Relaxations for binary polynomial optimization via signed certificates

TL;DR

This paper introduces a novel approach for binary polynomial optimization by leveraging signed certificates and min-cut algorithms, resulting in sparsity-preserving LP relaxations that improve computational efficiency.

Contribution

It develops parameterized LP representations for binary non-negative polynomials based on signed support patterns, enabling new hierarchies of relaxations for optimization.

Findings

Polynomial-time binary non-negativity checks via min-cut algorithms.

Construction of sparsity-preserving LP hierarchies for binary polynomial optimization.

Decomposition method for minimizing polynomials using signed certificates.

Abstract

We consider the problem of minimizing a polynomial $f$ over the binary hypercube. We show that, for a specific set of polynomials, their binary non-negativity can be checked in a polynomial time via minimum cut algorithms, and we construct a linear programming representation for this set through the min-cut-max-flow duality. We categorize binary polynomials based on their signed support patterns and develop parameterized linear programming representations of binary non-negative polynomials. This allows for constructing binary non-negative signed certificates with adjustable signed support patterns and representation complexities, and we propose a method for minimizing $f$ by decomposing it into signed certificates. This method yields new hierarchies of linear programming relaxations for binary polynomial optimization. Moreover, since our decomposition only depends on the support of $f$, the new hierarchies are sparsity-preserving.