Sublinear Circuits for Polyhedral Sets

Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.


Introduction
Let A be a non-empty finite subset of R n and R A denote the set of real vectors whose components are indexed by the set A.
For β ∈ A, write ) for the cone of vectors in R A whose entries sum to 0 and which may only have a negative entry in component β.Here, ν \β abbreviates the vector in R A\{β} which consists of all the components of ν except the component indexed with β.
In the context of non-negative polynomials and non-negative exponential sums, Murray and the authors [17] have recently introduced the following generalization of the simplicial circuits of an affine matroid.A non-zero vector ν * ∈ N β is called a sublinear circuit of A with respect to a given convex set X (for short, X-circuit) if (1) if ν → sup x∈X ((−Aν) T x) is linear on a two-dimensional cone in N β , then ν * is not in the relative interior of that cone.Here, A is treated as a linear operator A : R A → R n , ν → α∈A αν α .
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.Moreover, the condition ν * ∈ N β enforces that the convex hull of the support supp ν * := {α : ν * α = 0} forms a simplex (possibly of dimension less than n) and exactly one element in supp ν * is contained in the relative interior of this simplex.By the identification of Date: August 31, 2021.circuits ν with their supports, it is customary to call a subset A ⊂ A a simplicial circuit if its convex hull conv(A) forms a simplex and the relative interior relint conv(A) contains exactly one element of A. See Figure 1.
Sublinear circuits appear naturally in the study of non-negative polynomials and, more generally, of non-negative exponential sums α∈A c α exp(α T x).In the framework of exponential sums, Murray, Chandrasekaran and Wierman [16] have shown that the set of exponential sums (also denoted as signomials) α∈A c α exp(α T x) which have at most one negative term and which are non-negative on X can be characterized in terms of a relative entropy program.Sums of such exponential sums are non-negative as well.The cone of exponential sums which admit such a non-negativity certificate is called the X-SAGE cone (or conditional SAGE cone) supported on A and is denoted C X (A).Here, the acronym SAGE stands for Sums of Arithmetic-Geometric Exponentials [3].This cone yields non-negativity certificates for a subclass of polynomials and signomials and thus provides a complement to non-negativity certificates based on sums of squares.It is possible to combine these techniques, see [11].
The introduction and the study of sublinear circuits is motivated by the following guiding questions: (1) For f ∈ C X (A), there is often more than one way to write f as a sum of exponential sums which have only one negative term and which are non-negative on X. Are there distinguished representations among them?(2) Can the X-SAGE cone be naturally decomposed as a Minkowski sum of smaller subcones?(3) How can the convex geometric properties, such as the extremal rays, of the X-SAGE cone be characterized?Here, the second and the third question can be seen as geometric viewpoints of the first question.By [17], the conditional SAGE cone C X (A) can be decomposed as a Minkowski sum, where each non-trivial summand refers to the X-SAGE exponentials induced by a sublinear circuit (see Proposition 2.6 for a formal statement).Therefore, the sublinear circuits can be seen as a convex-combinatorial core underlying the conditional SAGE cone.In the unconstrained setting, the circuit viewpoint has been used prominently in the works of Reznick [21], Iliman and de Wolff [9] as well as Pantea, Koeppl and Craciun [20] on non-negative polynomials.
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 notion generalizes the reducibility notion for the unconstrained situation which was introduced in [12], see also [7].The reduced R n -circuits are the key concept to characterize the extremal rays of the unconstrained SAGE cone, since the reduced R n -circuits induce extremal rays.In generalization of this, the reduced sublinear circuits facilitate to study the extremal rays of the X-SAGE cone [17], see Proposition 2.8 for a formal statement.
From a more general point of view, sublinear circuits generalize the combinatorial concepts known from an affine matroid, by taking additionally into account a convex constraint set X.As such, sublinear circuits enlarge the tool set of convex-combinatorial techniques in algebraic geometry and algebraic optimization, see [2,4,10,13,22] for general background on the rich connections between these disciplines.
In the current paper, we study sublinear circuits for the situation that X is polyhedral.In this setting, the sublinear circuits can be exactly characterized in terms of the normal fan of a certain polyhedron, see Proposition 2.2.This induces a rich polyhedral-combinatorial structure and makes these sublinear circuits amenable to effective computations.For polyhedral X, the number of sublinear circuits is finite, and this gives decompositions of the X-SAGE cones into finitely many summands referring to the X-SAGE exponentials induced by a sublinear circuit.
