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 ${\tilde{C}}_{n,d}$ and ${\widetilde{\Sigma C}}_{n,d}$ respectively, then our main contribution is to prove that ${\tilde{C}}_{n,d}={\widetilde{\Sigma C}}_{n,d}$ 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 G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\Sigma C_{n,d}$ 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 ${\tilde{C}}_{2,6}\setminus {\widetilde{\Sigma C}}_{2,6}$ and ${\tilde{C}}_{3,4}\setminus {\widetilde{\Sigma C}}_{3,4}$ and forms in $C_{3,6}\setminus \Sigma C_{3,6}$ and $C_{4,4}\setminus \Sigma C_{4,4}$, and a general procedure for constructing forms in $C_{n,d+2}\setminus \Sigma C_{n,d+2}$ 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.

25 pages; minor editorial revisions made; formal certificates for computer assisted proofs of the paper added to arXiv

References in corpus (3)

• Symmetry groups, semidefinite programs, and sums of squares
• NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems
• A convex polynomial that is not sos-convex

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• Lifting for Simplicity: Concise Descriptions of Convex Sets
• Piecewise SOS-Convex Moment Optimization and Applications via Exact Semi-Definite Programs
• A convex form that is not a sum of squares
• A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity
• New examples of extremal positive linear maps