Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
The X-circuits Behind Conditional SAGE Certificates
Conditional SAGE certificates are a decomposition method to prove nonnegativity of a signomial or polynomial over some subset X of Euclidean real space. In the case when X is convex, membership in the signomial "X-SAGE cone" can be completely characterized by a relative entropy program involving the support function of X. Following promising computational experiments, and a recently proven completeness result for a hierarchy of X-SAGE relaxations for signomial optimization, we undertake a structural analysis of signomial X-SAGE cones. Our approach begins by determining a suitable notion of an "X-circuit," in such a way as to generalize classical affine-linear simplicial circuits from matroid theory. Our definition of an X-circuit is purely convex-geometric, with no reference to signomials or SAGE certificates. We proceed by using X-circuits to characterize the more elementary "X-AGE cones" which comprise a given X-SAGE cone. Our deepest results are driven by a duality theory for X-circuits, which is applicable to primal and dual X-SAGE cones in their usual forms, as well as to a certain logarithmic transform of the dual cone. In conjunction with a notion of reduced X-circuits this facilitates to characterize the extreme rays of the X-SAGE cones. Our results require no regularity conditions on X beyond those which ensure a given X-SAGE cone is proper; particularly strong conclusions are obtained when X is a polyhedron.