A multivariate polynomial is SOS-convex if its Hessian matrix H can be factored as H(x) = ST(x)S(x), where S is a matrix with polynomial entries in x, making H an SOS matrix polynomial. Equivalently, the form g(x,y) = yTH(x)y is a sum of squares of forms, highlighting the connection between SOS-convexity and polynomial sums of squares.
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 (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:5
p ( x ) = 32 x 1 8 + 118 x 1 6 x 2 2 + 40 x 1 6 x 3 2 + 25 x 1 4 x 2 4 − 43 x 1 4 x 2 2 x 3 2 − 35 x 1 4 x 3 4 + 3 x 1 2 x 2 4 x 3 2 − 16 x 1 2 x 2 2 x 3 4 + 24 x 1 2 x 3 6 + 16 x 2 8 + 44 x 2 6 x 3 2 + 70 x 2 4 x 3 4 + 60 x 2 2 x 3 6 + 30 x 3 8 {\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.6 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 2021.7
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 = 4.8 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).
See also
References
Helton, J. William; Nie, Jiawang (2010). "Semidefinite representation of convex sets". Mathematical Programming. 122 (1): 21–64. arXiv:0705.4068. doi:10.1007/s10107-008-0240-y. ISSN 0025-5610. S2CID 1352703. /wiki/ArXiv_(identifier) ↩
Ahmadi, Amir Ali; Parrilo, Pablo A. (2010). "On the equivalence of algebraic conditions for convexity and quasiconvexity of polynomials". 49th IEEE Conference on Decision and Control (CDC). pp. 3343–3348. doi:10.1109/CDC.2010.5717510. hdl:1721.1/74151. ISBN 978-1-4244-7745-6. S2CID 11904851. 978-1-4244-7745-6 ↩
Ahmadi, Amir Ali; Olshevsky, Alex; Parrilo, Pablo A.; Tsitsiklis, John N. (2013). "NP-hardness of deciding convexity of quartic polynomials and related problems". Mathematical Programming. 137 (1–2): 453–476. arXiv:1012.1908. doi:10.1007/s10107-011-0499-2. ISSN 0025-5610. S2CID 2319461. /wiki/ArXiv_(identifier) ↩
Ahmadi, Amir Ali; Parrilo, Pablo A. (2009). "A positive definite polynomial Hessian that does not factor". Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. pp. 1195–1200. doi:10.1109/CDC.2009.5400519. hdl:1721.1/58871. ISBN 978-1-4244-3871-6. S2CID 5344338. 978-1-4244-3871-6 ↩
Ahmadi, Amir Ali; Parrilo, Pablo A. (2009). "A positive definite polynomial Hessian that does not factor". Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. pp. 1195–1200. doi:10.1109/CDC.2009.5400519. hdl:1721.1/58871. ISBN 978-1-4244-3871-6. S2CID 5344338. 978-1-4244-3871-6 ↩
Blekherman, Grigoriy (2009-10-04). "Convex Forms that are not Sums of Squares". arXiv:0910.0656 [math.AG]. /wiki/ArXiv_(identifier) ↩
Saunderson, James (2022). "A Convex Form That is Not a Sum of Squares". Mathematics of Operations Research. 48: 569–582. arXiv:2105.08432. doi:10.1287/moor.2022.1273. S2CID 234763204. /wiki/Mathematics_of_Operations_Research ↩
Ahmadi, Amir Ali; Parrilo, Pablo A. (2013). "A Complete Characterization of the Gap between Convexity and SOS-Convexity". SIAM Journal on Optimization. 23 (2): 811–833. arXiv:1111.4587. doi:10.1137/110856010. ISSN 1052-6234. S2CID 16801871. /wiki/ArXiv_(identifier) ↩