Geometric proofs for convex hull defining formulations
Introduction
The connection between probability measures and integer programming has antecedents in [6] (in the “Remark” on page 186), and earlier (see Part 1 in [4]), with later development in [2] (beginning in Section 2.1) and in [8] (Chapters 3 and 4), where this connection is shown to generalize and contextualize the lifting methods of [1], [7], [6], [5]. In the course of the analysis in [8], a method is described by which an LP relaxation of an integer programming formulation can be proven to define the convex hull of its integer points if it can be shown that for each point in that is in the linear relaxation, it is possible to draw sets in an arbitrary measure space that have properties that match the logical properties of the integer feasible set in a certain way. The method is described as “geometric”, or as in [8], as the “proof by picture method”, as one can potentially demonstrate that a formulation is convex hull defining by physically drawing sets in the real line or plane with the requisite properties. In this work we will provide a compact description and rigorous justification of the method, and we will use it to resolve a conjecture that appeared in [3].
Section snippets
Binary integer programming and probability measures
First we review some basic definitions and facts. Definition 1 Given a set and a family of subsets of , the pair is said to be an “algebra on ” if is closed under unions, intersections and complementations, where complementation is defined as . Note that for any algebra . Given an algebra , we will refer to a set as being “in algebra ” if . A nonempty set in an algebra is called an “atom” of if for every , either or , i.e.
Proving the conjecture
We now have the machinery to prove the conjecture of [3]. The set for which we will be seeking a convex hull formulation is the set of 0,1 points in that satisfy a covering constraint , and for which exactly two coordinates have value 1. The coefficients of the covering constraint are listed in descending order . (The convex hull of is referred to in [3] as .)
In line with the notation of [3], we define to be the largest index such that
Concluding remarks and further work
Much of the appeal of the proof by picture method as an aid in proving the integrality of formulations is that it may be used to effectively transform the geometry of an dimensional problem to one or two dimensions, where most people have far better intuition. It is useful in particular where the feasible space has a simple set theoretic (i.e. logical) interpretation. Though we have not discussed this here, it can therefore also be helpful in supplying geometric intuition when dealing with
Acknowledgments
I would like to express my gratitude to Dan Bienstock (Columbia University), who has collaborated with me on topics related to this work for many years, and to my colleague Peter Malkin (BHP Billiton, Melbourne University). Their insightful comments and suggestions have significantly enhanced this work, and are much appreciated. I also gratefully acknowledge the ongoing support of BHP Billiton.
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