The $\mathcal{S}$-cone and a primal-dual view on second-order representability

The $\mathcal{S}$-cone provides a common framework for cones of polynomials or exponential sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmetic-geometric exponentials (SAGE). In this paper, we study the $\mathcal{S}$-cone and its dual from the viewpoint of second-order representability. Extending results of Averkov and of Wang and Magron on the primal SONC cone, we provide explicit generalized second-order descriptions for rational $\mathcal{S}$-cones and theirs duals.


Introduction
The question to characterize and to decide whether a polynomial or an exponential sum is non-negative occurs in many branches of mathematics and application areas. In the development of real algebraic geometry, the connection between the cone of nonnegative polynomials and the cone of sums of squares of polynomials plays a prominent role (see, for example, [3,13,17]). If a polynomial can be written as a sum of squares of polynomials, this provides a certificate for the non-negativity of the polynomial. Since the beginning of the current millennium, non-negativity certificates of polynomials have also seen much interest from the computational point of view and have strongly advanced the rich connections between real and convex algebraic geometry as well as polynomial optimization (see, for example, [11,12]).
Within the research activities on non-negativity certificates in the last years, the cones of sums of arithmetic-geometric exponentials (SAGE, introduced by Chandrasekaran and Shah [4]) and sums of non-negative circuit polynomials (SONC, introduced by Iliman and de Wolff [8]) have received a lot of attention (see, e.g., [1,5,7,14,15,18]). These cones provide non-negativity certificates based on the arithmetic-geometric inequality and are particularly useful in the context of sparse polynomials.
In [10], the authors of the current paper and Katthän have introduced a common generalization, called the S-cone, which facilitates to study the SAGE cone and the SONC cone within a uniform generalized setting. Formally, for two finite disjoint sets ∅ = A ⊆ with real coefficients c α , α ∈ A ∪ B. Our precondition A ∩ B = ∅ is a slight restriction to the setup in [10], in order to enable a little more convenient notation. One motivation for the class of functions (1.1) is that it allows to capture non-negativity of polynomials on R n and non-negativity of polynomials on the non-negative orthant R n + within a uniform setting. Moreover, global non-negativity of the summand α∈A c α |x| α is equivalent to global non-negativity of the exponential sum y → α∈A c α exp(α T y). Definition 1. 1. A function f of the form (1.1) is called an even AG function if for at most one α ∈ A, c α is negative and for all β ∈ B, c β is zero; and it is called an odd AG function if for all α ∈ A, c α is non-negative and for at most one β ∈ B, c β is nonzero.
f is called an AG function (arithmetic-geometric mean function), if f is an even AG function or an odd AG function. where cone denotes the conic (or positive) hull.
The SAGE and SONC cones arise as special cases of this cone, see Section 2. Both from the geometric and from the optimization point of view, it is of prominent interest to understand how the different classes of cones relate to each other and whether techniques for different cones can be fruitfully combined. In [9], Karaca, Darivianakis et al. have studied non-negativity certificates based on a combination of the SAGE cone with the cone of sums of squares. Concerning relations between the various cones, Averkov has shown that the SONC cone can be represented as a projection of a spectrahedron [1]. In fact, his proof applies the techniques from [2], which reveals that the SONC cone is even second-order representable. Wang and Magron gave an alternative proof based on binomial squares and A-mediated sets [19].
Here, we take the general view of the S-cone as well as a primal-dual viewpoint. Generalizing the results of Averkov and of Wang and Magron, we show that rational S-cones and their duals are second-order representable and provide explicit and direct descriptions. Our proof combines the techniques for the second-order cones techniques from [2] with the concepts and the duality theory of the S-cone from [10]. Our derivation is different from the approach of Wang and Magron, and it does not need binomial squares or A-mediated sets. Moreover, our second-order representation prevents the consideration of redundant circuits by using a characterization of the extreme rays of the S-cone from [10].
Beyond the specific representability result, the goal of the paper is to offer further insights into the use of the framework of the S-cone as a generalization of SONC and SAGE.

Preliminaries
Throughout the text, we use the notations N = {0, 1, 2, 3, . . .} and R + = {x ∈ R : x ≥ 0}. For a finite subset A ⊆ R n , denote by R A the set of |A|-dimensional vectors whose components are indexed by the set A. Moreover, we write and if one component of x is zero and the corresponding exponent is negative, then we set |x| α = ∞.
