Georgia Institute of Technology College of Sciences

A basic strategy for linear optimization over a complicated convex set is to try to express the set as the projection of a simpler convex set that admits efficient algorithms. This philosophy underlies all "lift-and-project" methods in optimization which attempt to find polyhedral or spectrahedral lifts of complicated sets. In this talk I will explain how the existence of a lift is equivalent to the ability to factorize a certain operator associated to the convex set through a cone. This theorem extends a result of Yannakakis who showed that polyhedral lifts of polytopes are controlled by the nonnegative factorizations of the slack matrix of the polytope. The connection between cone lifts and cone factorizations of convex sets yields a uniform framework within which to view all lift-and-project methods, as well as offers new tools for understanding convex sets. I will survey this evolving area and the main results that have emerged thus far.