A Complete Characterization of the Gap between Convexity and SOS-Convexity
Amir Ali Ahmadi, Pablo A. Parrilo

TL;DR
This paper characterizes when convex polynomials are also sos-convex, showing they coincide only in specific cases, and provides explicit examples and a construction method for non-sos convex polynomials.
Contribution
It proves the equivalence of three sos-based convexity conditions and fully characterizes when convex polynomials are sos-convex, extending Hilbert's results.
Findings
Convex and sos-convex polynomials are equivalent only in specific cases.
Explicit examples of convex but not sos-convex polynomials are provided.
A construction method for non-sos convex polynomials from nonnegative, non-sos forms is introduced.
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 variables of degree with and respectively, then our main contribution is to prove that if and only if or or . We also present a complete characterization for forms (homogeneous polynomials) except for the case which is joint work with G. Blekherman and is to be…
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Taxonomy
TopicsAdvanced Optimization Algorithms Research · Stability and Control of Uncertain Systems · Polynomial and algebraic computation
