A convex polynomial that is not sos-convex

A multivariate polynomial p(x) = p(x1, . . . , xn) is sos-convex if its Hessian H(x) can befactored as H(x) = M (x)M(x) with a possibly nonsquare polynomial matrix M(x). It iseasy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, theproblem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefiniteprogram, which can be solved efficiently. Motivated by this computational tractability, it hasbeen recently speculated whether sos-convexity is also a necessary condition for convexity ofpolynomials. In this paper, we give a negative answer to this question by presenting an explicitexample 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 programmingtechniques and the duality theory of semidefinite optimization. As a byproduct of our numericalprocedure, we obtain a simple method for searching over a restricted family of nonnegativepolynomials that are not sums of squares.