Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
65/2020

The X-circuits Behind Conditional SAGE Certificates

Riley Murray, Helen Naumann and Thorsten Theobald

Abstract

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.

Received:
Jun 15, 2020
Published:
Jun 18, 2020

Related publications

inJournal
2022 Journal Open Access
Riley Murray, Helen Naumann and Thorsten Theobald

Sublinear circuits and the constrained signomial nonnegativity problem

In: Mathematical programming, (2022)