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].Moreover, every univariate case can be transformed to one of the two conic cases R (unconstrained case), R + (one-sided infinity interval), or to the non-conic [−1, 1] (compact interval).In the multivariate case, the polyhedra R n (unconstrained case), R n + (non-negative orthant) and the cube [−1, 1] n provide prominent examples.In contrast to the unconstrained case and to the non-negative orthant, the cube [−1, 1] n provides a non-conic case.
The goal of the current paper is to develop techniques for handling sublinear circuits, which also provide an access towards approaching non-conic polyhedral sets.
Contributions. 1.We reveal some precise connections between sublinear circuits and their supports, see Lemma 3.3.In particular, we show that in general sublinear circuits are not uniquely determined by their supports, see Example 3.2.
2. We develop necessary and sufficient conditions for identifying X-circuits based on support conditions.See Theorems 4.1 and 4.5.
3. We give conditions for identifying reduced X-circuits which generalize the known characterizations for the unconstrained case.See Theorems 6.2 and 6.4.

4.
Building upon the criteria for sublinear circuits, we study the prominent cases of the non-negative orthant R n + and the cube [−1, 1] n in detail, in particular the planar case and with regard to small support sets.Specifically, for A = {(i, j) : 1 ≤ i, j ≤ 3} ⊂ R 2 and X = [−1, 1] 2 , there are 132 circuits and 24 reduced X-circuits, which we classify.See Sections 5 and 6.
5. As a specific consequence, we can exactly determine the extreme rays of the univariate [−1, 1]-SAGE cone, see Theorem 6.6.
For further recent work on the techniques for certifying non-negativity of signomials and polynomials based on the SAGE cone and its variants, see [1,5,18,23,24].
The paper is structured as follows.After collecting relevant concepts of sublinear circuits and non-negative signomials in Section 2, we study the connection of X-circuits and their supports in Section 3. Section 4 deals with necessary and sufficient conditions for sublinear circuits, and Section 5 focuses on the case of the cube [−1, 1] n .In Section 6, we provide criteria for reduced sublinear circuits, which gives as a consequence the characterization of the extreme rays of the [−1, 1]-SAGE cone.Section 7 concludes the paper.
Acknowledgment.We thank Riley Murray for valuable discussions and the anonymous referees for their helpful suggestions.

Preliminaries
Throughout the paper, the symbol 0 denotes the zero vector, 1 denotes the all-ones vector and [m] abbreviates the set {1, . . ., m} for m ∈ N.For a given convex subset X ⊂ R n , denote by σ X (y) = sup{y T x : x ∈ X} its support function.
2.1.X-circuits.For a non-empty convex set X and finite A ⊂ R n , we consider X-circuits as defined in the Introduction, where we note that the two defining conditions can also be expressed in terms of the support function: then (1) becomes the condition σ X (−Aν ⋆ ) < ∞ and in (2), the mapping ν → σ X (−Aν) occurs.
A sublinear circuit λ ∈ N β is called normalized if λ β = −1, in which case the condition 1 T λ = 0 in the definition of N β implies α =β λ α = 1.For normalized X-circuits, we usually employ the symbol λ, whereas we use the symbol ν for X-circuits which are not necessarily normalized.For X ⊂ R n , denote by Λ X (A, β) the set of normalized X-circuits of A with negative entry corresponding to β ∈ A and by Λ X (A) := β∈A Λ X (A, β) the set of all normalized X-circuits of A. For example, in the univariate case X = R with A = {α 1 , . . ., α m } ⊂ R, the normalized X-circuits are where e (i) denotes the i-th unit vector in R m .It is possible that a given support set A ⊂ R n has no R n -circuits, but then every α ∈ A is an extreme point of conv A.
