Relaxations for binary polynomial optimization via 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 (i.e. on the hypercube) can be checked in polynomial time via minimum cut algorithms, from which we construct a linear programming representation for this set of polynomials. We categorize binary polynomials according to their signed support patterns and develop parameterized linear programming representations for binary non-negative polynomials. This allows the construction of signed certificates of binary non-negativity with adjustable signed support patterns and representation complexities; and we propose a method for minimizing $f$ by decomposing it as a sum of signed certificates. This method yields new hierarchies of linear programming relaxations for binary polynomial optimization. Moreover, since our decomposition depends only on the support of $f$, the new hierarchies are sparsity-preserving.