Set characterizations and convex extensions for geometric convex-hull proofs

TL;DR

This paper simplifies Zuckerberg's geometric convex-hull proof method by introducing set characterizations, extending its applicability from 0/1-polytopes to general convex sets, and demonstrating its practical usefulness.

Contribution

It presents a lightweight, accessible approach to Zuckerberg's proof technique, extending its scope to arbitrary polyhedra and convex sets, with new strategies and characterizations.

Findings

Extended framework applies to general convex sets.

Characterizes convex hulls of Boolean and bilinear functions.

Demonstrates wide applicability with illustrative examples.

Abstract

In the present work, we consider Zuckerberg's method for geometric convex-hull proofs introduced in [Geometric proofs for convex hull defining formulations, Operations Research Letters 44(5), 625-629 (2016)]. It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg's proof technique, building on ideas from [Extended formulations for convex hulls of some bilinear functions, Discrete Optimization 36, 100569 (2020)]. We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg's proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.