Example 2.1.In the context of the conditional SAGE cone, we can assume without loss of generality, that the convex set X is closed.In the one-dimensional case, up to translation and additive inversion, each closed, convex set is of the form X (1) = R, X (2) For the support set A = {0, 1, 2}, it is instructive to list the sublinear circuits with respect to the three sets X (1) , X (2) and X (3) .The set Λ (1) of X (1) -circuits is R + (1, −2, 1) T , which is a special case of (2.1).The set Λ (2) of X (2) -circuits is and this is a special case of Proposition 2.3 below.In particular, the element (−1, 1, 0) T is not an X (1) -circuit, because The set Λ (3) of X (3) -circuits is which is a special case of Proposition 3.1 proven in Section 3. Note that, for example, the element (1, −1, 0) T is not an X (2) -circuit, as For polyhedral X, the sublinear circuits can be characterized in terms of normal fans of polyhedra.We refer the reader for background on normal fans to [25,Chapter 7] (for the bounded case of polytopes), [8,Section 5.4] or [22,Chapter 2].For each face F of a polyhedron P , let N P (F ) = {w : z T w = σ P (w) ∀ z ∈ F } be the associated outer normal cone.
The support function σ P of a polyhedron P is linear on every outer normal cone, and the linear representation may be given by σ P (w) = z T w for any z ∈ F .The outer normal fan of P is the collection of all outer normal cones, O(P ) = {N P (F ) : F is a face of P }.
For a convex cone K ⊂ R n , denote by K * := {c ∈ R n : c T x ≥ 0 for all x ∈ K} the dual cone and by K • := −K * the polar.For a set S ⊂ R n , let rec(S) := {t : ∃s ∈ S such that s + λt ∈ S ∀ λ ≥ 0} denote its recession cone.Using these notations, the support of O(P ) coincides with rec(P ) • .The full-dimensional linearity domains of the support function σ P are the outer normal cones of the vertices of P (see also [6,Section 1]).Proposition 2.2.[17] Let X be a polyhedron.Then As a consequence, there are only finitely many normalized X-circuits.
If X is a polyhedral cone, the situation simplifies, because the support function σ X (−Aν) of a circuit ν can only attain the values zero and infinity.Namely, since O the X-circuits ν ∈ N β are precisely the edge generators of the polyhedral cone {ν ∈ N β : σ X (−Aν) ≤ 0}.
In the univariate case with A = {α 1 , . . ., α m } ⊂ R, the sublinear circuits for the univariate cone [0, ∞) have been determined in [17]: are the vectors either of the form λ = e (k) − e (j) for j < k or of the form Note that the X-circuits of the second form are exactly the R-circuits from (2.1).
Remark 2.4.By Proposition 2.2, the X-circuits of A are the outer normal vectors to facets of polyhedra P = −A T X + N • β (for some β).As N β is pointed, P is always full-dimensional.Example 2.5.If X is a convex cone, then the second condition in the definition of X-circuits simplifies, because the support function evaluates to 0 whenever it is finite.Consider the conic sets + with respect to the support set A = {(0, 0) T , (0, 4) T , (4, 0) T , (1, 1) T }, as illustrated in Figure 2. Three points of A are vertices of the convex hull of A, and the point (1, 1) T is contained in the relative interior of the convex hull of A. The set Λ (1) of X (1) -circuits is R + (2, 1, 1, −4) T , and the set Λ (2) of X (2) -circuits is Similar to the arguments for the one-dimensional-cases in Example 2.1, the X (2) -circuit (0, 1, 3, −4) T is not an X (1) -circuit, as the resulting support function with respect to X (1) is not finite anymore: 2.2.Non-negativity of signomials.We consider the cone C X (A) of X-SAGE signomials supported on A [16], which was informally introduced in the Introduction.For β ∈ A, set called the X-AGE cone supported on A with respect to β.By [16], C X (A) decomposes as All signomials in C X (A, λ) are non-negative over X.Moreover, for polyhedral X, the conditional SAGE cone can be naturally decomposed into the Minkowski sum of a finite set of λ-witnessed cones, where λ runs over the normalized X-circuits.
Proposition 2.6.[17] Let X ⊂ R n be a polyhedron and Λ X (A) be nonempty.Then the conditional SAGE cone C X (A) decomposes as the finite Minkowski sum Reduced circuits and non-negativity of signomials.In general, the representation in Proposition 2.6 can include redundancies.Using a reducibility concept of circuits, which takes into account the value σ X (−Aν) of an X-circuit ν, an irredundant representation can be given.The extended form of an is called the circuit graph of (A, X).Whenever we consider the circuit graph or the reduced sublinear circuits defined subsequently, we will tacitly assume that the functions x → exp(α T x), α ∈ A, are linearly independent on X.Then the set G X (A) is pointed and closed (see [17]).
