Geometric interpretation of the condition number

In this section, we give a geometric interpretation of the number λ. More precisely, we relate this number to with the radius of the largest ball (in the Hilbert seminorm) contained in a certain tropical Metzler spectrahedral cone in dimension2n. We start with the definition of the Hilbert seminorm and the balls in this seminorm.

Definition 8.7. We define the Hilbert seminorm of a vector x ∈ Rⁿ as ∥x∥H := t(x)−b(x), wheret(x) := max_k∈[n]x_k and b(x) := min_k∈[n]x_k.

Lemma 8.8. Hilbert seminorm is, indeed, a seminorm. More precisely, it satisfies the condi-tions ∥x∥H⩾0,∥λx∥H=|λ|x, and∥x+y∥H⩽∥x∥H+∥y∥H for all x, y∈Rⁿ and λ∈R.

8.1. Archimedean feasibility problems 143

x₁ x₂

Figure 8.1: Hilbert ball (forx0 = 0).

x1

x₂

Figure 8.2: Tropical spectrahedron and its max-imal inner Hilbert ball (for x₀= 0).

Proof. The first two conditions are trivially satisfied. The prove the third one, note that for all k, l∈[n]we have

|xk+yk−xl−yl|⩽|xk−xl|+|yk−yl|. Hence,

∥x+y∥H= max

k,l∈[n]|xk+yk−xl−yl|⩽ max

k,l∈[n]|xk−xl|+ max

k,l∈[n]|yk−yl|=∥x∥H+∥y∥H. Remark 8.9. We point out that ∥ · ∥H is not a norm, because it does not satisfy the condition

∥x∥H = 0 ⇐⇒ x = 0. Indeed, we have ∥x∥H = 0 if and only if x has equal coordinates, x=λ(1,1, . . . ,1).

Definition 8.10. If x ∈ Rⁿ and r ∈ R_⩾0, then we define the Hilbert ball centered at x with radiusr, denoted B_H(x, r), as

B_H(x, r) :={y∈Rⁿ:∥x−y∥H⩽r}.

Ifx∈Tⁿis a point with nonempty support K ⊂[n], then we extend this definition by putting B_H(x, r) :={y∈Tⁿ:the support of y is equal to K and ∥x_K−y_K∥H⩽r}.

Since ∥ · ∥H is invariant by adding a the same constant to all coordinates, Hilbert balls are unbounded.

Example 8.11. Figure 8.1 depicts a Hilbert ball (its center is marked by a dot). Figure 8.2 depicts the tropical Metzler spectrahedral cone from Example 4.15 and the largest Hilbert ball that is included inside this cone. This example also shows that such ball is not unique (because one can slide the center of the ball in the direction of the top right corner).

In the next proposition we show that Hilbert balls are tropical cones. This proposition is well known, as it is a particular application of the tropical spectral theory, see e.g. [BCOQ92, Chapter 3.7], [Ser07]. We present a self-contained proof for the sake of completeness.

144 Chapter 8. Condition number of stochastic mean payoff games Proposition 8.12. The setBH(x, r)∪{−∞}is a tropical cone. Moreover, we have the equality

B_H(x, r)∪ {−∞}={^⊕

k∈[n]

λ_k⊙(rϵ_k+x) : λ₁, . . . , λ_n∈T}, (8.1) where ϵ_k denotes the kth vector of the standard basis inRⁿ.

Proof. LetK ⊂[n]denote the support ofx. To prove thatB_H(x, r)∪ {−∞}is a tropical cone, we first show that BH(x, r)∪ {−∞} is equal to the set of all y ∈Tⁿ that satisfy the following system of inequalities:

∀k, l∈K, k̸=l, r+yk⩾(xk−xl) +yl

∀k /∈K, y_k⩾−∞

∀k /∈K, −∞⩾y_k.

(8.2) Indeed, y = −∞ satisfies this system. Moreover, if y ∈ B_H(x, r), then y_k = −∞ for all k /∈ K. Furthermore, for all k, l ∈ K we have |x_k−y_k−x_l+y_l| ⩽ r, which implies that (xk−xl) +yl ⩽r+yk. Conversely, ify ∈Tⁿsatisfies the system above, then it is either equal to

−∞, or its support is equal toK. Indeed, its support cannot be greater thanK by the last two sets of inequalities. Moreover, if the support ofyis nonempty but smaller thanK, then there are two coordinatesk, l∈K such thaty_k=−∞andy_l̸=−∞, which gives a contradiction with the first inequality in (8.2). If support ofy is equal toK, then the inequalityr+y_k⩾(x_k−x_l) +y_l implies x_k−y_k−x_l+y_l ⩽r and, by exchanging kand l,|x_k−y_k−x_l+y_l|⩽r. Thus

∥xK−yK∥H= max

k,l∈K|xk−yk−xl+yl|⩽r

andy∈B_H(x, r). Hence, the system of inequalities (8.2) describesB_H(x, r)∪{−∞}. Moreover, it is immediate to check that each of these inequalities defines a tropical cone. Hence,BH(x, r)∪ {−∞} is also a tropical cone.

