NOTES ON THE ESTIMATION OF INTERNATIONAL MIGRATION
DE INFORMACION SOBRE LA RESIDENCIA DE LOS HIJOS SOBREVIVIENTES
3.4.1
Existence and Uniqueness of Enclosed Solutions
In this section, useful results for verifying existence and uniqueness of enclosed so- lutions are presented. The first result is simply a generalization of the converse of Miranda’s theorem (Cor. 5.3.8 in [101]) for existence of enclosed solutions.
Theorem 3.4.1. Let X ∈ IDx and P ∈ IDp. Let H ≡ [hL, hU] be an inclusion
function, and an interval extension, of h on X × P . For some i, choose ˜xi ∈ Xi and
set Zi := [ ˜xi, ˜xi] and Zj := Xj for j ̸= i. If for some l, hLl(Z, P )hUl (Z, P ) > 0 holds,
Proof. Suppose not. Then there exists a solution z to h(z, p) = 0 in Z for some
p∈ P . Since Zi = ˜xi, the point x = [x1, x2, . . . , ˜xi, . . . , xnx]
T must satisfy h(x, p) = 0
for some (x1, . . . , xi−1, xi+1, . . . , xnx, p)∈ X1×. . .×Xi−1×Xi+1×. . .×Xnx×P , since
H is an inclusion function and an interval extension of h. This implies 0∈ H(Z, P )
which implies hL
l(Z, P )hUl (Z, P )≤ 0 for every l = 1, 2, . . . , nx, a contradiction.
The previous result offers an efficient way to rule out a partial enclosure scenario. For instance, if a partial enclosure is suspected in dimension i, then it is expected that xi crosses a boundary of Xi; either its upper bound xUi , lower bound xLi, or both.
Therefore, setting ˜xi := xLi or ˜xi := xUi in Theorem 3.4.1 allows one to verify if xi does
not leave the lower or upper bound of Xi, respectively. Of course, in order to verify
this for all i = 1, 2, . . . , nx, Theorem 3.4.1 must be applied 2nx times. A simpler way
to exclude the possibility of a partial enclosure is given in the next result.
Theorem 3.4.2. Let X0 ∈ ID
x, P ∈ IDp and {Xk} be a sequence of intervals gener-
ated by a parametric interval method (3.11), starting at X0. Suppose that a continuous
solution branch x : P → Dx of (3.2) exists. Then the quantities referred to exist and
for k ∈ N:
ˆ
xUi (P )≥ x0,Li ≥ ˆxLi(P ) ⇒ xk,Li = x0,Li , i = 1, 2, . . . , nx.
Proof. The minimum and maximum of a continuous function on a compact set exist
and define the image set. By hypothesis ˆxUi (P )≥ x0,Li ≥ ˆxLi (P ) so that by continuity of the solution, x0,Li forms part of a solution for some ˆp ∈ P . Also, x0,Li ∈ X0
i
by definition. Hence, applying Theorem 3.3.8 on [ˆp, ˆp] yields x0,Li ∈ Xik, ∀k and
thus xk,Li ≤ x0,Li , ∀k must hold. However, by construction xk,Li ≥ x0,Li , ∀k. Thus xk,Li = x0,Li , ∀k for i = 1, 2, . . . , nx.
Corollary 3.4.3. Suppose the hypotheses of Theorem 3.4.2 are satisfied. Then for
k ∈ N:
1. xk,Li > x0,Li ⇒ xi(p)̸= x0,Li , ∀p ∈ P,
hold for i = 1, 2, . . . , nx.
The practical application of this result is that when applying a parametric inter- val Newton-type method, if any improvement is observed on any of the bounds, it is known that a solution curve does not cross that bound. Thus, if it is not known whether X0 is a partial enclosure of a solution curve, one can apply a parametric in-
terval Newton-type method and potentially guarantee no partial enclosure. As we will see in Section 3.5, Theorem 3.4.1 and Corollary 3.4.3 will be applied within Algorithm 3.1 to find a position to bisect the current interval so as to avoid a partial enclosure situation altogether. The following result provides a much sharper, computationally verifiable test for existence and uniqueness of enclosed solutions as compared to the standard test of Theorem 3.3.10.