Definition 2.7.An X-circuit ν is called reduced if its extended form generates an extreme ray of G X (A).Denote by Λ ⋆ X (A) the set of normalized reduced X-circuits.For polyhedral X, the conditional SAGE cone C X (A) can be decomposed into the Minkowski sum of a finite set of λ-witnessed cones, where λ runs over the reduced X-circuits.Proposition 2.8.[17] Let X ⊂ R n be a polyhedron and Λ X (A) be non-empty.Then the conditional SAGE cone C X (A) decomposes as the finite Minkowski sum Moreover, there does not exist a proper subset In the univariate case with A = {α 1 , . . ., α m } sorted ascendingly, we have (2) − e (1) }.See [7] for Λ ⋆ R (A) and [17] for Λ ⋆ [0,∞) (A).

X-circuits and their supports
In this section, we study the relationship between X-circuits and their supports.We begin with a study of the compact univariate case [−1, 1].This complements the known cases R from the Introduction and [0, ∞) from Proposition 2.3.

and only if it is of the following form:
(1) λ = e (j) − e (i) Proof.Fix j ∈ [n] and write N j := N (α j ) for short.By Proposition 2.2, the X-circuits are the vectors spanning the rays in the outer normal cone of the polyhedron Hence, a point w is contained in P if and only if By eliminating µ, this is equivalent to Eliminating θ then gives We claim that none of the inequalities in the definition of P is redundant.Namely, for each inequality 2), the point e (i) + e (j) + e (k) satisfies this particular inequality with equality and all of the other inequalities strictly.Similarly, for the inequalities in (3.1), it suffices to consider the point α j e (i) + α i e (j) in case i < j and α i e (i) + α j e (j) in case i > j.By Remark 2.4, the polyhedron P is full-dimensional.
Hence, by Proposition 2.2, the normalized X-circuits in N j are exactly the ones given in the statement of the theorem.
The supports of X-circuits.As stated in the Introduction, in the classical case of affine circuits, the normalized circuits are uniquely determined by their supports.Moreover, as a consequence of Theorem 3.1, in the case X = [−1, 1], the normalized X-circuits are uniquely determined by their signed supports.As explained in the following, this phenomenon does not extend to sublinear circuits for arbitrary sets.
In the case of sublinear circuits supported on two elements, the two non-zero entries are additive inverses of each other, so that, for a given β and a given support, indeed this signed support uniquely determines the circuit up to a positive factor.In order to exhibit the mentioned phenomenon, we present a counterexample with support size 3.

Example 3.2. Let
We show that for β := α 1 , there are two non-proportional circuits which are supported on all three elements of A. Specifically, we construct an example, in which ν (1) := (−2, 1, 1) T and ν (2) := (−3, 1, 2) T are sublinear circuits.Note that both of them have the same signed support, but they are not multiples of each other.Observe that We set up X in such a way that (−1, −1) T and (−1, −2) T are normal vectors of X.For example, choose X as the cone in R 2 spanned by (−1, 1) T and (2, −1) T .We obtain , it can be verified (for example, using a computer calculation) that ν (1) and ν (2) are indeed sublinear circuits, and they are the only ones having a negative component ν β up to scaling by a positive factor.

Necessary and sufficient conditions
In this section, we obtain some criteria for elements ν ∈ β∈A N β to be X-circuits of some fixed set X.These criteria only involve the supports rather than the exact values of the coefficients.
X-circuits of polyhedral cones X.As discussed after Proposition 2.2, in the case of polyhedral cones X we always have σ X (−Aλ) = 0 whenever this value is finite.Since we will reduce the determination of the sublinear circuits Λ X (A) for a cone X in some prominent cases to the classical affine circuits Λ R n (A) (which of course is also a case of a polyhedral cone), we first look at an example for the latter case.
