Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

Abstract

For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the \(\lceil \frac{n+2d-1}{2}\rceil \) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that \(\frac{n+2d-1}{2} \) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.