Lifts of convex sets and cone factorizations

TL;DR

This paper explores the conditions under which convex sets can be represented as linear images of affine slices of convex cones, linking geometric lifts to operator factorizations and extending Yannakakis's theorem.

Contribution

It generalizes the connection between convex set lifts and factorizations, introducing the concept of cone rank and providing new tools for analyzing cone lifts of convex sets.

Findings

Equivalence between convex set lifts and operator factorizations.

Extension of Yannakakis's theorem to general convex sets.

Introduction of the cone rank concept for convex sets.

Abstract

In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.