In the following, we examine sublinear circuits for various sets X ⊂ R n (for some n ∈ N) and support sets of the form A = {(i, j) : 1 ≤ i, j ≤ k}, k ∈ N. In these situations, we can write a sublinear circuit ν as a matrix M (ν) ∈ R k×k such that M (ν) i,j = ν (i,j) for all (i, j) ∈ A. Example 4.4.For X = R 2 and support A = {(i, j) : 1 ≤ i, j ≤ 3}, there are 16 sublinear circuits (up to multiples).Namely, there are 8 sublinear circuits with support size 3 (all of them have non-zero entries 1, −2, 1; they appear in the three rows, the three columns and the two diagonals of the 3 × 3-matrix).Moreover, there are the following 8 sublinear circuits of support size 4. Here, the upper left entry of the matrices refers to the support point (1, 1): as well as the 90-degree, 180-degree and 270-degree rotations about the (2, 2)-element of these matrices.As 0 ∈ int R 2 and rec(R 2 ) * = {0}, this reflects in particular the statements of Theorem 4.1 and Lemma 4.2.
Next we consider the sublinear circuits of the non-negative orthant R n + .For a non-empty subset S ⊂ [n] and a support point α ∈ A ⊂ R n , we write α S for the projection of α onto the components of S, i.e., α S := (α s ) s∈S .We also set A S := {α S : α ∈ A} and for a matrix M with n rows, we set M S as the submatrix of M defined by the rows with indices in S, which in particular yields M S λ = (Mλ) S .
; that is, the entry "1" appears in "lower right" quadrant of the entry "−1".(2) 16 normalized sublinear circuits in which the entries 1  2 , −1, 1 2 appear in columns 1,2,3, respectively, such that the entry −1 appears above the line through the two entries 1  2 .(3) 16 normalized sublinear circuits in which the entries 1  2 appear in rows 1,2,3, respectively, such that the −1 appears left to the line containing the two entries 1  2 .(4) 8 R 2 -circuits of cardinality 4, which are the normalized versions of the ones from Example 4.4.Since the diagonal and the anti-diagonal are counted both in cases ( 2) and (3), we have to subtract 2, which gives 27 + 16 + 16 + 8 − 2 = 65.The following table shows in row i and column j the number of sublinear circuits with ν − = {(i, j)}.
1 2 3 1 8 14 2 2 14 21 2 3 2 2 0 Exemplarily, for the case ν − = {(1, 2)}, there are five circuits of type (1) as well as the following nine (in the subsequent list not normalized) sublinear circuits ν with ν − = {(1, 2)}, i.e., the component with index (1, 2) is the negative component.As before, the upper left entry of the matrices refer to the support point (1, 1): The following theorem characterizes the connection between the X-circuits and the R ncircuits for more general polyhedral cones X. Theorem 4.8.Let X = pos{v (1) , . . ., v (k) } be an n-dimensional polyhedral cone spanned by the vectors v (1) , . . ., v (k) , where k ≥ n.Then Proof.Fix β ∈ A and denote by W the k × n-matrix whose rows are the transposed vectors (v (1) ) T , . . ., (v (k) ) T .Hence, X * = {x ∈ R n : W x ≥ 0}.The set Λ X (A, β) is the set of normalized vectors spanning the extreme rays of the cone and the set Λ n R (A, β) is the set of normalized vectors spanning the extreme rays of the cone Since the matrix W has rank n, the linear mapping x → W x is injective, and thus its kernel is {0}.Hence, K R n = {ν ∈ N β : W Aν = 0}.The cone K R n is contained in the cone K X .As a consequence, if λ ∈ N β is not contained in the right hand side of (4.2), it is not contained in the left hand side.Conversely, let λ ∈ N β be contained in the right hand side of (4.2).Then Aλ = 0 and W Aλ = 0. Assume there exists a decomposition into a convex combination λ = θ 1 λ (1) + θ 2 λ (2)  with W Aλ (1) = 0. Since W Aλ = 0 and W Aλ (1) ≥ 0, at least one component of W A(λ − θ 1 λ (1) ) = W Aθ 2 λ (2) is smaller than zero.This is a contradiction.Hence, λ is contained in the left hand side of (4.2).

The n-dimensional cube X = [−1, 1] n
We discuss the sublinear circuits of the n-dimensional cube [−1, 1] n , which is a prominent case of a compact polyhedron.Throughout the section, we assume X = [−1, 1] n for some fixed n ∈ N and A ⊂ R n non-empty and finite.We can already apply some of the former statements to gain knowledge of the structure of X-circuits.For example, as rec(X) * = R n = − rec(X) * , Lemma 4.2 implies that every element supported on exactly two points is an X-circuit.Hence, we examine the structure of those X-circuits λ ∈ Λ X (A) that have more than two support points.We begin with a necessary criterion.α j ≤ β j for all α ∈ λ + or α j ≥ β j for all α ∈ λ + , (5.1) Note that the precondition expresses that there exists a vertex v of [−1, 1] n such that for all α ∈ λ + , the maximal face of the function x → (β − α) T x contains v.
We provide a slightly more general version of Lemma 5.1, whose proof is analogous.
Then λ is not an X-circuit of A.
1 2 3 1 8 17 8 2 17 32 17 3 8 17 8 The subsequent list gives the 17 (not necessarily normalized) X-circuits ν with ν − = {(1, 2)}, i.e., the component with index (1, 2) is the negative component.As before, the upper left entry of the matrices refer to the support point (1, 1): The case k = 4.In the case A = {(i, j) : 1 ≤ i, j ≤ 4}, a computer calculation shows that there are 980 normalized X-circuits, which come in the following classes with regard to λ − : 1 Note that in this case, the criteria of this and the previous section are not sufficient to determine the set of sublinear circuits solely from these criteria.

Reducibility and extremality
By Proposition 2.8, the reduced sublinear circuits provide an irredundant decomposition of conditional SAGE cones.In this section, we discuss some criteria and key examples for reduced sublinear circuits.As an application of the criteria, we will determine the extremals of the [−1, 1]-SAGE cone in Theorem 6.6.
For the classical case of affine circuits supported on a finite set A, the following exact characterization in terms of the support is known.For example, with regard to the two matrices in (4.1) of Example 4.4, the left one is not reduced, but the right one is.The following theorem gives a generalization for the necessary direction of Proposition 6.1 to the constrained situation, where X is a non-empty, convex set in R n .Theorem 6.2.Let λ ∈ Λ X (A, β).If there exists β ′ ∈ A \ supp λ and some normalized λ ′ ∈ N β ′ where (λ ′ ) + ⊂ supp(λ) and Aλ ′ = γAλ for some γ ≥ 0, then λ is not reduced.
Before providing the proof within this section, we discuss its consequences.
Proof.The first statement follows by applying Theorem 6.2 with β ′ ∈ conv(supp λ) \ supp λ, (λ ′ ) + are the vertices of supp λ and γ = 0.The second one is a direct consequence of Proposition 6.1 the fact that for X = R n all X-circuits λ have the property Aλ = 0.
For every X-circuit e (i+1) − e (i) with i > 1, the X-circuit e (i+1) − e (1) satisfies the precondition of Theorem 6.2 and for every X-circuit e (i−1) − e (i) with i < m, the X-circuit e (i−1) − e (m) satisfies the precondition of Theorem 6.2.Hence, all those X-circuits are not reduced.We see that for all i ∈ [m], there is precisely one normalized X-circuit λ that appears in the listed set of possible reduced X-circuits.As rec(X) * = R, there exists at least one X-AGE signomial where the i-th coefficient is negative, hence C X (A, α i ) = ∅ for all i ∈ [m].As C X (A, α i ) is the union of several λ-witnessed X-AGE cones and those cones can be solely represented by reduced X-circuits (compare [17], Sections 4 and 5), for every i ∈ [m] there exists at least one reduced X-circuit in C X (A, α i ).With this, the statement follows.
As a consequence of the results in this section, we can give an exact characterization of the extreme rays of the [−1, 1]-SAGE cone.Theorem 6.6.Let X = [−1, 1] and A = {α 1 , . . ., α m } be sorted ascendingly, where m ≥ 3. The extremal rays of C X (A) are the following: ( , where We first deal with the atomic extreme rays, that is, extreme rays which are supported on a single element.These extreme rays are not captured by the X-circuit view.Lemma 6.7 (Atomic extreme rays of C X (A) for compact sets X).Let X ⊂ R n be a compact set and A ⊂ R n finite with |A| ≥ 2.Then, there are no atomic extreme rays of C X (A).
