| Monday, March 18th, 2013 | 4pm-5pm | Burnside 1205 |
In optimization one often represents convex sets in terms of convex cones. Such representations or 'lifts' of a convex set are especially useful if the cone admits efficient algorithms for linear optimization over its affine slices, as in the classical cases of linear and semidefinite programming. Despite the fact that these techniques are widely used, there are many aspects (particularly, existence and efficiency) that are still poorly understood. In this talk we discuss the relationship between conic representations of convex sets, and a special "conic" factorization of an operator associated to the convex set, generalizing earlier results of Yannakakis on polyhedral lifts of polytopes and nonnegative factorizations. 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 (e.g., psd rank of a convex set), as well as techniques for lower bounding these ranks. We will provide a gentle introduction to these techniques, emphasizing geometric intuition, open questions as well as recent results. Based on joint work with Joao Gouveia, Hamza Fawzi, and Rekha Thomas.