Computing the volume of the convex hull of the graph of a trilinear monomial using mixed volumes

TL;DR

This paper introduces a new method based on mixed volumes to compute the convex hull volume of a trilinear monomial's graph, providing a potentially more extensible approach for optimization problems.

Contribution

The paper presents an alternative, theory-based method for calculating convex hull volumes, expanding the tools available for mixed integer nonlinear optimization.

Findings

New mixed volume approach for volume computation

Potential for extending convexification techniques

Comparison with existing formulas suggests efficiency

Abstract

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, $y=x_1x_2x_3$, over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.