As a first step towards the proof of the tropical Helton–Nie conjecture, we study the case of real tropical cones. We will show that semilinear real tropical cones are characterized as sublevel sets of semilinear, monotone, homogeneous operators. The next definitions and lemmas introduce these class of cones and operators and their basic properties.
Definition 5.23. We say that a set X ⊂Rn is a real tropical cone if for every x, y ∈ X and everyλ, µ∈Rwe have (λ⊙x)⊕(µ⊙y)∈X.
Remark 5.24. We point out that a real tropical cone is nothing but the main stratum of a tropical cone as defined in Section 5.1. Indeed, if Y is a tropical cone, then Y ∩Rn is a real tropical cone, whereas if X is a real tropical cone, thenX∪ {−∞}is a tropical cone.
Definition 5.25. We say that a function F:Rn →Rm is piecewise affine if there exists a set of full-dimensional polyhedra W(1), . . . ,W(p) ⊂ Rn satisfying ∪ps=1W(s) = Rn and such that the restriction ofF toW(s) is affine, i.e.,F|W(s)(x) =A(s)x+b(s)for some matrix A(s)∈Rm×n and vectorb(s) ∈Rm. We shall say that the family (W(s), A(s), b(s))s is apiecewise description of the function F.
Remark 5.26. We point out that our definition implies that piecewise affine functions are con-tinuous (because the polyhedraW(1), . . . ,W(p) are closed).
The following minimax representation result was proved by Ovchinnikov [Ovc02]. In the sequel, we use the notationF(x) = (F1(x), . . . , Fm(x)).
Theorem 5.27 ([Ovc02]). Suppose that the function F:Rn →Rm is piecewise affine, and let (W(s), A(s), b(s))s∈[p] be a piecewise description of F. Then, for every k ∈ [m] there exists a number 2p ⩾Mk⩾1 and a family {Ski}i∈[Mk] of subsets of[p]such that for allx∈Rn we have
Fk(x) = min
i∈[Mk]max
s∈Ski
(A(s)k x+b(s)k ).
Definition 5.28. We say that a function F: Rn → Rm is semilinear if its graph {(x, y) ∈ Rn×m:y=F(x)}is a semilinear set.
The next lemma shows that continuous semilinear functions are piecewise affine.
5.2. Real tropical cones as sublevel sets of dynamic programming operators 89 Lemma 5.29. Suppose that the continuous function F: Rn → Rm is semilinear. Then, it is piecewise affine. Moreover, it admits a piecewise description (W(s), A(s), b(s))s∈[p] such that the polyhedra W(s) are semilinear, and the matricesA(s) are rational.
Proof. SinceFis continuous and semilinear, the graph ofF is a closed semilinear set. Therefore, by Lemma 3.12, it is a finite union of semilinear polyhedra. Let {(x, y) :Bx+Cy⩾d}, where B ∈ Qp×n, C ∈Qp×m, d ∈Rp be one of these polyhedra. If we fix x, then, by the definition of a graph, the polyhedron consisting of all y such that Cy ⩾ d−Bx reduces to a point y.
Thus, ifJ ⊂[p]denotes the maximal set such that CJy=dJ−BJx, then this system of affine equalities has a unique solution. Hence, there exists a setI ⊂J,|I|=m such that the matrix CI ∈Qm×m is invertible and satisfiesy =CI−1(dI−BIx) =CI−1dI−CI−1BIx. In other words, the graph ofF is a finite union of polyhedra of the form
W ={(x, y) :Bx+Cy⩾d, y=CI−1dI−CI−1BIx},
whereCI is an invertible submatrix of C. As a result, ifπ:Rn+m →Rndenotes the projection on the firstncoordinates, andx∈π(W)is any point, then we haveF(x) =CI−1dI−CI−1BIx. By eliminating the polyhedraπ(W)that are not full dimensional,1we obtain a piecewise description of F satisfying the expected requirements.
Definition 5.30. We say that a selfmapF:Rn→Rn is monotone ifF(x)⩽F(y) as soon as x⩽y, where⩽denotes the coordinatewise partial order overRn. Such a function is said to be (additively) homogeneous if F(λ+x) =λ+F(x) for all λ∈ R and x ∈Rn. Here, ifz ∈ Rn, thenλ+z stands for the vector with entriesλ+zk.
The following observation is well known [CT80].
Lemma 5.31. Every monotone homogeneous operator is nonexpansive in the supremum norm.
Proof. Observe that x ⩽ ∥x−y∥∞+y. Therefore, we get F(x) ⩽ F(∥x−y∥∞+y) = ∥x− y∥∞+F(y). Analogously, F(y)⩽∥x−y∥∞+F(x) and ∥F(x)−F(y)∥∞⩽∥x−y∥∞.
