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A convex polynomial that is not sos-convex | | Amir Ali Ahmadi
; Pablo A. Parrilo
; | | Date: |
6 Mar 2009 | | 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 has been recently speculated 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. Interestingly, our example is found with software using sum of
squares programming techniques and the duality theory of semidefinite
optimization. As a byproduct of our numerical procedure, we obtain a simple
method for searching over a restricted family of nonnegative polynomials that
are not sums of squares. | | Source: | arXiv, 0903.1287 | | Services: | Forum | Review | PDF | Favorites | |
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