Sum-of-squares hierarchies for binary polynomial optimization

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