A complete characterization of the gap between convexity and sos-convexity

Abstract

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming, whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in n variables of degree d with C̃n,d and ∑C̃n,d respectively, then our main contribution is to prove that C̃n,d = ∑C̃ n,d if and only if n = 1 or d = 2 or (n, d) = (2, 4). We also present a complete characterization for forms (homogeneous polynomials) except for the case (n, d) = (3, 4), which is joint work with Blekherman and is to be published elsewhere. Our result states that the set Cn,d of convex forms in n variables of degree d equals the set ∑Cn,d of sos-convex forms if and only if n = 2 or d = 2 or (n, d) = (3, 4). To prove these results, we present in particular explicit examples of polynomials in C̃2,6\ ∑C̃2,6 and C̃3,4\∑C̃3,4 and forms in C3,6\∑C3,6 and C4,4\ ∑C4,4, and a general procedure for constructing forms in C n,d+2\∑Cn,d+2 from nonnegative but not sos forms in n variables and degree d. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp., forms) are sos-convex exactly in cases where nonnegative polynomials (resp., forms) are sums of squares, as characterized by Hilbert. © 2013 Society for Industrial and Applied Mathematics.