  Certifying function nonnegativity is a ubiquitous problem in computational
mathematics, with especially notable applications in optimization. We study the
question of certifying nonnegativity of signomials based on the recently
proposed approach of Sums-of-AM/GM-Exponentials (SAGE) decomposition due to the
second author and Shah. The existence of a SAGE decomposition is a sufficient
condition for nonnegativity of a signomial, and it can be verified by solving a
tractable convex relative entropy program. We present new structural properties
of SAGE certificates such as a characterization of the extreme rays of the
cones associated to these decompositions as well as an appealing form of
sparsity preservation. These lead to a number of important consequences such as
conditions under which signomial nonnegativity is equivalent to the existence
of a SAGE decomposition; our results represent the broadest-known class of
nonconvex signomial optimization problems that can be solved efficiently via
convex relaxation. The analysis in this paper proceeds by leveraging the
interaction between the convex duality underlying SAGE certificates and the
face structure of Newton polytopes. While our primary focus is on signomials,
we also discuss how our results provide efficient methods for certifying
polynomial nonnegativity, with complexity independent of the degree of a
polynomial.