To prove the second part of the claim, fixl∈[n]and consider the point y^(l):=rϵl+x. Note that the support of y coincides withK. Furthermore, for allk^′, l^′ ∈K we have

|x_k′−y^(l)_k′ −x_l′+y^(l)_l′ |=

{r ifk^′ ̸=l^′ and l∈ {k^′, l^′} 0 otherwise.

Hence ∥x_K −y_K^(l)∥H = max_k′,l^′∈K|x_k′−y^(l)_k′ −x_l′+y^(l)_l′ | ⩽ r and y^(l) ∈ B_H(x, r). Therefore, using the fact that B_H(x, r)∪ {−∞} is a tropical cone, we have

{^⊕

k∈[n]

λ_k⊙(rϵ_k+x) :λ₁, . . . , λ_n∈T} ⊂B_H(x, r)∪ {−∞}.

To prove the opposite inclusion, take y ∈B_H(x, r) and let λ_l := y_l−x_l−r for all l ∈K and λ_l:=−∞ otherwise. Then, for anyl∈K the vector λ_l⊙(rϵ_l+x)is given by

(λ_l⊙(rϵ_l+x)⁾_k=









y_k ifk=l

yl−xl+xk−r ifk̸=l, k∈K

−∞ ifk /∈K .

By (8.2) we havey_k⩾max_l∈K(y_l−x_l+x_k−r)for allk∈K. Hencey=^⊕_k∈[n]λ_k⊙(rϵ_k+x).

8.1. Archimedean feasibility problems 145

-0.5 -0.25 0 0.25

-1.25 -1

x₁ x2

Figure 8.3: The largest inner Hilbert ball of the tropical spectrahedron from Example 7.22.

We can now interpret the quantity λis terms of the Hilbert balls included inside a spectra-hedron. As before, letQ⁽¹⁾, . . . , Q⁽ⁿ⁾∈T^m×m be a sequence of symmetric Metzler matrices and let

λ:= sup{λ∈R:Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is nontrivial}.

Lemma 8.13. Suppose that λ ⩾ 0. Then, for every λ ∈ R such that 0 ⩽ λ ⩽ λ, the set S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) contains a Hilbert ball of radius λ.

Proof. Let S := S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) and let x ∈ Tⁿ\ {−∞} be a point that belongs to the set Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) (we recall that if λ is finite, then such a point exists even for λ = λ, see Theorem 7.18). Therefore, for every i ∈ [m] we have Q⁺_ii(x) ⩾ λ⊙Q⁻_ii(x) and for every pair i, j∈[m],i < j we haveQ⁺_ii(x)⊙Q⁺_jj(x)⩾(λ⊙Qij(x))^⊙2. Fix k∈[n]and consider the point y:=λϵ_k+x. By the definition ofy, for everyi, j∈[m],i⩽j we haveQ⁻_ij(y)⩽λ⊙Q⁻_ij(x) and Q⁺_ii(y)⩾Q⁺_ii(x). Hencey ∈ S. Since the choice of kwas arbitrary and S is a tropical cone (as proved in Example 5.12), Proposition 8.12 shows that B_H(x, λ)⊂ S.

The next example shows that the converse of Lemma 8.13 is, in general, false.

Example8.14. Consider the tropical Metzler spectrahedral coneSfrom Example 7.22. As noted in Example 6.2, the associated game has constant value and this value is given by χ = 1/56.

Hence, by Theorem 7.21 we have λ = 1/28. However, the largest Hilbert ball contained in S has radius1/20, as depicted in Fig. 8.3. The ball is centered at (−1/20,−11/10,0).

Even though the converse of Lemma 8.13 does not hold in general, it it true if the entries of the matricesQ⁽¹⁾, . . . , Q⁽ⁿ⁾ have a special sign pattern.

Assumption F. For every k∈[n]and every pairi, j∈[m]such thati < j and |Q^(k)_ij |is finite, we have Q^(k)_ii =Q^(k)_jj =−∞.

Lemma 8.15. Suppose that the matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ satisfy Assumption F. Then, the set S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) contains a Hilbert ball of radius λ∈R_⩾0 if and only if Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is nontrivial.

Proof. The “if” part of the claim follows from Lemma 8.13. To prove the opposite inclu-sion, denote S := S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) and suppose that B_H(x, λ) ⊂ S. We will show that x ∈ Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾). To do so, first take anyi∈[m]and consider the inequalityQ⁺_ii(x)⩾Q⁻_ii(x).

