Spectrahedral Lifts of Convex Sets

Abstract: Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher dimensions called a lift of the original set. This is a brief survey of recent developments in the topic of lifts of convex sets. Our focus will be on lifts that arise from affine slices of real positive semidefinite cones known as psd or spectrahedral lifts. The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set. This leads to several research directions and results that lie at the intersection of convex geometry, combinatorics, real algebraic geometry, optimization, computer science and more.