@inproceedings{02dff3202d7d4e9790e553c6bfc48028,
title = "A positive definite polynomial Hessian that does not factor",
abstract = "The notion of sos-convexity has recently been proposed as a tractable sufficient condition for convexity of polynomials based on a sum of squares decomposition of the Hessian matrix. A multivariate polynomial p(x) = p(x 1,...,xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = MT (x)M(x) with a possibly nonsquare polynomial matrix M(x). The problem of deciding sos-convexity of a polynomial can be reduced to the feasibility of a semidefinite program, which can be checked efficiently. Motivated by this computational tractability, it has been speculated whether every convex polynomial must necessarily be sos-convex. In this paper, we answer this question in the negative by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex.",
author = "Ahmadi, {Amir Ali} and Parrilo, {Pablo A.}",
year = "2009",
doi = "10.1109/CDC.2009.5400519",
language = "English (US)",
isbn = "9781424438716",
series = "Proceedings of the IEEE Conference on Decision and Control",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "1195--1200",
booktitle = "Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009",
address = "United States",
note = "48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 ; Conference date: 15-12-2009 Through 18-12-2009",
}