A Convex Form That Is Not a Sum of Squares

James Saunderson ()
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James Saunderson: Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3800, Australia

Mathematics of Operations Research, 2023, vol. 48, issue 1, 569-582

Abstract: Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree 4 in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree that are near-constant on the unit sphere are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere.

Keywords: Primary: 90C22; secondary: 90C25; 52A41; convexity; sums of squares; octonions; Cauchy-Schwarz inequality; semidefinite programming (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:1:p:569-582

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