Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions

TL;DR

This paper analyzes the approximation quality of McCormick relaxations for bilinear functions, establishing bounds that grow with the problem size and characterizing cases of exact relaxation.

Contribution

It provides asymptotic lower and upper bounds on the approximation factor for McCormick relaxations of bilinear functions, answering an open question and improving existing bounds.

Findings

The factor c cannot be bounded independently of n.

For a random bilinear function, c asymptotically almost surely exceeds √n/4.

An improved upper bound of 600√n for c is established.

Abstract

We investigate how well the graph of a bilinear function $b:[0,1]^n\to\mathbb{R}$ can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number $c$ such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most $c$ times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor $c$ cannot be bounded by a constant independent of $n$. More precisely, we show that for a random bilinear function $b$ we have asymptotically almost surely $c\geqslant\sqrt n/4$. On the other hand, we prove that $c\leqslant 600\sqrt{n}$, which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions $b$ for which the McCormick relaxation is equal to the convex hull.