2.1. The S-cone, SAGE and SONC. We explain that the S-cone generalizes the SAGE cone and the SONC cone and collect some basic properties of the three cones. The SAGE cone. Let A be a non-empty, finite set. An exponential sum supported on A is a function of the form can be identified with the space of exponential sums supported on A by means of the substitution |x i | = exp(y i ). For finite A ⊆ R n , A A and β ∈ A \ A , the SAGE cone C SAGE (A) is defined as [4]). We observe that the S-cone C S (A, ∅) can be identified with C SAGE (A) using the substitution (2.1). C SAGE (A) is a closed convex cone in R A . The membership problem for this convex cone can be formulated as a relative entropy program ( [14], see also Proposition 2.2 below). The SONC cone. Here, let the non-empty finite set A be contained in N n . Let where relint denotes the relative interior of a set. For singleton sets A = {α}, the sets (A, β) are formally of the form ({α}, α). By convention, we write these circuits simply as (α), and with this convention, the set {(α) : α ∈ (2N) n } ∩ A is contained in I(A). defines the cone of SONC polynomials with support A (see [1,8]).
The cone C SONC (A) is a closed convex cone, and it can be recognized as a special case of a rational S-cone by observing (see [10]). Using the results from [14], membership in the SONC cone can also be formulated in terms of a relative entropy program.
The S-cone. The S-cone from Definition 1 offers a uniform setting for the SAGE and the SONC cones. We collect some further properties of the S-cone. For a non-empty finite set A ⊆ R n and β ∈ N n \ ((2N) n ∪ A) let be the cone of non-negative odd AG functions supported on (A, β), and similarly for β ∈ R n \ A let be the cone of non-negative even AG functions supported on (A, β). By definition, Note that non-negative even AG functions correspond exactly to the AGE functions (arithmetic-geometric exponentials) in [4] and [14].
The following alternative representation allows to express the S-cone in terms of the SAGE cone. Here, |d| denotes the absolute value of the vector d ∈ R B , taken componentwise.
Proposition 2.1. [10] Let ∅ = A ⊆ R n , B ⊆ N n \ (2N) n be finite and disjoint. Then, and immediately observe C S (A) = α∈A P even A\{α},α . Hence, for our purpose it suffices to study the cone P even A,β of even AG functions and use the results of this cone for the odd case in Section 4.
Using the relative entropy function and the circuit number, the cones P even A,β and P odd A,β can be characterized in terms of convex optimization problems. For a finite set ∅ = A ⊆ R n , denote by D : the relative entropy function. D can also be extended to R A + × R A + → R ∪ {∞} via the conventions 0 · ln 0 y = 0 for y ≥ 0 and y · ln y 0 = ∞ for y > 0. Non-negativity of an (even or odd) AG function f with coefficients c α and c β can be characterized through the product α∈A (c α /λ α ) λα and c β (see [10,Theorem 2.7]). For an affinely independent ground set A, this product is called the circuit number of f (see [8]). In particular, for an even AG function, this non-negativity characterization in terms of the circuit number is given by The following characterization of P even A,β and P odd A,β in terms of the relative entropy function and in terms of the circuit number is a direct consequence of Theorem 2.7 of [10].
(1) If f is an even AG function, then (2) If f is an odd AG function, then If A is a set of affinely independent vectors and β ∈ relint A, then λ is unique. We call the corresponding AG function a circuit function, the tuple (A, β) the circuit and identify the unique λ with the above declared characteristics λ = λ(A, β).

Duality theory.
Studying the duality theory has been initiated in [4] (for SAGE), [6] (for SONC) and [10] (for the S-cone). We can identify the dual space of with coefficients c ∈ R A and an element v ∈ R A , we consider the natural duality pairing Using this notation, the dual cone (C S (A)) * is defined as The following statement expresses the dual S-cone in terms of the dual SAGE cone.
Proof. We use (2.5), which provides a characterization for the primal cone C S (A, B) in terms of an existential quantification. Consider its lifted cone The dual cone of the right-hand cone in (2.10) is the set where e (β) denotes the unit vector with respect to β ∈ B. As intersection and Minkowski sum are dual operations, we obtain Identifying the S-cone with its coefficients, we can express C S (A, B) * in terms of the lifted cone C S (A, B) by Convexity then implies the second characterization (2.9).
Hence, as in the primal case, it suffices to study even AG functions in the dual situation. We will make use of a representation of the dual of the S-cone from [10]. For this, observe that similar to the SONC case in (2.2), one can also consider circuits in the case of the SAGE cone. In slight variation of (2.2), for a finite set ∅ = A ⊆ R n , the set of circuits supported on A is the set Two examples of circuits are the pairs (A, β) with A = {0, 6} and β = {2} (see Figure  1) and (A , β ) with A = {(0, 0) T , (4, 2) T , (2, 4) T } and β = (1, 1) T (see Figure 2). Thereby, the dual S-cone C S (A) can be represented as follows.
can be represented as a projection of the feasible set of a second-order program.
