Optimal cutting planes from the group relaxations

TL;DR

This paper evaluates the strength of valid inequalities in group models by maximizing the volume of the cut off orthant, identifying GMI cuts as optimal in both finite and infinite group contexts.

Contribution

It introduces a volume-based criterion for inequality strength and proves GMI cuts are optimal under this criterion for finite and infinite group models.

Findings

GMI cuts maximize the volume of the orthant cut in finite group models.

GMI cuts are also optimal in the infinite group model case.

The volume concept is extended to infinite-dimensional spaces.

Abstract

We study quantitative criteria for evaluating the strength of valid inequalities for Gomory and Johnson's finite and infinite group models and we describe the valid inequalities that are optimal for these criteria. We justify and focus on the criterion of maximizing the volume of the nonnegative orthant cut off by a valid inequality. For the finite group model of prime order, we show that the unique maximizer is an automorphism of the {\em Gomory Mixed-Integer (GMI) cut} for a possibly {\em different} finite group problem of the same order. We extend the notion of volume of a simplex to the infinite dimensional case. This is used to show that in the infinite group model, the GMI cut maximizes the volume of the nonnegative orthant cut off by an inequality.