A convex polynomial that is not sos-convex

TL;DR

This paper provides a counterexample demonstrating that some convex polynomials are not sos-convex, challenging the assumption that sos-convexity is necessary for polynomial convexity.

Contribution

It constructs the first explicit example of a convex polynomial that is not sos-convex, using sum of squares programming and semidefinite optimization.

Findings

Counterexample of a convex but not sos-convex polynomial

Demonstrates sos-convexity is not necessary for convexity

Introduces a method for finding non-sos polynomials

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.