A Convex Form That Is Not a Sum of Squares
本文给出了一个具体的272变量四次凸型，它不是平方和，并利用该构造改进了球面上四次型优化的基本平方和松弛的近似质量。
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. Funding: This work was supported by the Australian Research Council [Grant DE210101056].