Exact SDP Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems

For binary polynomial optimization problems (POPs) of degree d with n variables, we prove that the ⌈(n+ d− 1)/2⌉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 ⌈(n + d − 2)/2⌉. 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 ⌈(n+d−1)/2⌉th SDP relaxation for general binary POPs and the ⌈(n+d−2)/2⌉th SDP relaxation for even-degree binary POPs to obtain the exact optimal values.