2023/05/24 by Mareike Dressler, Dressler, Mareike, Salma Kuhlmann +3
Computer Science · Mathematics · #26C10 #52A99 #52B99 #90C23 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Polynomial and algebraic computation #Primary: 14P99 #Secondary: 12D15
paper · pdf · doi:10.48550/arxiv.2305.14848
openalex publication_date 2023/05/24 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28
In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplications nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert's 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of both the SOS and the SONC cone. This follows by exploiting a new necessary condition for membership in the SONC cone.