A positive definite polynomial Hessian that does not factor
Name
Ahmadi-2009-A positive definite polynomial Hessian that does not factor.pdf
Size
366.25 KB
Format
Adobe PDF
Checksum (MD5)
b36481e0955099593dc672e9bd1e715c
Author(s)
Ahmadi, Amir Ali
Parrilo, Pablo A.
Date Issued
January 2010
Journal
Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009
Publisher
Institute of Electrical and Electronics Engineers
Citation
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.
Version
Final published version
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[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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/CDC.2009.5400519
