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We consider operators acting on convex subsets of the unit hypercube. Theseoperators are used in constructing convex relaxations of combinatorialoptimization problems presented as a 0,1 integer programming problem or a 0,1polynomial optimization problem. Our focus is mostly on operators that, whenexpressed as a lift-and-project operator, involve the use of semidefinitenessconstraints in the lifted space, including operators due to Lasserre andvariants of the Sherali--Adams and Bienstock--Zuckerberg operators. We studythe performance of these semidefinite-optimization-based lift-and-projectoperators on some elementary polytopes --- hypercubes that are chipped (atleast one vertex of the hypercube removed by intersection with a closedhalfspace) or cropped (all $2^n$ vertices of the hypercube removed byintersection with $2^n$ closed halfspaces) to varying degrees of severity$\rho$. We prove bounds on $\rho$ where these operators would perform badly onthe aforementioned examples. We also show that the integrality gap of thechipped hypercube is invariant under the application of severallift-and-project operators of varying strengths.

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