For a symmetric 2 × 2-matrix, positive semidefiniteness can be formulated as a secondorder condition. Let S n + be the subset of symmetric n × n-matrices which are positive semidefinite. By [1], there exists some m ∈ N so that the cone of SONC polynomials C SONC (A) supported on A can be written as the projection of the spectrahedron (S 2 + ) m ∩ H for some affine space H.

A second-order representation for the cone of non-negative AG functions and its dual
In order to provide a second-order representation for the S-cone and its dual, the main task is to capture the cone of non-negative AG functions and its dual. For a comprehensive collection of techniques for handling second-order cones, we refer to [2].
Throughout the section, let (A, β) be a fixed circuit and rational barycentric coordinates λ ∈ R A + , which represent β as a convex combination of A. That is, β = α∈A λ α α and α∈A λ α = 1. Let p ∈ N denote the smallest common denominator of the fractions λ α for α ∈ A, i.e., λ α = pα p with p α ∈ N for all α ∈ A and p is minimal.
With the given circuit (A, β) ∈ I(A), we associate a set of dual circuit variables where k ∈ [ log 2 (p) −1] and i ∈ [2 log 2 (p) −k ]. The collection of these log 2 (p) −1 k=1 2 log 2 (p) −k = 2 log 2 (p) − 2 variables is denoted as y A,β or shortly as y. Further, denote the restriction of a vector v ∈ R A to the components of A ⊆ A by v |A .
is a block diagonal matrix consisting of the blocks the singleton block (v β ), as well as 2 log 2 (p) −1 blocks of the form u y 1,l y 1,l w for l ∈ [2 log 2 (p) −1 ], (3.4) where in each of these blocks u and w represent a variable of the set {v α : α ∈ A} ∪ {v β } such that altogether each v α appears p α times and v β appears 2 log 2 (p) − p times.
In this definition, the exact order of appearances of the variables in {v α : α ∈ A}∪{v β } is not uniquely determined. However, since this order of appearances will not matter, we will speak of the dual circuit matrix.
Remark 3.2. Each block of the type (3.4) contains two (not necessarily identical) variables from the set {v α : α ∈ A} ∪ {v β }. Since α∈A λ α = 1, we have α∈A p α = p and hence the total number of occurrences of variables from the set {v α : α ∈ A} ∪ {v β } in the blocks of type (3.4) is which is twice the number of blocks of type (3.4).
Note that every y k,i only serves as an auxiliary variable to make the non-linear constraints ln(v β ) ≤ α∈A λ α ln(v α ) of the dual S-cone description from Proposition 2.4 linear. In the end, we will only multiply those constraints to obtain the original ones. In particular, factors v β serve to cover cases where p is not a power of 2. For the purpose of the second-order descriptions, it does not matter in which order the variables appear in the blocks (3.4), because only the product of these blocks will be considered.
The goal of this subsection is to show the following characterization of the cone of non-negative even AG functions P even A,β supported on the circuit (A, β). Here, positive semidefiniteness of a symmetric matrix is denoted by 0.

Theorem 3.3. The dual cone (P even
A,β ) * of the cone of non-negative even AG functions P even A,β supported on the circuit (A, β) ∈ I(A) is the projection of the spectrahedron Here, the second-order representability follows immediately from the representation (3.5) in connection with Lemma 2.5. Let us consider an example for the theorem. In [1], Averkov considered the size of the blocks in the SDP-representation of SONCpolynomials but does not give a number or bound on the number of blocks. Here, for the S-cone, we provide a bound on the number of inequalities of a second-order representation, which also gives a bound on the number of 2 × 2-blocks in a semidefinite representation. The bound depends on the smallest common denominator of the barycentric coordinates representing the inner exponent of a circuit as a convex combination of the outer ones.  for u, w ∈ {v α : α ∈ A} ∪ {v β }, such that v α appears p α times for every α ∈ A and v β appears 2 log 2 (p) − p times.
The next lemma prepares one inclusion of Theorem 3.3.
Finally, we can conclude the proof of Theorem 3.3.