Theorem 3.4.4 (Existence and Uniqueness). Let Z ∈ IDx, P ∈ IDp, and z ∈
Z. Suppose Y−1 ≡ m(Jx(Z, P )) is nonsingular. Let Φ ∈ {N, K} be defined as in
Definitions 3.3.3 or 3.3.5. Let A = |Y|rad(Jx(Z, P )). Let λmax = maxi{|λi|} be
the magnitude of the extremal eigenvalue(s) of A. Suppose it is guaranteed (say by Theorem 3.4.1 or Corollary 3.4.3) that no solution branch intersects the bounds of Z. If λmax < 1, Φ(z, Z, ¯p) ⊂ int(Z), z ∈ int(Z), and ¯p ∈ P , then there exists a unique
solution branch in Z for every p∈ P .
Proof. From the hypotheses, the following hold:
1. if solution branches exist in Z, they are contained in its interior, 2. since Φ(z, Z, ¯p)⊂ int(Z), a unique solution exists in Z at ¯p
3. since λmax < 1, by Theorem 3.2.28 it follows that Jx(Z, P ) is nonsingular. Therefore, the hypotheses of Proposition 5.1.4 and Theorem 5.1.3 in [101] are satisfied. It follows that there exists a unique solution branch in Z.
Remark 3.4.5. Guaranteeing that no solution branch intersects the bounds of Z can
be done by either applying Theorem 3.4.1 2nx times (at each bound) or by applying a
parametric interval method and verifying Z∗ ⊂ Z0 strictly, where Z∗ is the converged interval and Z0 is the initial interval; a result of Corollary 3.4.3.
3.4.2
Convergence
An important property of enclosures of locally unique solution branches of (3.2) gener- ated by parametric interval methods, is their convergence behavior under partitioning
P . The results in this section have important implications when using this bounding
information within global optimization applications such as in the next chapter and in [128, 134]. The reader should be aware that the results presented in this section may be not entirely complete and are the topic of future research. The following result will be important in later continuity arguments.
Lemma 3.4.6. Let A ∈ IRn×n with m(A) nonsingular. If m(A)−1A is nonsingular,
then for B ∈ IA, m(B)−1B is nonsingular and has diagonal elements that do not contain zero.
Proof. By Corollary 4.1.3 in [101], since m(A)−1A is nonsingular, m(B)−1B is non-
singular. By Theorem 4.1.1 in [101], diagonal elements of m(B)−1B do not contain
0.
Lemma 3.4.7. Let m[Jx(X0, P )] be nonsingular for some (X0, P ) ∈ IDx × IDp.
Suppose Y0Jx(X0, P ) is nonsingular with =m[Jx(X0, P )]−1, then the interval-valued
functions N, K : Dx × IDx × IDp → IRnx, defined in Definitions 3.3.3, 3.3.4, and
3.3.5, are continuous on their domains.
Proof. By continuous differentiability of h on Dx × Dp, h and Jx are continuous. By Lemma 3.4.6, if Y0J
x(X0, P ) is nonsingular for some (X0, P ) ∈ IDx× IDp, and Y0 = m[J
x(X0), P ]−1, then YkJx(Xk, P ) is nonsingular for every k and its diagonal elements do not enclose 0. By Theorem 2.1.1 in [101], K : Dx× IDx× IDp as in Def.
3.3.4 and Def. 3.3.5 and N : Dx× IDx× IDp as in Def. 3.3.3 are continuous.
Theorem 3.4.8. Let X0 ∈ ID
x be such that there exists a locally unique solution x(p) ∈ X0 for every p ∈ P1 ∈ IDp with Jx(X0, P1) nonsingular. Let {Pl} ∈ IP1 define a nested sequence of parameter intervals such that ∩∞
l=1Pl = [ˆp, ˆp]. Let
Def. 3.3.6, with Φ = N as in Definition 3.3.3, using finite-precision rounded in- terval arithmetic starting at the initial interval X0. Then lim
l→∞klim→∞Φ(x
k, Xk, Pl) =
lim
k→∞llim→∞Φ(x
k, Xk, Pl) = [x(ˆp), x(ˆp)].
Proof. Let Gk(Pl) = Xk = Φ(xk−1, Xk−1, Pl) and let ϵ
tol be the precision to which
intervals are rounded. Consider lim
k→∞llim→∞G
k(Pl). Since J
x(X0, P ) is nonsingular,
m(Jx(X0, P )) is nonsingular. Therefore m(Jx(Xk, P ))−1Jx(Xk, P ) is nonsingular for every k from Lemma 3.4.6. Furthermore, from the same Lemma, the diagonal el- ements of m(Jx(Xk, P ))−1Jx(Xk, P ) do not contain 0. From Lemma 3.4.7, N is continuous. It is clear from continuity of N (xk, Xk,· ) on IDp that lim
k→∞llim→∞G
k(Pl) =
lim
k→∞G
k(ˆp), which is a simple reduction of the parametric case to the non-parametric
case. Since Jx(X0, ˆp) is nonsingular, we have m(Jx(Xk, ˆp))−1Jx(Xk, ˆp) nonsingular and its diagonals do not contain 0. Then, from Theorem 5.2.6 in [101], Gk(ˆp) →
G∗(ˆp) = Φ(x∗, X∗(ˆp), ˆp) = X∗(ˆp) = [x(ˆp), x(ˆp)] = x(ˆp). By Theorem 3.2.18, since
{Gk(Pl)} is a nested sequence of intervals, it is convergent. Since IRnx is a complete metric space by Theorem 3.2.23, {Gk(Pl)} is a Cauchy sequence in IRnx. Thus, for every ϵtol > 0, there exists an M such that for every n, m ≥ M, Pl ∈ IDp we have
dH(Gn(Pl), Gm(Pl)) < ϵtol. Therefore, by Theorem 7.8 in [117], {Gk(Pl)} converges
uniformly on IDx. It follows directly that lim
l→∞klim→∞G
k(Pl) = lim
k→∞llim→∞G
k(Pl) = x(ˆp).
An analogous result holds for the parametric interval method using the K opera- tor.
Theorem 3.4.9. Let X0 ∈ ID
x be such that there exists unique solution x(p) ∈ X0
for every p∈ P1 ∈ ID
p. Let {Pl} ∈ IP define a nested sequence of parameter inter-
vals such that ∩∞l=1Pl = [ˆp, ˆp]. Let {Xk} be the nested sequence of intervals generated
by the parametric interval method, Def. 3.3.6 with Φ = K as in Definition 3.3.5, using finite-precision rounded interval arithmetic with xk = m(Xk). Let λ
max= maxi{|λi|}
be the magnitude of the extremal eigenvalue(s) of the matrix |Y0J
x(X0, P )− I|. If
λmax < 1, then lim
l→∞klim→∞Φ(x
k, Xk, Pl) = lim
k→∞llim→∞Φ(x
Proof. Let Gk(Pl) = Xk= Φ(xk−1, Xk−1, Pl) and let ϵ
tolbe the precision to which in-
tervals are rounded. Similar to Theorem 3.4.8, from Lemma 3.4.6, Ak in (3.10) is non-
singular for each k and its diagonal elements do not contain 0. Therefore by Lemma 3.4.7, we have continuity of K on P . It immediately follows that lim
k→∞llim→∞G
k(Pl) =
lim
k→∞G
k(ˆp), which is the reduction to the non-parametric case. Since J
x(X0, ˆp) is nonsingular and λmax < 1, from Theorem 5.2.2 in [101], it follows that Gk(ˆp) →
G∗(ˆp) = K(x∗, X∗(ˆp), ˆp)∩ X∗(ˆp) = X∗(ˆp) = [x(ˆp), x(ˆp)] = x(ˆp). In an analogous
argument to Theorem 3.4.8, it follows that lim
l→∞klim→∞G
k(Pl) = G∗(ˆp) = x(ˆp).