A positive definite polynomial Hessian that does not factor
Author(s)Ahmadi, Amir Ali; Parrilo, Pablo A.
MetadataShow full item record
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[subscript 1],...,x[subscript n])is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M[superscript T](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.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009
Institute of Electrical and Electronics Engineers
Ahmadi, A.A., and P.A. Parrilo. “A positive definite polynomial Hessian that does not factor.” Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on. 2009. 1195-1200. © 2010 Institute of Electrical and Electronics Engineers.
Final published version
INSPEC Accession Number: 11149467