  We describe a generalization of the Sums-of-AM/GM Exponential (SAGE)
relaxation methodology for obtaining bounds on constrained signomial and
polynomial optimization problems. Our approach leverages the fact that relative
entropy based SAGE certificates conveniently and transparently blend with
convex duality, in a manner that Sums-of-Squares certificates do not. This more
general approach not only retains key properties of ordinary SAGE relaxations
(e.g. sparsity preservation), but also inspires a novel perspective-based
method of solution recovery. We illustrate the utility of our methodology with
a range of examples from the global optimization literature, along with a
publicly available software package.
