- Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators Yu Hin (Gary) Au(aumsoe.edu) Levent Tunçel(ltunceluwaterloo.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 ) Citation: Download: [PDF]Entry Submitted: 08/26/2016Entry Accepted: 08/28/2016Entry Last Modified: 08/26/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.