Proof.As in Lemma 4.2, we use invariance of the X-circuits under translation of X and can w.l.o.g.assume 0 ∈ X.Let α = β ∈ A arbitrary.Assume that f = c α exp(α T x) with c α > 0 is extremal.We observe that λ ∈ N β with λ α = 1 = −λ β is an X-circuit inducing the ray where s ≥ 0 is finite and such that σ X (−Aλ)) = s.Hence, the X-AGE signomials sum to f , contradicting the extremality of f .
Proof of Theorem 6.6.Let A = {α 1 , . . ., α m } be sorted ascendingly.By Lemma 6.7, there are no atomic extreme rays, and by Theorem 6.5 and Proposition 2.8, all the extreme rays are supported on two or three elements.We start by considering the 2-term case.By Lemma 6.5, the only candidates for the extreme rays are the ones given in the cases (1) and (2).Since these cases are symmetric, it suffices to consider case (1), i.e., f (x) = exp(α 2 x) − exp(α 1 − α 2 ) exp(α 1 x).Any conic combination of 3-term AGE functions and of functions of case (2) has a lowest-exponent term with positive coefficient.Hence, f cannot be written as a convex combination of 3-term AGE functions and of functions of case (2).Thus, f indeed is extremal.Now consider the 3-term case.By Lemma 6.5, the only candidates for extreme rays are of the form f (x) = c i−1 exp(α i−1 x) + c i exp(α i x) + c i+1 exp(α i+1 x) with c i−1 > 0, c i+1 > 0 and c i < 0. The proof in [17,Theorem 6.1] shows that f must have a zero in [−1, 1] and that the location x * of the zero is where λ i−1 and λ i+1 are defined as in case (3) of the theorem.This gives the defining condition for c i as well as the inequality conditions in case (3).Any decomposition of f cannot involve a 2-term AGE function.For x * ∈ (−1, 1), this follows from the strict positivity of the 2-term AGE functions of type (1) and (2).For the boundary situations x * ∈ {−1, 1}, we can additionally use the derivative condition f ′ (x * ) = 0 to exclude the 2-term AGE functions.
It remains to show that the 3-term AGE function f cannot be decomposed in terms of 3-term AGE functions.However, since f has a zero in [−1, 1] and thus in R, it induces an extremal ray of the cone C R (A) and cannot be decomposed using only 3-term AGE functions by [12,Proposition 4.4].
Example 6.8.The reduced sublinear circuits for the cube [−1, 1] 2 .We consider again the support A = {(i, j) : 1 ≤ i, j ≤ k} for some k ∈ N. In the case k = 3, there are 24 normalized reduced X-circuits, which come in the following classes: (1) 12 sublinear circuits with entries 1, −1, namely, as well as the 90-degree, 180-degree and 270-degree rotation of thesis matrix.Note that, when starting from the set of all sublinear circuits λ for [−1, 1] 2 , Theorem 6.2 is applicable to rule out that λ is reduced in a number of cases.For example, the matrices represent sublinear circuits λ and λ ′ with Aλ = (−1/2, 0) T and Aλ ′ = (−3/2, 0) T , to which Theorem 6.2 can be applied in order to show that λ is not reduced.Also note that all reduced R 2 -circuits for the support set A are also reduced [−1, 1] 2circuits.Namely, since for all other [−1, 1] 2 -circuits λ, we have σ X (−Aλ) = 0, those circuits cannot be used to decompose an R 2 -circuit (which has σ X (−Aλ) = 0).In the case k = 4 with 16 support points, a computer calculation shows that there are 72 reduced sublinear circuits.

Conclusion and outlook
We have studied the connection of sublinear circuits and their supports and the sublinear circuits for polyhedral sets X.Since for polyhedral sets, the number of X-circuits is finite, this allows to apply polyhedral and combinatorial techniques.In particular the X-SAGE cones can be decomposed into a finite number of power cones, which arise from the reduced sublinear circuits.
For non-polyhedral sets X, in general the number of X-circuits is not finite anymore.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.
(a) 8 with entry −1 in a corner and entry +1 beside or below the corner, (b) 4 with entry −1 in a non-corner boundary entry and entry +1 in the central, interior entry, (2) 8 sublinear circuits, where the sequence1  2 , −1, 1 2 appears in a row (3 possibilities), in a column (3 possibilities) or on the diagonal or the antidiagonal, (3) 4 sublinear circuits supported on 4 elements, namely