Abstract
Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.
Highlights
Let A be a non-empty finite subset of Rn and RA denote the set of real vectors whose components are indexed by the set A
Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality
The goal of the current paper is to develop techniques for handling sublinear circuits, which provide an access towards approaching non-conic polyhedral sets
Summary
Let A be a non-empty finite subset of Rn and RA denote the set of real vectors whose components are indexed by the set A. A reducibility concept for sublinear circuits provides a non-redundant decomposition of the conditional SAGE cone in terms of reduced circuits. This reducibility notion generalizes the reducibility notion for the unconstrained situation which was introduced in [12], see [7]. 4. Building upon the criteria for sublinear circuits, we study the prominent cases of the non-negative orthant Rn+ and the cube [−1, 1]n in detail, in particular the planar case and with regard to small support sets.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
