On the strength of recursive McCormick relaxations for binary polynomial optimization

TL;DR

This paper explores the effectiveness of recursive McCormick relaxations in binary polynomial optimization, showing they are implied by a more general, efficiently solvable LP relaxation called the extended flower relaxation.

Contribution

It proves that recursive McCormick relaxations are implied by the extended flower relaxation, which can be solved efficiently for fixed-degree problems.

Findings

Recursive McCormick relaxations are implied by the extended flower relaxation.

Extended flower relaxation can be solved in strongly polynomial time for fixed degree.

The results connect convexification techniques with linear programming relaxations.

Abstract

Recursive McCormick relaxations have been among the most popular convexification techniques for binary polynomial optimization problems. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence, and finding an optimal recursive sequence amounts to solving a difficult combinatorial optimization problem. In this paper, we prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation that is a natural generalization of the flower relaxation introduced by Del Pia and Khajavirad 2018, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.