# Extended formulations for convex hulls of graphs of bilinear functions

## TL;DR

This paper develops systematic methods to derive small extended formulations for the convex hulls of bilinear functions on the unit cube, focusing on specific graph structures like cycles and cliques.

## Contribution

It introduces a systematic approach to identify sufficient BQP facets for convex hulls of bilinear functions, extending previous results and providing new formulations for specific graph classes.

## Key findings

- Small-sized extended formulations for cycle graphs with arbitrary weights
- Extended formulations for clique and almost clique graphs with unit weights
- Use of geometric methods to prove convex hull characterizations

## Abstract

We consider the problem of characterizing the convex hull of the graph of a bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$. Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of $f$ that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters \textbf{44} (2016), 625--629] to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights.