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Certifying function nonnegativity is a ubiquitous problem in computationalmathematics, with especially notable applications in optimization. We study thequestion of certifying nonnegativity of signomials based on the recentlyproposed approach of Sums-of-AM/GM-Exponentials (SAGE) decomposition due to thesecond author and Shah. The existence of a SAGE decomposition is a sufficientcondition for nonnegativity of a signomial, and it can be verified by solving atractable convex relative entropy program. We present new structural propertiesof SAGE certificates such as a characterization of the extreme rays of thecones associated to these decompositions as well as an appealing form ofsparsity preservation. These lead to a number of important consequences such asconditions under which signomial nonnegativity is equivalent to the existenceof a SAGE decomposition; our results represent the broadest-known class ofnonconvex signomial optimization problems that can be solved efficiently viaconvex relaxation. The analysis in this paper proceeds by leveraging theinteraction between the convex duality underlying SAGE certificates and theface structure of Newton polytopes. While our primary focus is on signomials,we also discuss how our results provide efficient methods for certifyingpolynomial nonnegativity, with complexity independent of the degree of apolynomial.

Newton Polytopes and Relative Entropy Optimization