Quantifying Double McCormick

TL;DR

This paper quantifies and compares the strength of different double McCormick relaxations for trilinear monomials over nonnegative boxes by calculating their volumes, providing insights into the optimal grouping strategy.

Contribution

It precisely quantifies the relaxation strength of all groupings in double McCormick and characterizes the best relaxation, applicable to complex formulations with auxiliary variables.

Findings

Calculated the volume of relaxations for all groupings.

Identified the optimal double McCormick relaxation.

Results apply to auxiliary variables and complex expressions.

Abstract

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by product, we characterize the best double McCormick relaxation. We wish to emphasize that, in the context of spatial branch-and-bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.