Valuation of interior and regions of strict feasibility

To finish this chapter, we consider the problem of characterizing the image by valuation of the interior of a spectrahedron. As previously, let Q⁽¹⁾, . . . ,Q⁽ⁿ⁾ ∈ K^m×m be a sequence of symmetric matrices, denoteQ(x) :=x1Q⁽¹⁾+· · ·+xnQ⁽ⁿ⁾ and letS :={x∈Kⁿ_⩾0:Q(x)≽0} be the associated spectrahedral cone. We want to characterize the set val⁽int(S)⁾. It turns out that the analysis done in the previous section extends to this problem if the matrices Q^(k) are Metzler, but does not solve the case of non-Metzler matrices. To partially handle this case, we study the image by valuation of strictly feasible points of S, and show that our techniques extend naturally to this setting. LetQ^(k):=sval(Q^(k)) for all k∈[n].

First, we suppose that the matrices Q^(k) are Metzler. The following lemma extends the claim of Lemma 4.17.

Lemma 4.41. Suppose that the matrices Q^(k) are Metzler and let T be as in Definition 4.16.

Then cl_R(T ∩Rⁿ)⊂val⁽int(Sⁱⁿ)⁾.

Proof. Ifx∈ T ∩Rⁿ, then the proof of Lemma 4.17 shows thatx∈Sⁱⁿ for any x∈val⁻¹(x)∩ Kⁿ>0. In particular, the ball]t^x1,2t^x1[×· · ·×]t^xn,2t^xn[belongs toSⁱⁿand hencex∈val⁽int(Sⁱⁿ)⁾. The set int(Sⁱⁿ)is semialgebraic by Proposition 2.15 and hence cl_R(T ∩Rⁿ)⊂val⁽int(Sⁱⁿ)⁾by Theorem 3.1.

As a corollary, we characterize val⁽int(Sⁱⁿ)⁾for matrices that satisfy the conditions of The-orem 4.20.

Corollary 4.42. Suppose that the matricesQ^(k)are Metzler and that they satisfy the conditions of Theorem 4.19. Then

val⁽int(Sⁱⁿ)⁾=S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩Rⁿ.

Proof. We trivially have val⁽int(S)⁾ ⊂ val(S ∩Kⁿ>0) ⊂ S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩Rⁿ. Furthermore, the proof of Theorem 4.19 shows that S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) ∩Rⁿ = cl_R(T ∩Rⁿ), and we have cl_R(T ∩Rⁿ)⊂val⁽int(S)⁾ by Lemma 4.41.

One may think that the claim of Corollary 4.42 extends to the case of non-Metzler matrices if we replace the conditions of Theorem 4.19 by the analogous conditions of Theorem 4.28. The following example shows that this is not true.

Example 4.43. Take the matrices Q⁽⁰⁾ :=

[−∞ a

a −∞

]

, Q⁽¹⁾:=

[−∞ b

b −∞

]

, Q⁽²⁾:=

[−∞ ⊖c

⊖c −∞

] ,

wherea, b, c∈R. The setS(Q⁽⁰⁾, Q⁽¹⁾, Q⁽²⁾)fulfills the conditions of Theorem 4.28. Moreover, it is the set of all points(x₀, x₁, x₂)∈T³such thatmax{a+x₀, b+x₁}=c+x₂. Nevertheless, if we

4.4. Valuation of interior and regions of strict feasibility 79 take any matricesQ⁽⁰⁾,Q⁽¹⁾,Q⁽²⁾such thatsval(Q^(k)) =Q^(k), then the associated spectrahedron is a plane,

S ={(x₀,x₁,x₂)∈K³_⩾0:Q⁽⁰⁾₁₂x₀+Q⁽¹⁾₁₂x₁ =Q⁽²⁾₁₂x₂}.

This shows that Corollary 4.42 does not carry over to non-Metzler matrices. Indeed, S has empty interior but S(Q⁽⁰⁾, Q⁽¹⁾, Q⁽²⁾)∩Rⁿ is nonempty.

