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

Yu Hin (Gary) Au(au***at***msoe.edu)
Levent Tunšel(ltuncel***at***uwaterloo.ca)

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.

Keywords: combinatorial optimization, lift-and-project methods, integrality gap, design and analysis of algorithms with discrete structures, integer programming, semidefinite programming, convex relaxations

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Integer Programming (0-1 Programming )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )


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Entry Submitted: 08/26/2016
Entry Accepted: 08/28/2016
Entry Last Modified: 08/26/2016

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