Approximate cone factorizations and lifts of polytopes

TL;DR

This paper introduces methods to construct inner and outer convex approximations of polytopes using approximate cone factorizations of their slack matrices, extending Yannakakis's result to approximate settings with efficient second order cone representations.

Contribution

It generalizes the connection between slack matrix factorizations and polytope lifts to approximate factorizations, providing robust approximations and extending to generalized slack matrices.

Findings

Approximations behave well under polarity.

Efficient representations using second order cones.

Relationship between factorization quality and approximation quality.

Abstract

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 Yannakakis 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.