TY  - JOUR
A1  - Murray, Riley
A1  - Naumann, Helen
A1  - Theobald, Thorsten
T1  - Sublinear circuits and the constrained signomial nonnegativity problem
T2  - Mathematical programming : Series A, Series B
N2  - 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.
KW  - Sums of arithmetic-geometric exponentials
KW  - Positive signomials
KW  - Exponential sums
KW  - Sums of nonnegative circuit polynomials (SONC)
KW  - Positive polynomials
KW  - Multiplicative convexity
KW  - Log convex sets
Y1  - 2022
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/69563
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-695634
SN  - 1436-4646
N1  - Open Access funding enabled and organized by Projekt DEAL.
N1  - Mathematics Subject Classification: Primary 14P05, 90C23, Secondary
05B35
VL  - 198
IS  - 1
SP  - 471
EP  - 505
PB  - Springer
CY  - Berlin ; Heidelberg
ER  -