A convex polynomial that is not sos-convex |
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Authors: | Amir Ali Ahmadi Pablo A. Parrilo |
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Affiliation: | 1. Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
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Abstract: | A multivariate polynomial p(x)?=?p(x 1, . . . , x n ) is sos-convex if its Hessian H(x) can be factored as H(x)?= M T (x) M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it is natural to study whether sos-convexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex. |
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