The X-circuits Behind Conditional SAGE Certificates
Riley Murray, Helen Naumann, and Thorsten Theobald
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Submission date: 15. Jun. 2020
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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.