# On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation

## TL;DR

This paper develops volume-based rules for selecting branching points in spatial branch-and-bound algorithms for trilinear monomials, improving relaxation tightness and guiding optimal variable and point choices.

## Contribution

It introduces simple, volume-based criteria for optimal branching in convex-hull relaxations of trilinear monomials, advancing previous heuristic methods.

## Key findings

- Established rules for optimal branching variable and point based on volume measures.
- Compared new rules with existing software practices, demonstrating potential improvements.
- Provided analytical insights into relaxation tightness for trilinear monomials.

## Abstract

In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three possible "double-McCormick relaxations" with the tight convex-hull relaxation. Here, again using volume as a measure, for the convex-hull relaxation of trilinear monomials, we establish simple rules for determining the optimal branching variable and optimal branching point. Additionally, we compare our results with current software practice.