Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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.  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 $\rho$.  prove bounds on $\rho$ where these operators would perform badly on the aforementioned examples.  also show that the integrality gap of the chipped hypercube is invariant under the application of several lift-and-project operators of varying strengths.
['Yu Hin Au', 'Levent Tunçel']
I am writing a research paper on a idea Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators
[{"from": "human", "value": "I am writing a research paper on a idea Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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.  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 $\\rho$.  prove bounds on $\\rho$ where these operators would perform badly on the aforementioned examples.  also show that the integrality gap of the chipped hypercube is invariant under the application of several lift-and-project operators of varying strengths."}]