Proof of Theorem 3.3. Let p be defined as in Definition 3.1 and λ ∈ R A denote the barycentric coordinates representing β as a convex combination of A, i.e., λ α = pα p with p α ∈ N for all α ∈ A. By (2.3) and Proposition 2.4, we have Applying Lemmas 3.7 and 3.8, we obtain that C * A,β (x, v β ) 0 if and only if v ∈ P * A,β . Our derivation of the second-order representation of the dual cone (P even A,β ) * also suggests a simple way to derive a second-order cone representation of the primal cone P even A,β . For the dual cone, Proposition 2.4 gives -besides non-negativity-constraints on v α for α ∈ A and on v β -the condition ln(v β ) ≤ α∈A λ α ln(v α ) for every circuit (A, β) ∈ I(A). Those conditions can -as done in the previous proof -be stated as The conditions for the primal cone can be reformulated similarly. Namely, by (2.6), an even circuit function f with coefficient vector c is non-negative if and only if −c β ≤ α∈A (c α /λ α ) λα , which we write as This motivates to carry over the definition of the dual circuit matrix to the primal case as follows. Since c β may be negative (in contrast to the dual case), we introduce the primal circuit variables, or simply circuit variables, where k ∈ [ log 2 (p) ] and i ∈ [2 log 2 (p) −k ]. As in the dual case, we refer to these 1 + log 2 (p) k=1 2 log 2 (p) −k = 2 log 2 (p) variables as x A,β or shortly as x. Definition 3.9 (Circuit matrix). The circuit matrix C A,β (c |A∪{β} , x β , x) is the block diagonal matrix consisting of the blocks the two singleton blocks as well as 2 log 2 (p) −1 blocks of the form u x 1,l x 1,l w for l ∈ [2 log 2 (p) −1 ], (3.12) where u, w ∈ {c α : α ∈ A} ∪ { α∈A (λ α ) λα x β }, such that c α appears p α times for every α ∈ A and α∈A (λ α ) λα x β appears 2 log 2 (p) − p times.
Note that for a circuit (A, β), the product α∈A (λ α ) λα is always non-zero, because β ∈ relint conv A and A consists of affinely independent vectors.
In contrast to the dual cone, there is no sign constraint on c β in the primal cone. If p is not a power of 2, then x β appears on the main diagonal of (3.12). In our coupling of x β with c β , the constraint x β + c β ≥ 0 results in −c β ≤ x β and thus reflects these sign considerations.
Note that the primal cone consists of circuit functions, whereas in our definition of the dual cone, the elements are coefficient vectors. Therefore, the projection regarded in Theorem 3.3 only delivers the coefficients of the circuit functions rather than the cone itself.
Theorem 3.10. The set of coefficients of the cone P even A,β of non-negative even circuit polynomials supported on the circuit (A, β) coincides with the projection of the spectrahedron on (c |A , c β ). The cone P even A,β is second-order representable. The last equality constraint in (3.13) is redundant and can be omitted. We include it here, because this formulation is needed in Section 4 for the description of the S-cone supported on the full set A.
Proof. First, let (c, x) ∈ P even A,β . The positive semidefiniteness of the 2 × 2-blocks in C A,β (c |A∪{β} , x β , x) imply the inequalities The two 1×1-blocks from (3.11) give the inequalities x log 2 (p) ,1 ≥ α∈A λ λα α x β and x β ≥ −c β . They imply −c β α∈A λ λα α ≤ x β α∈A λ λα α ≤ x log 2 (p) ,1 . Hence, similar to Lemma 3.7, . This is equivalent to which, together with the considerations before the chain of inequalities, yields (−c β ) p ≤ α∈A (c α /λ α ) pα and further c |A∪{β} ∈ P even A,β . For the converse inclusion, we remind the reader that λ α > 0 for all α ∈ A. We set x β := x log 2 (p) ,1 α∈A 1 λα λα and, similar to the proof of Lemma 3.8, define x inductively by x 1,l = √ uw for those u, w which occur in the block with x 1,l , Analogous to that proof, the construction of x gives C A,β (c A∪{β} , x β , x) 0. Second-order representability is then an immediate consequence in view of Lemma 2.5. A given vector (c 0 , c 1 , c 2 ) is contained in P A,B if and only if Similar to Lemma 3.5, we can determine the number of blocks.

A second-order representation of the S-cone and its dual
In Section 3, we obtained second-order representations of the subcones of non-negative even circuit functions and their duals, under the condition that the barycentric coordinates are rational. We now assume that A and B are rational and derive an explicit secondorder representation of the rational S-cone C S (A, B) and its dual. In the primal case, those cones are obtained via projection and Minkowski sum, and in the dual case, they arise from projection and intersection. First we consider the lifted cones for the dual case.
