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Study of the cone of sums of squares plus sums of nonnegative circuit forms

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2025

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Dressler, Mareike

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https://doi.org/10.1515/advgeom-2024-0031
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Advances in Geometry. De Gruyter. 2025, 25(1), S. 127-146. ISSN 1615-715X. eISSN 1615-7168. Verfügbar unter: doi: 10.1515/advgeom-2024-0031

Zusammenfassung

In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplication nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert’s 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.

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Fachgebiet (DDC)
510 Mathematik

Schlagwörter

Nonnegative polynomial, sum of squares, sum of nonnegative circuit polynomials, Hilbert’s 1888 Theorem, polynomial optimization

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ISO 690DRESSLER, Mareike, Salma KUHLMANN, Moritz SCHICK, 2025. Study of the cone of sums of squares plus sums of nonnegative circuit forms. In: Advances in Geometry. De Gruyter. 2025, 25(1), S. 127-146. ISSN 1615-715X. eISSN 1615-7168. Verfügbar unter: doi: 10.1515/advgeom-2024-0031
BibTex
@article{Dressler2025-01-29Study-72193,
  title={Study of the cone of sums of squares plus sums of nonnegative circuit forms},
  year={2025},
  doi={10.1515/advgeom-2024-0031},
  number={1},
  volume={25},
  issn={1615-715X},
  journal={Advances in Geometry},
  pages={127--146},
  author={Dressler, Mareike and Kuhlmann, Salma and Schick, Moritz}
}
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