Experimental validation of volume-based comparison for double-McCormick relaxations

TL;DR

This paper experimentally validates that volume-based measures effectively predict the quality of polyhedral relaxations in global optimization problems involving trilinear monomials, confirming theoretical insights with practical results.

Contribution

It provides empirical evidence supporting the use of volume formulae to predict relaxation quality in complex optimization problems, bridging theory and practice.

Findings

Volume formulae accurately predict relaxation strength.

Predictions align with experimental results on box cubic problems.

Volume-based comparison is practical for global optimization.

Abstract

Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region.