Let us note, however, that we have the following partial extension of Lemma 4.41 to the case of non-Metzler matrices. (This extension will be useful in Section 7.4 where we relate the problem of deciding the feasibility to the problem of stochastic mean payoff games.)

Lemma 4.44. Let Q⁽¹⁾, . . . , Q⁽ⁿ⁾ be any tropical symmetric matrices. Put Σ := {(i, j) ∈ [m]²: i < j} and ♢ := ∅ and consider the set TΣ,♢ defined as in Lemma 4.26. Then, we have cl_R(TΣ,♢∩Rⁿ)⊂val⁽int(Sⁱⁿ)⁾.

Proof. By our choice of (Σ,♢) and Lemma 4.41, the set cl_R(TΣ,♢ ∩ Rⁿ) is included in the valuation of the interior of the set of pointsx∈Kⁿ_⩾0 defined by the inequalities

∀i,Qii(x)⩾0,

∀i < j,Q_ii(x)Q_jj(x)⩾(m−1)²(Q⁺_ij(x) +Q⁻_ij(x))². This set is included in Sⁱⁿ because (Q⁺_ij(x) +Q⁻_ij(x))²⩾(Q_ij(x))² for allx.

Remark 4.45. Let us point out why the lemma above does not extend to other choices of Σ.

Indeed, if we take a different Σ, fix x ∈ ∩_♢cl_R(TΣ,♢ ∩Rⁿ), and try to repeat the proof of Lemma 4.26, then we can construct a pointz∈Kⁿ_>0 that belongs to the interior of the set

{y∈Kⁿ_⩾0:∀i,Q_ii(y)⩾0∧ ∀(i, j)∈Σ,Q_ii(y)Q_jj(y)⩾(m−1)²(Q_ij(y))²} (4.9) and satisfiesQ_ij(z) = 0for all (i, j)∈Σ^∁. However, this does not imply thatz belongs to the interior of Sⁱⁿ, as shown by Example 4.43. In this example, if we take Σ := ∅, then the set given by (4.9) is equal toK³_⩾0, but Sⁱⁿis a plane (and z is a point that belongs to this plane).

This problem is avoided in cases where the weak inequalities that define (4.9) can be replaced strict inequalities, as discussed in the sequel.

Let us switch our attention to the problem of characterizing the valuation of strictly feasible points of a spectrahedron. To do so, let us recall that a matrix A ∈ K^m_⩾0^×m is called positive definite if it is positive semidefinite and invertible. By the completeness of the theory of real closed fields, this is equivalent to demanding that all principal minors ofAare positive, that the inequality x^⊺Ax>0is true for allx̸= 0 and so on (see, e.g., [Mey00, Section 7.6]). Moreover, let us recall the following definition.

Definition 4.46. We say that S is strictly feasible if there exists a point x ∈Kⁿ>0 such that the matrixQ(x) is positive definite.

Let S⁺⁺⊂Kⁿ>0 be the set of all strictly feasible points ofS, i.e., S⁺⁺:={x∈Kⁿ>0:Q(x) is positive definite}.

It is easy to check that S⁺⁺ is convex. Even more, if S⁺⁺ is nonempty, then it is equal to the interior of S.Therefore, it may seem that studying val(S⁺⁺) is very similar to studying val⁽int(S)⁾. However, there are spectrahedra that cannot be strictly feasible for trivial reasons.

For instance, if there exists i∈[m] such that Q^(k)_ii = 0 for all k, then the set S⁺⁺ is trivially empty (because the matrix Q(x) has a zero entry on its diagonal). Therefore, it is natural to make the following assumption.

80 Chapter 4. Tropical spectrahedra Assumption B. For every i∈[m], there existsk∈[n]such thatQ^(k)_ii ̸= 0.

Let us point out that the matrices given in Example 4.43 do not satisfy Assumption B. In what follows, we show that the behavior of Example 4.43 cannot be reproduced by matrices that satisfy Assumption B. To start, we point out that the notion of diagonal dominance extends to the case of positive definite matrices.

Lemma 4.47. Suppose that a symmetric matrix A ∈ R^m×m is strictly diagonally dominant, i.e., satisfies the inequality Aii>^∑_j̸=i|Aij|for all i∈[m]. Then A is positive definite.

