SOS-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 polynomial of degree large than four 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]

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.[6] 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).

• Hilbert's seventeenth problem
• Polynomial SOS
• Sum-of-squares optimization