Lifts of Non-Compact Convex Sets and Cone Factorizations

This paper generalizes the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, the authors 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. The authors 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. The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.