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A Convex Form That Is Not a Sum of Squares

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.

Ketersediaan

Informasi Detail

Judul Seri

MATHEMATICS OF OPERATIONS RESEARCH; Vol.48 No.1 February 2023

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 569-582

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

SUMS OF SQUARES
CONVEXITY
SEMIDEFINITE PROGRAMMING
OCTONIONS
CAUCHY-SCHWARZ INEQUALITY

Info Detail Spesifik

https://doi.org/10.1287/moor.2022.1273

Pernyataan Tanggungjawab

James Saunderson

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