Set characterizations and convex extensions for geometric convex-hull proofs

Abstract

In the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 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 Gupte et al. (Discrete Optim 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.

Set characterizations and convex extensions for geometric convex-hull proofs

“…Our conclusions can be found in Section 7. Finally, in the online supplement [BS20] to this article, we provide several further examples for the application of our framework in the context of stable-set problems, mixed-integer models for piecewise linear functions as well as interval matrices and give some proofs omitted in Section 6.…”

“…Constraints inducing global couplings between the variables make it harder to use. In the online supplement [BS20], we give further examples for the use of this technique in the context of clique and stable-set problems.…”

“…We think that the possibility to consider arbitrary right-hand sides in an algorithmic fashion makes the Zuckerberg approach a valuable tool for proving total unimodularity. A further example for this concept is given in the online supplement [BS20].…”