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Approximate cone factorizations and lifts of polytopes. (English) Zbl 1319.52010
Summary: In this paper, we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of M. Yannakakis [J. Comput. Syst. Sci. 43, No. 3, 441–466 (1991; Zbl 0748.90074)] that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave well under polarity and have efficient representations using second order cones. We establish a direct relationship between the quality of the factorization and the quality of the approximations, and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron.

##### MSC:
 52A23 Asymptotic theory of convex bodies 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
##### Keywords:
approximations; polytopes; cone factorizations; slack matrix
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##### References:
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