Nonnegativity of signomials with Newton simplex over convex sets

TL;DR

The paper examines signomials with Newton polytopes equal to a simplex, where vertex terms have positive coefficients and interior terms may be negative, and studies their nonnegativity over a convex set $X$. It develops a signed X-SAGE framework, proving that such a signomial is nonnegative on $X$ if and only if it lies in the signed X-SAGE cone $C_X(\mathcal{A},\mathcal{B})$, which yields a convex relative-entropy program using the support function $\sigma_X$ for verification. The contributions include a complete characterization, a tractable nonnegativity test for constrained settings, dual cone representations, and both univariate and polynomial variants, along with necessary conditions demonstrated via counterexamples. This work extends unconstrained SAGE certificates to constrained domains, providing an efficient certificate method for a broad class of sparse signomials in optimization contexts. The results reveal hidden convexity structure and connect AM/GM-based certificates to duality and polyhedral geometry, offering practical tools for sparse signomial optimization.

Abstract

We study a class of signomials whose positive support is the set of vertices of a simplex and which may have multiple negative support points in the simplex. Various groups of authors have provided an exact characterization for the global nonnegativity of a signomial in this class in terms of circuit signomials and that characterization provides a tractable nonnegativity test. We generalize this characterization to the constrained nonnegativity over a convex set $X$. This provides a tractable $X$-nonnegativity test for the class in terms of relative entropy programming and in terms of the support function of $X$. Our proof methods rely on the convex cone of constrained SAGE signomials (sums of arithmetic-geometric exponentials) and the duality theory of this cone.

Figures (3)

• Figure 1: The support and the Newton polytope of $f$ in Example \refex: introduction
• Figure 2: The support points of $f$ in Example \ref{['ex:one-dim1']}
• Figure 3: The support points of $f$ in Example \ref{['ex:two-dim2']}

Theorems & Definitions (20)

• Example 1.1
• Theorem 1.2
• Theorem 2.1: chandrasekaran-shah-2016mcw-partial-dualization
• Proposition 2.2: Theorem 1 and Corollary 1 in mcw-partial-dualization
• Proposition 2.3: mcw-partial-dualization
• Remark 2.4
• Lemma 3.1
• proof
• Remark 3.2
• proof : Proof of Theorem \ref{['th:main']}