Sublinear Circuits and the Constrained Signomial Nonnegativity Problem

TL;DR

This paper introduces the concept of $X$-circuits for analyzing the structure of conditional SAGE signomials over convex sets, providing new tools for nonnegativity proofs and polynomial optimization.

Contribution

It generalizes the theory of circuits to convex sets, develops a duality framework, and offers a geometric interpretation for decomposing nonnegative signomials.

Findings

$X$-circuits exhibit rich combinatorial properties for polyhedral $X$.

The duality theory enables optimal power cone reconstruction.

Characterization of extreme rays of conditional SAGE cones.

Abstract

Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset $X$ of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets $X$. We introduce the $X$-circuits of a finite subset $\mathcal{A} \subset \mathbb{R}^n$, which generalize the simplicial circuits of the affine-linear matroid induced by $\mathcal{A}$ to a constrained setting. The $X$-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral $X$, in which case the set of $X$-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of $X$-circuits transparently reveals when an $X$-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler $X$-nonnegative signomials. We develop a duality theory for $X$-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when $X$ is polyhedral. In conjunction with a notion of reduced $X$-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.