For binary polynomial optimization problems (POPs) of degree $d$ with $n$ variables, we prove that the $\lceil(n+d-1)/2\rceil$th semidefinite (SDP) relaxation in Lasserre's hierarchy of the SDP relaxations provides the exact optimal value. If binary POPs involve only even-degree monomials, we show that it can be further reduced to $\lceil(n+d-2)/2\rceil$. This bound on the relaxation order coincides with the conjecture by Laurent in 2003, which was recently proved by Fawzi, Saunderson and Parrilo, on binary quadratic optimization problems where $d=2$. We also numerically confirm that the bound is tight. More precisely, we present instances of binary POPs that require solving at least the $\lceil(n+d-1)/2\rceil$th SDP relaxation for general binary POPs and the $\lceil(n+d-2)/2\rceil$th SDP relaxation for even-degree binary POPs to obtain the exact optimal values.
METR 2016-01, Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan, January 2016