Take k ∈ [n] such that Q⁻_ii(x) = Q^(k)_ii +x_k and consider the point y := λϵ_k +x. By our

146 Chapter 8. Condition number of stochastic mean payoff games choice of k we have Q⁺_ii(y) = Q⁺_ii(x) and Q⁻_ii(y) = λ⊙Q⁻_ii(x). Since y ∈ BH(x, λ), we have y ∈ S and hence Q⁺_ii(x) ⩾ λ⊙Q⁻_ii(x). Similarly, for all i, j ∈ [m], i < j we con-sider the inequality Q⁺_ii(x)⊙Q⁺_jj(x) ⩾ (Q_ij(x))^⊙2. If Q_ij(x) = −∞, then we trivially have Q⁺_ii(x)⊙Q⁺_jj(x)⩾(λ⊙Q_ij(x))^⊙2. Otherwise, we takek∈[n]such thatQ_ij(x) =Q^(k)_ij +x_k ̸=−∞

and set y := λϵ_k +x. By Assumption F, we have Q⁺_ii(y) = Q⁺_ii(x), Q⁺_jj(y) = Q⁺_jj(x), and Qij(y) = λ⊙Qij(x). As previously, this gives Q⁺_ii(x)⊙Q⁺_jj(x) ⩾ (λ⊙Qij(x))^⊙2. Hence x∈ Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾).

Given Lemma 8.15, we can now take arbitrary symmetric Metzler matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ and construct another tropical Metzler spectrahedral cone such thatλcorresponds to the radius of its largest Hilbert ball. To do so, we define the matricesQ˜⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾∈T^(m+n)_± ^×(m+n) as

Q˜^(k)_ij :=



















Q^(k)_ij ifi, j ⩽m,k⩽n, and Q^(k)_ij ∈T₋, Q^(k−n)_ij ifi=j⩽m,k > n, and Q^(k−n)_ij ∈T+, 0 ifi=j=m+kand k⩽n ,

⊖0 ifi=j=m+k−nand k > n ,

−∞ otherwise.

In this way, the setSλ( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾) is described by the constraints

∀i⩽m, Q⁺_ii(xn+1, . . . , x2n)⩾λ⊙Q⁻_ii(x1, . . . , xn),

∀k⩽n, x_k⩾λ⊙x_n+k

∀i, j∈[m], i < j, Q⁺_ii(xn+1, . . . , x2n)⊙Q⁺_jj(xn+1, . . . , x2n)⩾(λ⊙Q⁻_ij(x1, . . . , xn))^⊙2.

(8.3)

Furthermore, the matricesQ˜⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾ satisfy Assumption F.

Proposition 8.16. The set S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is a projection of S( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾) on the first n coordinates. Furthermore, for any λ > 0, the set Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is nontrivial if and only if S( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾) contains a Hilbert ball of radiusλ/2.

Proof. Denote S := S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) and S˜ := S( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾). To prove that S is the projection ofS˜, note that for everyi∈[m]we haveQ⁺_ii(x₁, . . . , x_n)⩾Q⁺_ii(x_n+1, . . . , x_2n). Hence, Sis included in the projection ofS. Moreover, for any˜ (x1, . . . , xn)∈ S we can definexn+k:=xk

for all k∈ [n] and the point (x1, . . . , x2n) belongs to S˜. This proves the first claim. To prove the second one, suppose thatSλ:=Sλ(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)is nontrivial, take a pointx∈ Sλ\ {−∞}

and define x_n+k := −^λ₂ +x_k for all k ∈ [n]. Then Q⁺_ii(x_n+1, . . . , x_2n) = −^λ₂ +Q⁺_ii(x₁, . . . , x_n) for all i∈ [m]. Hence, (x1, . . . , x2n) belongs to Sλ/2( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾) and S˜ contains a Hilbert ball of radiusλ/2 by Lemma 8.15. Conversely, if S˜contains a Hilbert ball of radiusλ/2, then Sλ/2( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾)is nontrivial by Lemma 8.15 and the fact that the matricesQ˜⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾ satisfy Assumption F. Take a pointx∈T²ⁿ\{−∞}that belongs toSλ/2( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾). Then, we have Q⁺_ii(x₁, . . . , x_n)⩾ ^λ₂ +Q⁺_ii(x_n+1, . . . , x_2n) and (x₁, . . . , x_n)belongs to Sλ.

Remark 8.17. The proof shows that every point inSλ( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾)\ {−∞}is the projection of the center of some Hilbert ball of radiusλ/2 contained inS( ˜Q⁽¹⁾, . . . ,Q˜⁽²ⁿ⁾).