Sublinear circuits and the constrained signomial nonnegativity problem

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 $${\mathcal {A}}\subset {\mathbb {R}}^n$$ A ⊂ R n , which generalize the simplicial circuits of the affine-linear matroid induced by $${\mathcal {A}}$$ 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.

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

“…In [26], the following concept of sublinear circuits has been developed to resolve these questions. For β ∈ T , recall the notion N β = {ν ∈ R T : ν \β ≥ 0, α∈T ν α = 0}.…”

“…Then a vector ν * ∈ N β is an X-circuit if and only if (ν * , σ X (−T ν * )) spans an extreme ray of P . For the case of polyhedral X, this characterization of X-circuits is straightforward to see and for non-polyhedral X, the convex-geometric details can be found in [26,Theorem 3.6].…”

“…The geometry of the SAGE cone is governed by the simplicial circuits of the affine-linear matroid induced by the support T [12,24,36]. In order to exhibit the geometry of the conditional SAGE cone, we spell out and study the sublinear circuits of a finite point set in R n , which generalize the simplicial circuits to a constrained setting [26].…”

Symmetry Reduction in AM/GM-Based Optimization

“…These techniques rely on the fact that every SONC form p (and similarly, SAGE forms) can be written as a sum of nonnegative circuit polynomials supported on the support of p ( [38], see also [27,32]). The AM/GM techniques can also be extended to constrained settings ( [13,28,29,37]). For the second-order representability of the SAGE cone and the SONC cone see [4,24,30] and for combining the SONC cone with the cone of sums of squares see [12].…”

The Duality of SONC: Advances in Circuit-based Certificates

“…Minimal circuits are also referred to as reduced circuits, see e.g. [MNT22], or as w-thin, see [Rez89]. Here, we follow the notation introduced in [FdW19].…”