very broad class of cuts for mixed integer programming. In general, any cut that can be derived from a disjunctive argument can be considered a disjunctive cut. Here we consider specifically cuts that are valid inequalities for some simple disjunctive relaxation of the mixed integer program. Such a relaxation can e.g. be obtained by relaxing the integrality condition on all but a single variable. The liftand-project procedure developed in the early nineties is a systematic way to generate an optimal (in a specific sense) disjunctive cut for a given disjunctive relaxation. It involves solving a higher dimensional cut generating linear program (CGLP) and has been developed for the simplest possible disjunctions; those requiring that a simple variable be either zero or one. In our work we consider the problem of efficiently generating disjunctive cuts for any given disjunction. That is, once we are presented with a disjunctive relaxation of a mixed integer program, how can we efficiently generate one or more cuts that cuts off an optimal solution to the LP relaxation? This problem naturally falls into two cases: Two-term disjunctions, as those the original lift-and-project procedure was designed to solve, and more general multiple-term disjunctions. For the two-term disjunctions we show how one can effectively reduced the CGLP, but the main result is that we show a precise correspondence between the lift-andproject cuts obtained from the CGLP and simple disjunctive cuts from rows of the LP relaxation simplex tableau. The implication is that lift-and-project cuts from the high dimensional CGLP can be obtained directly from the LP relaxation. Furthermore, if integrality on all variables are considered then this becomes a correspondence between strengthened lift-and-project cuts and Gomory’s mixed integer cuts. Using this correspondence we present a procedure to efficiently generate an optimal mixed integer Gomory cut (optimal in the sense of the CGLP) through pivots in the simplex tableau of the LP relaxation. In the case of multiple-term disjunctions we present procedures that provide an optimal solution to the high dimensional CGLP, by solving the cut problem in the original space without recourse to the many auxiliary variables present in the CGLP. Finally, we propose a procedure that generates a set of facets of the convex hull of a given disjunctive set.