47 research outputs found
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 of
real space. In this article, we undertake the first structural analysis of
conditional SAGE signomials for convex sets . We introduce the -circuits
of a finite subset , which generalize the
simplicial circuits of the affine-linear matroid induced by to a
constrained setting. The -circuits exhibit particularly rich combinatorial
properties for polyhedral , in which case the set of -circuits is
comprised of one-dimensional cones of suitable polyhedral fans.
The framework of -circuits transparently reveals when an -nonnegative
conditional AM/GM-exponential can in fact be further decomposed as a sum of
simpler -nonnegative signomials. We develop a duality theory for
-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 is polyhedral. In conjunction with a
notion of reduced -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.Comment: 30 pages. V2: The title is new, and Sections 1 and 2 have been
rewritten. Section 1 contains a summary of our results. We improved one
result, and consolidated some other