SOS-convexity


A multivariate polynomial is SOS-convex if its Hessian matrix H can be factored as H = STS where S is a matrix which entries are polynomials in x. In other words, the Hessian matrix is a SOS matrix polynomial.
An equivalent definition is that the form defined as g = yTHy is a sum of squares of forms.

Connection with convexity

If a polynomial is SOS-convex, then it is also convex. Since establishing whether a polynomial is SOS-convex amounts to solving a semidefinite programming problem, SOS-convexity can be used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic quartic polynomial of degree four is convex is a NP-hard problem.
The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial that is sum-of-squares and given by:
In the same year, Grigoriy Blekherman proved in a non-constructive manner that there exist convex forms that is not representable as sum of squares. An explicit example of a convex form that is not a sum of squares was claimed by James Saunderson in 2021.

Connection with non-negativity and sum-of-squares

In 2013 Amir Ali Ahmadi and Pablo Parrilo showed that every convex homogeneous polynomial in n variables and degree 2d is SOS-convex if and only if either n = 2 or 2d = 2 or n = 3 and 2d = 4. Impressively, the same relation is valid for non-negative homogeneous polynomial in n variables and degree 2d that can be represented as sum of squares polynomials.