Proof. Matrix A is positive semidefinite by Lemma 4.9. It is nonsingular by [Mey00, Exam-ple 4.3.3].

As a corollary, we get the following results that can be proven as in Lemma 4.8 and Corol-lary 4.11.

Lemma 4.48. Let A∈K^m×m be a symmetric matrix. Suppose that A has positive entries on its diagonal and that the inequality A_iiA_jj > (m−1)²A²_ij holds for all pairs (i, j) such that i̸=j. Then A is positive definite.

Corollary 4.49. Let A ∈ T^m_±^×m be a symmetric matrix such that Aii ∈ T+ for all i and A_ii⊙A_jj > A^⊙_ij² for alli < j. Let A∈K^m×m be any symmetric matrix such thatsval(A) =A.

Then A fulfills the conditions of Lemma 4.48. (In particular, it is positive definite.)

We will now give the analogues of Lemmas 4.17 and 4.26 for the regions of strict feasibility.

As usual, we first consider the case of Metzler matrices.

Lemma 4.50. Suppose that the matricesQ⁽¹⁾, . . . , Q⁽ⁿ⁾ are Metzler and satisfy Assumption B.

Let T be as in Definition 4.16. If x∈ T ∩Rⁿ andx∈val⁻¹(x)∩Kⁿ_>0 is any lift, thenx∈S⁺⁺. Moreover, we have cl_R(T ∩Rⁿ)⊂val(S⁺⁺).

Proof. The proof of Lemma 4.17 shows that if x ∈ T ∩Rⁿ, andx∈val⁻¹(x)∩Kⁿ>0 is any lift of x, then the matrix A:= sval(Q(x)) is such that A_ii =Q⁺_ii(x) ⩾Q⁻_ii(x) for all i. Moreover, by Assumption B (and the fact that x∈Rⁿ) the diagonal entries of A are finite,Aii∈T+ for all i. Furthermore, the proof of Lemma 4.17 shows thatA_ii⊙A_jj > A^⊙_ij² for all i < j. Hence, by Corollary 4.49, x ∈ S⁺⁺. Moreover, since S⁺⁺ is semialgebraic, its image by valuation val(S⁺⁺) is closed inRⁿ (Theorem 3.1) and the claim follows.

In the following lemma we abandon the assumption that the matrices are Metzler.

Lemma 4.51. Suppose that the matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ satisfy Assumption B. Let TΣ,♢ be as in Lemma 4.26. Then, we have the inclusion

∪

Σ

∩

♢

cl_R⁽TΣ,♢∩Rⁿ⁾⊂val(S⁺⁺).

Proof. Fix anyΣ. By repeating the proof of Lemma 4.26 (replacing Lemma 4.17 by Lemma 4.50)

4.4. Valuation of interior and regions of strict feasibility 81 we find a pointz∈val(x)⁻¹∩Kⁿ>0 such that²

∀i,Q_ii(z)>0,

∀(i, j)∈Σ,Q_ii(z)Q_jj(z)>(m−1)²(Q_ij(z))²,

∀(i, j)∈Σ^∁,Qij(z) = 0.

HenceQ_ii(z)Q_jj(z)>(m−1)²(Q_ij(z))² for alli < j andz is strictly feasible by Lemma 4.48.

Corollary 4.52. Suppose that the matrices Q⁽¹⁾, . . . ,Q⁽ⁿ⁾ satisfy Assumption B and the con-ditions of Theorem 4.28. Then

val(S⁺⁺) =S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩Rⁿ.

Proof. The proof of Theorem 4.28 shows that val(S⁺⁺) ⊂val(S∩Kⁿ_>0) = S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩ Rⁿ and that S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩Rⁿ = ^∪_Σ^∩_♢cl_R⁽TΣ,♢∩Rⁿ⁾. Hence, the claim follows from Lemma 4.51.