Taking all circuits (A, β) into account would induce a highly redundant representation. To avoid those redundancies, we make use of the following characterization from [10] of the extreme rays of the S-cone.
For finite and disjoint sets ∅ = A, B ⊆ R n , the set of reduced circuits contained in A ∪ B is the set Less formally, this is the set of all circuits with outer exponents in A and inner exponents in B without additional support points contained in the convex hull of A.
Note that for A ⊆ R n and B ⊆ N n \ (2N) n disjoint and finite, the set R(A, A) is exactly the set of even reduced circuits and the set R(A, B) the set of odd reduced circuits. The set R(A, A ∪ B) denotes the set of all reduced circuits (A, β) with A ⊆ A and β ∈ A ∪ B. A circuit function supported on a reduced circuit in R(A, A∪B) has non-negative coefficients corresponding to exponents in A and a possibly negative coefficient corresponding to a single exponent in A ∪ B.
The question whether a circuit is reduced or not depends on the ground set A.  Figure 4).
The following proposition is a direct consequence of Theorem 3.5(d) in [10]. Using this decomposition theorem, we can exclude many circuits from our consideration. Thus, the second-order-program will be much smaller than the one considering all circuits.
In Section 3, we only considered even circuits. To use Lemma 2.1 and obtain the conditions for odd circuits as well, we extend the dual circuit variables for odd circuits to (y β , (y k,i ) k,i ) for k ∈ [2 log 2 (p) − 1] and i ∈ [2 log 2 (p)−k ]. We call them y A,β nevertheless for a fixed circuit (A, β) ∈ R (A, B).
For the dual case, we consider the coordinates which consist of (A,β)∈R(A,A∪B) 2 log 2 (p A,β ) − 1 components, where p A,β denotes the smallest common denominator of the barycentric coordinates λ A,β of the circuit (A, β) representing β as a convex combination of A.
For the primal case, we consider which consist of (A,β)∈R(A,A∪B) 2 log 2 (p A,β ) components. Using Lemma 2.1, we can use our earlier characterizations of P even A,β to obtain the following second-order characterization for P odd A,β . (1) Let f be an odd AG function supported on (A, β) with coefficient vector c. f is non-negative if and only if there exists x ∈ R 2 log 2 (p) such that C A,β (c |A , x β , x) 0 and x β c β c β x β 0. (4.1) (2) A vector v ∈ R A,β is contained in P odd A,β * if and only if there exist y ∈ R 2 log 2 (p) −2 and y β ∈ R such that C * A,β (v |A , y β , y) 0 and Note that, as a consequence of the application of Lemma 2.1, the second argument of C * A,β (v |A , y β , y) is y β now instead of v β that we had in Theorem 3.3. Proof.
(1) The semidefinite condition on the matrix (4.1) is equivalent to x β ≥ 0 and |c β | ≤ x β . Hence, altogether we obtain f ∈ P odd A,β if and only if |c β | ≤ α∈A c α λ λα for barycentric coordinates λ ∈ R A + decomposing β as a convex combination of A. This is exactly Proposition 2.2(b).
(2) If v ∈ (P odd A,β ) * , then, in the notation of Theorem 2.9, there exists some u such that (v, u) ∈ (P even A,β ) * and u ≥ |v β |. In particular, u ≥ 0 is necessary for containment in P even A,β * . The semidefinite constraints (4.2) are equivalent to y β ≥ 0 and the latter inequality u ≥ |v β |, and the constraint C * A,β (v |A , y β , y) 0 is equivalent to (v, y β ) ∈ P even A,β * by Theorem 3.3.
For every odd reduced circuit (A, β) ∈ R(A, B), define the block diagonal matrix C * A,β (v |A∪{β} , y β , y) consisting of the dual circuit matrix C * A,β (v |A∪{β} , y β , y) and (4.1) for the dual cone. Considering all the reduced circuits, these lifting matrices define the lifted cone Here, for every (A, β) ∈ R(A, A), P even A,β is the set from Theorem 3.10. Corollary 4.3.
(1) The dual of the rational S-cone C * S (A, B) is the projection on the coordinates v ∈ R A,B of C * (A, B).  that v α appears (p A,β ) α times for each α ∈ A and y A,β β appears 2 log 2 (p A,β ) − (p A,β ) α times, (4) ||v β || 2 ≤ y A,β β , and for every reduced even circuit (A, β) ∈ R(A, A) the conditions of Theorem 3.3.