Extended formulations for the multilinear polytope of acyclic hypergraphs

TL;DR

This paper explores the multilinear polytope in binary polynomial optimization, linking hypergraph acyclicity to the complexity of its facial structure and providing polynomial-size extended formulations for certain acyclic hypergraphs.

Contribution

It characterizes acyclic hypergraphs that admit polynomial-size extended formulations of the multilinear polytope, advancing understanding of their geometric complexity.

Findings

Polynomial-size extended formulations exist for certain acyclic hypergraphs.

Acyclic hypergraphs are characterized by their impact on the facial structure complexity.

The work connects hypergraph properties with optimization polytope representations.

Abstract

This article provides an overview of our joint work on binary polynomial optimization over the past decade. We define the multilinear polytope as the convex hull of the feasible region of a linearized binary polynomial optimization problem. By representing the multilinear polytope with hypergraphs, we investigate the connections between hypergraph acyclicity and the complexity of the facial structure of the multilinear polytope. We characterize the acyclic hypergraphs for which a polynomial-size extended formulation for the multilinear polytope can be constructed in polynomial time.