2This proof requires to check that the set

{y∈Kⁿ>0:∀i,Qii(y)>0∧ ∀(i, j)∈Σ,Qii(y)Qjj(y)>(m−1)²(Qij(y))²} is convex. Note that this is true because for every(i, j)∈Σthe set

{y∈Kⁿ>0:Qii(y)>0,Qjj(y)>0,Qii(y)Qjj(y)>(m−1)²(Qij(y))²} is the set of strictly feasible points of a spectrahedron defined by matrices of size2×2.

82 Chapter 4. Tropical spectrahedra

CHAPTER 5

Tropical analogue of the Helton–Nie conjecture

As discussed in Section 1.1, an important question in semidefinite optimization consists in char-acterizing the sets that arise as projections of spectrahedra [Nem07]. Helton and Nie [HN09]

conjectured that every convex semialgebraic set is a projected spectrahedron. The conjecture has been recently disproved by Scheiderer [Sch18b], who showed that the cone of positive semi-definite forms cannot be expressed as a projection of a spectrahedron, except in some particular cases.

Theorem 5.1 ([Sch18b]). The cone of positive semidefinite forms of degree 2d in n variables can be expressed as a projection of a spectrahedron only when2d= 2orn⩽2or(n,2d) = (3,4).

In this section we study the tropicalizations of convex semialgebraic sets and we show the following theorem, which may be thought of as a “Helton–Nie conjecture for valuations.”

Theorem 5.2. Let K be a real closed valued field equipped with a nontrivial and convex valua-tion val:K_→Γ ∪ {−∞} and suppose thatS ⊂Kⁿ is a convex semialgebraic set. Then, there exists a projected spectrahedron S^′⊂Kⁿ such that val(S) =val(S^′).

As in Chapter 3, in order to prove Theorem 5.2 we first study the case of Puiseux seriesK = Kand then use model theory to generalize the result to other fields. Along the way, we obtain a more precise characterization of sets that arise as tropicalizations of convex semialgebraic sets overK.

84 Chapter 5. Tropical analogue of the Helton–Nie conjecture Definition 5.3. We say that a setS ⊂Tⁿ is a tropicalization of a convex semialgebraic set if there exists a convex semialgebraic set S ⊂Kⁿ such thatval(S) =S.

Definition 5.4. We say that a tropical Metzler spectrahedron S ⊂Tⁿ is real if it is included inRⁿ.

Our characterization of tropicalizations of convex semialgebraic sets is given in the next result.

Theorem 5.5. Fix a setS_⊂Tⁿ. Then, the following conditions are equivalent:

(a) Sis a tropicalization of a convex semialgebraic set;

(b) Sis tropically convex and has closed semilinear strata;

(c) S is tropically convex and every stratum of S is a projection of a real tropical Metzler spectrahedron;

(d) Sis a projection of a tropical Metzler spectrahedron;

(e) there exists a projected spectrahedronS ⊂Kⁿ_⩾0 such thatval(S) =S.

The rest of this chapter is organized as follows. In Section 5.1 we recall some basic notions of tropical convexity. Then, in Sections 5.2 and 5.3 we study real tropical cones and show that these objects can be described by monotone homogeneous operators and by graphs. Then, we show Theorems 5.2 and 5.5. This is done is three steps. First, we prove a simpler variant of Theorem 5.5 for real tropical cones (Section 5.4), then we prove both theorems for Puiseux series (Section 5.5), and finally we extend the result to more general fields (Section 5.6). This chapter is based on the article [AGS19].

Before starting, let us point out that the result of Scheiderer generalizes to all real closed fields. In other words, the Helton–Nie conjecture is false over every such field.

Corollary 5.6 (of [Sch18b, Corollary 4.25]). Let K be a real closed field. Then, the cone of positive semidefinite forms of degree 2din n variables over K can be expressed as a projection of a spectrahedron over K only when 2d= 2 or n⩽2 or (n,2d) = (3,4).

Proof. Fix the integers d,m,n, and p. The statement “the cone of positive semidefinite forms of degree 2d innvariables over K is the projection of a spectrahedron inK^p associated with matrices of sizem×m” is a sentence in the language of ordered ringsLor. Since the theory of real closed fields is complete (Theorem 2.110), this sentence is true over R if and only if it is true over K.