Sum-of-squares hierarchies for binary polynomial optimization

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.