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.

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.