The algebraic boundary of the sonc cone

Abstract

We describe the algebraic boundary of the cone of sums of nonnegative circuit polynomials (sonc). This cone is generated by monomials and singular circuit polynomials. We interpret a singular circuit polynomial as compositions of an agiform in the sense of Reznick and a real torus action. Our main result is that the algebraic pieces of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each piece is contained in a rational variety called the positive discriminant, defined as the image of a Horn-Kapranov-type map. We give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.We recover that the sonc cone equals the sage cone, and we prove that that the sonc cone is equal to the nonnegativity cone in codimension at most two. We also derive a combinatorial characterization of support sets for which the sonc cone is equal to the nonnegativity cone, disproving in the process a conjecture by Chandrasekaran, Murray, and Wierman.

Sublinear circuits for polyhedral sets

“…It remains a future task to study necessary and sufficient criteria for sublinear circuits of structured non-polyhedral sets, such as sets with symmetry; for recent work on symmetric SAGE-based optimization see [14]. In a different direction, Forsgård and de Wolff [7] have characterized the boundary of the SAGE cone through a connection between circuits and tropical geometry. It also remains for future work to establish a generalization of this, aiming at connecting the conditional SAGE cone and sublinear circuits to tropical geometry.…”

“…One step further, a reducibility concept for sublinear circuits provides a non-redundant decomposition of the conditional SAGE cone in terms of reduced circuits. This reducibility concept generalizes the reducibility concept for the unconstrained situation which was introduced in [12], see also [7]. The reduced sublinear circuits are the key concept to study the extremal rays of the X-SAGE cone [17].…”

“…For an X-circuit ν, let ν + := {α : ν α ≥ 0} and ν − denote the single index β with ν β < 0. First recall that in the classical case of affine matroids, any simplicial circuit ν supported on at least three elements has no other support point except ν − contained in the relative interior of the convex hull of all its support points, and the coefficients of ν + are positive multiples of the barycentric coordinates of β, i.e., relint conv(supp(ν)) ∩ ν + = ∅ and Aν = 0 (see, e.g., [7]). In the following theorem, we give a generalization of this property to the case of X-circuits.…”

“…In the special case X = R n , the first condition is equivalent to Aν * = 0, which together with the second condition tells us that ν * is a circuit of the affine matroid with ground set A ⊂ R n (see, for example, [7,19]). Note that these R n -circuits are uniquely determined (up to scaling) by their supports.…”

“…Among the class of polyhedra, polyhedral cones exhibit particularly nice properties and were in the focus of attention in earlier treatments. Note that, as a very particular case, the unconstrained setting X = R n , which is treated in [7,12,15], also falls into the class of polyhedral cones. The univariate case R + was studied in detail in [17].…”