Sublinear circuits and the constrained signomial nonnegativity problem
- Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset X of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets X. We introduce the X-circuits of a finite subset A⊂Rn , which generalize the simplicial circuits of the affine-linear matroid induced by A to a constrained setting. The X-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral X, in which case the set of X-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of X-circuits transparently reveals when an X-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler X-nonnegative signomials. We develop a duality theory for X-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when X is polyhedral. In conjunction with a notion of reduced X-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.
Author: | Riley MurrayORCiD, Helen NaumannGND, Thorsten TheobaldORCiDGND |
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URN: | urn:nbn:de:hebis:30:3-695634 |
DOI: | https://doi.org/10.1007/s10107-022-01776-w |
ISSN: | 1436-4646 |
Parent Title (English): | Mathematical programming : Series A, Series B |
Publisher: | Springer |
Place of publication: | Berlin ; Heidelberg |
Document Type: | Article |
Language: | English |
Date of Publication (online): | 2022/02/09 |
Date of first Publication: | 2022/02/09 |
Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
Release Date: | 2023/08/10 |
Tag: | Exponential sums; Log convex sets; Multiplicative convexity; Positive polynomials; Positive signomials; Sums of arithmetic-geometric exponentials; Sums of nonnegative circuit polynomials (SONC) |
Volume: | 198 |
Issue: | 1 |
Page Number: | 35 |
First Page: | 471 |
Last Page: | 505 |
Note: | Open Access funding enabled and organized by Projekt DEAL. |
Note: | Mathematics Subject Classification: Primary 14P05, 90C23, Secondary 05B35 |
HeBIS-PPN: | 512574472 |
Institutes: | Informatik und Mathematik |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
MSC-Classification: | 52-XX CONVEX AND DISCRETE GEOMETRY / 52Axx General convexity / 52A20 Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] |
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C30 Nonlinear programming | |
Sammlungen: | Universitätspublikationen |
Licence (German): | Creative Commons - CC BY - Namensnennung 4.0 International |