Lifts of Non-compact Convex Sets and Cone Factorizations

TL;DR

This paper extends the cone factorization framework to a wider class of convex sets, linking their geometric properties to algebraic factorizations via slack operators and cone lifts.

Contribution

It generalizes the existing factorization theorem to include non-compact convex sets, incorporating recession cones and broader conditions for cone lifts.

Findings

Generalized cone lift characterization for broader convex sets

Introduction of slack operators for non-compact convex sets

Enhanced conditions for cone factorizations and lifts

Abstract

In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.