Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators

We consider operators acting on convex subsets of the unit hypercube. These operators are used in constructing convex relaxations of combinatorial optimization problems presented as a 0,1 integer programming problem or a 0,1 polynomial optimization problem. Our focus is mostly on operators that, when expressed as a lift-and-project operator, involve the use of semidefiniteness constraints in the lifted space, including operators due to Lasserre and variants of the Sherali-Adams and Bienstock-Zuckerberg operators. We study the performance of these semidefinite-optimization-based lift-and-project operators on some elementary polytopes - hypercubes that are chipped (at least one vertex of the hypercube removed by intersection with a closed halfspace) or cropped (all 2n vertices of the hypercube removed by intersection with 2n closed halfspaces) to varying degrees of severity ρ. We prove bounds on ρ where the Sherali-Adams operator (strengthened by positive semidefiniteness) and the Lasserre operator require n iterations to compute the integer hull of the aforementioned examples, as well as instances where the Bienstock-Zuckerberg operators require Ω(n) iterations to return the integer hull of the chipped hypercube. We also show that the integrality gap of the chipped hypercube is invariant under the application of several lift-and-project operators of varying strengths.