Sum-of-squares hierarchies for binary polynomial optimization

Lucas Slot, Monique Laurent

Abstract: We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube \({{\mathbb {B}}^{n}=\{0,1\}^n}\). This hierarchy provides for each integer \(r \in {\mathbb {N}}\) a lower bound \(\smash {f_{({r})}}\) on the minimum \(f_{\min }\) of f, given by the largest scalar \(\lambda \) for which the polynomial \(f - \lambda \) is a sum-of-squares on \({\mathbb {B}}^{n}\) with degree at most 2r. We analyze the quality of these bounds by estimating the worst-case error \(f_{\min }- \smash {f_{({r})}}\) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed \(t \in [0, 1/2]\), we can show that this worst-case error in the regime \(r \approx t \cdot n\) is of the order \(1/2 - \sqrt{t(1-t)}\) as n tends to \(\infty \). Our proof combines classical Fourier analysis on \({\mathbb {B}}^{n}\) with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds \(\smash {f_{({r})}}\) and another hierarchy of upper bounds \(\smash {f^{({r})}}\), for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube \(\mathbb ({\mathbb {Z}}/ q{\mathbb {Z}})^{n}\). Furthermore, our results apply to the setting of matrix-valued polynomials.