Reformulating the disjunctive cut generating linear program

Abstract

Lift-and-project cuts can be obtained by defining an elegant optimization problem over the space of valid inequalities, the cut generating linear program (CGLP). A CGLP has two main ingredients: (i) an objective function, which invariably maximizes the violation with respect to a fractional solution ${\bar{x}}$ to be separated; and (ii) a normalization constraint, which limits the scale in which cuts are represented. One would expect that CGLP optima entail the best cuts, but the normalization may distort how cuts are compared, and the cutting plane may not be a supporting hyperplane with respect to the closure of valid inequalities from the CGLP. This work proposes the reverse polar CGLP (RP-CGLP), which switches the roles conventionally played by objective and normalization: violation with respect to ${\bar{x}}$ is fixed to a positive constant, whereas we minimize the slack for a point p that cannot be separated by the valid inequalities. Cuts from RP-CGLP optima define supporting hyperplanes of the immediate closure. When that closure is full-dimensional, the face defined by the cut lays on facets first intersected by a ray from ${\bar{x}}$ to p, all of which corresponding to cutting planes from RP-CGLP optima if p is an interior point. In fact, these are the cuts minimizing a ratio between the slack for p and the violation for ${\bar{x}}$. We show how to derive such cuts directly from the simplex tableau in the case of split disjunctions and report experiments on adapting the CglLandP cut generator library for the RP-CGLP formulation.