Relationships among different SOS-based relaxations for quadratic distance problems

The objective of the paper is to investigate relationships among different convex relaxations for quadratic distance problems. The main motivation is that a number of problems in robust control can be cast as minimum distance problems from a point to a polynomial surface. It is proven that two families of relaxations proposed in the literature, both based on sum of squares, are equivalent: the former exploits properties of homogeneous forms, while the latter relies on the Positivstellensatz theorem. It is also shown that two different relaxations based on Positivstellensatz present different levels of conservativeness. The results presented in the paper provide useful insights on the trade off between computational burden and conservativeness of the considered relaxations.