Determine whether mixed SOS+SONC convexity is genuinely more general and computationally testable

Determine whether the mixed SOS+SONC first-order convexity condition defines a class strictly larger than the corresponding SOS-convexity and first-order SONC-convexity classes, and whether membership in that class admits computationally effective tests.

Background

The mixed condition requires the first-order remainders associated with the objective and constraints to lie in the SOS+SONC cone. The paper shows that this condition suffices for first-level exactness, but does not determine whether it provides a genuinely broader certificate-based convexity framework.

The authors also do not establish algorithms or tractable criteria for testing the condition. These two questions—strict enlargement of the known classes and effective computational verification—are explicitly identified as unresolved.