2020/03/08 by Allen Houze Wang, Wang, Allen Houze, Priyank Jaini +5
Engineering · Mathematics · #Advanced Control Systems Optimization #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.2003.03731
openalex publication_date 2020/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, the conditional SAGE certificate has been proposed as a sufficient condition for signomial positivity over a convex set. In this article, we show that the conditional SAGE certificate is complete. That is, for any signomial f(x) = ∑j=1ℓcj exp(Ajx) defined by rational exponents that is positive over a compact convex set X, there is p ∈ ℤ+ and a specific positive definite function w(x) such that w(x)p f(x) may be verified by the conditional SAGE certificate. The completeness result is analogous to Positivstellensatz results from algebraic geometry, which guarantees representation of positive polynomials with sum of squares polynomials. The result gives rise to a convergent hierarchy of lower bounds for constrained signomial optimization over an arbitrary compact convex set that is computable via the conditional SAGE certificate.