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

TL;DR

This paper investigates the effectiveness of semidefinite lift-and-project operators on elementary polytopes, revealing limitations and invariances in their performance for certain convex relaxations.

Contribution

It provides bounds on when these operators perform poorly on chipped and cropped hypercubes, and shows invariance of the integrality gap under various lift-and-project methods.

Findings

Operators perform badly beyond certain severity bounds.

The integrality gap remains unchanged under multiple lift-and-project operators.

Bounds are established for the failure of these operators on elementary polytopes.

Abstract

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 $2^n$ vertices of the hypercube removed by intersection with $2^n$ closed halfspaces) to varying degrees of severity $\rho$. We prove bounds on $\rho$ where these operators would perform badly on the aforementioned examples. 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.