Characterize the mixed SOS+SONC convexity condition

Characterize the mixed SOS+SONC convexity condition defined by membership of the first-order remainders of the objective and constraint polynomials in the SOS+SONC cone, and determine its relationship with SOS-convexity and first-order SONC-convexity.

Background

The paper observes that the KKT proof of first-level exactness does not require the first-order Taylor remainders to be purely SONC: membership in the larger SOS+SONC cone is sufficient. This yields a mixed convexity condition that may include cases not covered by either SOS-convexity or first-order SONC-convexity.

The structural relationship and practical significance of this mixed condition are not established. In particular, the authors leave its systematic study, including comparisons with the existing convexity notions, for future work.

References

A systematic study of this mixed convexity condition and its relationship with SOS-convexity and first-order SONC-convexity is left for future work.

— A Bounded Degree SOS Plus SONC Hierarchy for Polynomial Optimization  (2609.25954 - Dressler et al., 22 Sep 2026) in Remark 4.9, Remark “mixedConvexitycertificate”