Helly systems and certificates in optimization

TL;DR

This paper introduces a unified framework using Hoffman's Helly systems to understand certificates of optimality and infeasibility in optimization, providing bounds and clarifying the combinatorial nature of existing techniques.

Contribution

It formalizes the concept of certificates within Helly systems and establishes bounds on their sizes across different settings, revealing the combinatorial basis of some methods.

Findings

Established bounds on certificate sizes in various settings

Identified purely combinatorial techniques in optimization proofs

Unified understanding of optimality and infeasibility certificates

Abstract

Inspired by branch-and-bound and cutting plane proofs in mixed-integer optimization and proof complexity, we develop a general approach via Hoffman's Helly systems. This helps to distill the main ideas behind optimality and infeasibility certificates in optimization. The first part of the paper formalizes the notion of a certificate and its size in this general setting. The second part of the paper establishes lower and upper bounds on the sizes of these certificates in various different settings. We show that some important techniques existing in the literature are purely combinatorial in nature and do not depend on any underlying geometric notions.