Exact Semidefinite Programming Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems
研究了二元多项式优化问题中Lasserre半定规划松弛的精确阶数，证明了松弛阶数的上界并验证了其紧性，对求解此类问题有理论指导意义。
For binary polynomial optimization problems (POPs) of degree $d$ with $n$ variables, we prove that the $\lceil(n+d-1)/2\rceil$th semidefinite programming (SDP) relaxation in Lasserre's hierarchy of 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.