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A Convex Form That Is Not a Sum of Squares

Author

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  • James Saunderson

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

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.

Suggested Citation

  • James Saunderson, 2023. "A Convex Form That Is Not a Sum of Squares," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 569-582, February.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:1:p:569-582
    DOI: 10.1287/moor.2022.1273
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