Determine whether SONC is closed under further natural restrictions

Determine whether the SONC cone remains closed under natural polynomial restrictions beyond variable identifications and constant restrictions, in order to characterize the structural robustness of SONC certificates.

Background

The paper proves that the SONC cone is preserved under several structured operations, including diagonal scaling, nonnegative linear variable identification or projection, and constant restriction. These closure properties are used in the development of SONC-convexity and its first-level exactness results.

The authors leave unresolved whether additional natural restriction operations preserve SONC membership. Resolving this question would clarify which transformations can safely be used in certificate construction and would deepen the structural theory of the SONC cone.

References

Third, it would be interesting to study whether the SONC cone remains closed under further natural restrictions, beyond the variable identifications and constant restrictions considered here, as this would shed more light on the structural robustness of the cone.

— A Bounded Degree SOS Plus SONC Hierarchy for Polynomial Optimization  (2609.25954 - Dressler et al., 22 Sep 2026) in Conclusion, second paragraph of open questions