Kolokoltsov showed that every monotone homogeneous operatorF has a minimax represen-tation as a dynamic programming operator of a zero-sum game [Kol92]. WhenF is semilinear, the following lemma and its corollary show that we have a finite representation of the same nature.
Lemma 5.32. Suppose that F:Rn →Rn is piecewise affine, monotone, homogeneous and let (W(s), A(s), b(s))s∈[p] be any piecewise description of F. Then, every matrix A(s) is stochastic.
Proof. Take anyx∈int(Ws). Letybe the sum of the columns ofA(s). SinceF is homogeneous, for anyρ >0 small enough we have F(ρ+x) =A(s)x+b(s)+ρy=ρ+F(x). In other words, the sum of every line ofA(s) is equal to1. Letϵkdenote thekth vector of standard basis inRn. SinceF is monotone, forρ >0small enough we haveF(x+ρϵk) =A(s)x+b(s)+ρA(s)ϵk⩾F(x).
In other words, the matrixA(s)has nonnegative entries in itskth column. Sincekwas arbitrary, A(s) is stochastic.
1The fact that a projection of a polyhedron is also a polyhedron follows from the Fourier–Motzkin elimination, see [Sch87, Section 12.2].
90 Chapter 5. Tropical analogue of the Helton–Nie conjecture
Figure 5.2: A real tropical cone from Example 5.35 (for x3 = 0).
Corollary 5.33. If F:Rn → Rn is semilinear, monotone, and homogeneous, then it can be written in the form
∀k, Fk(x) = min
i∈[Mk]max
s∈Ski
(A(s)k x+b(s)k ), (5.1) where A(1), . . . , A(p) ∈ Qn×n is a sequence of stochastic matrices, b(s) ∈ Rn for all s ∈ [n], Mk⩾1 for all k∈[n], and Ski is a subset of[p] for everyk∈[n]and i∈[Mk].
Proof. Lemma 5.31 shows thatF is continuous. Let(W(s), A(s), b(s))s∈[p]a piecewise description of F as provided by Lemma 5.29. In particular, every matrix A(s) is rational. Furthermore, it is stochastic by Lemma 5.32. Therefore, the claim follows from Theorem 5.27.
We now characterize the class of closed, semilinear, real tropical cones. In this context,
“closed” means “closed in the standard topology ofRn.”
Proposition 5.34. A set S⊂Rn is a closed, semilinear, real tropical cone if and only if there exists a semilinear, monotone, homogeneous operatorF:Rn→Rnsuch thatS={x∈Rn:x⩽ F(x)}.
Proof. To prove the first implication, we consider two cases. If S is empty, then we take F(x) = x−(1, . . . ,1). Otherwise, we define F by Fk(x) := sup{yk:y ∈ S, y ⩽ x} for all k ∈ [n]. We claim that every supremum is attained. Indeed, the set {y ∈ S:y ⩽ x} is nonempty (take an arbitrary z ∈S, and consider y := λ+z for λ∈R small enough), closed, and bounded by x. Hence Fk(x) := max{yk:y ∈ S, y ⩽ x} for all k ∈ [n]. Observe that, sinceS is semilinear, the graph of the operatorF is definable in the language Log. Therefore, F is semilinear by Lemma 2.103. Besides, F is obviously monotone. It is also homogeneous because if y ∈S, then λ+y = (λ⊙y)⊕(λ⊙y) ∈ S for all λ∈ R. Moreover, the inclusion S⊂ {x∈Rn:x⩽F(x)} is straightforward. To prove the inverse inclusion, letx∈Rn be such that x ⩽F(x) and lety(k) ∈S be a point attaining the maximum in Fk(x). Then, the point y:=y(1)⊕ · · · ⊕y(n) is an element of Ssmaller than or equal tox. HenceF(x) =y⩽x. Since we supposed thatx⩽F(x), we have x=y∈S.
Conversely, fix a semilinear, monotone, homogeneous operatorF and take the setS={x∈ Rn: x ⩽ F(x)}. This set is is definable in Log and hence semilinear. Moreover, S is closed because F is continuous. To prove that this is a real tropical cone, fix a pair λ, µ ∈ R and x, y∈S. SinceF is monotone and homogeneous, we haveF(max{λ+x, µ+y})⩾F(λ+x) = λ+F(x)⩾λ+xand similarlyF(max{λ+x, µ+y})⩾µ+y. Hencemax{λ+x, µ+y} ∈S.
5.3. Description of real tropical cones by directed graphs 91