Limitations on the expressive power of convex cones without long chains of faces

Understand

A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace.

• In some cases, it is possible to replace a conic formulation using a certain cone, with a 'lifted' conic formulation using another cone that is higher-dimensional, but simpler, in some sense.
• One situation in which this can be computationally advantageous is when the higher-dimensional cone is a Cartesian product of many 'low-complexity' cones, such as second-order cones, or small positive semidefinite cones.
• This paper studies obstructions to a convex cone having a lifted representation with such a product structure.