SOS-convexity

A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = S^T(x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x.^[1] In other words, the Hessian matrix is a SOS matrix polynomial.

An equivalent definition is that the form defined as g(x,y) = y^TH(x)y is a sum of squares of forms.^[2]

Connection with convexity

If a polynomial is SOS-convex, then it is also convex.^{[citation needed]} 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 (or higher even degree) is convex is a NP-hard problem.^[3]

The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009.^[4] The polynomial is a homogeneous polynomial that is sum-of-squares and given by:^[4]

$${\displaystyle p(x)=32x_{1}^{8}+118x_{1}^{6}x_{2}^{2}+40x_{1}^{6}x_{3}^{2}+25x_{1}^{4}x_{2}^{4}-43x_{1}^{4}x_{2}^{2}x_{3}^{2}-35x_{1}^{4}x_{3}^{4}+3x_{1}^{2}x_{2}^{4}x_{3}^{2}-16x_{1}^{2}x_{2}^{2}x_{3}^{4}+24x_{1}^{2}x_{3}^{6}+16x_{2}^{8}+44x_{2}^{6}x_{3}^{2}+70x_{2}^{4}x_{3}^{4}+60x_{2}^{2}x_{3}^{6}+30x_{3}^{8}}$$

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.^[5] An explicit example of a convex form (with degree 4 and 272 variables) that is not a sum of squares was claimed by James Saunderson in 2021.^[6]

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 (a) n = 2 or (b) 2d = 2 or (c) n = 3 and 2d = 4.^[7] 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 (See Hilbert's seventeenth problem).