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.