Complete Hierarchy of Relaxation for Constrained Signomial Positivity

In this article, we prove that the Sums-of-AM/GM Exponential (SAGE) relaxation generalized to signomial over a constrained set is complete, with a compactness assumption. The high-level structure of the proof is as follows. We first apply variable change to convert a set of rational exponents to polynomial equations. In addition, we make the observation that linear constraints of the variables may also be converted to polynomial equations after variable change. Note that any convex set may be expressed as a set of linear constraints. Further, we use redundant constraints to find reduction to Positivstellensatz. We rely on Positivstellensatz results from algebraic geometry to obtain a decomposition of positive polynomials. Lastly, we explicitly show that the decomposition is of a form certifiable by SAGE.