TY - GEN

T1 - A positive definite polynomial Hessian that does not factor

AU - Ahmadi, Amir Ali

AU - Parrilo, Pablo A.

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77950845514&partnerID=8YFLogxK

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U2 - 10.1109/CDC.2009.5400519

DO - 10.1109/CDC.2009.5400519

M3 - Conference contribution

AN - SCOPUS:77950845514

SN - 9781424438716

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 1195

EP - 1200

BT - Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009

T2 - 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009

Y2 - 15 December 2009 through 18 December 2009

ER -