Geometry of Injection Region

5.3 Main Results

5.3.2 Geometry of Injection Region

In order to study the relationship between GNF and CGNF, it is beneficial to explore the geometry of the feasible set of GNF. Hence, we investigate the geometry of the injection regionP and the box-constrained injection region P ∩ B in this part.

GNF-Theorem 1. Consider two arbitrary points pˆn and p˜n in the injection region P.

The box B(ˆp_n,p˜_n) is contained in P.

The proof of this theorem is based on four lemmas, and will be provided later in this subsection. To understand this theorem, consider the injection region P depicted in Fig- ure 5.2(a) corresponding to the illustrative example given in Section 5.3.1. If any arbitrary box is drawn inR² in such a way that its upper right corner and lower left corner both lie in the green area, then the entire box must lie in the green area completely. This can be easily proved in this special case and is true in general due to Theorem 1. However, this result does not hold for the injection region given in Figure 5.3(b) because the assumption of monotonicity offij(·)’s is violated in this case. The result of Theorem 1 can be generalized to the box-constrained injection region, as stated below.

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Figure 5.4: (a) A particular graphG. (b) The matrix~ M(¯p_d,p˜_d) corresponding to the graph G~ in Figure (a). (c) The (j,(i, j))^th entry ofM(¯pd,p˜d) (shown as “*”) is equal to the slope of the line connecting the point (¯p_ij,p¯_ji) to (˜p_ij,p˜_ji).

Corollary 1. Consider two arbitrary points pˆ_n and p˜_n belonging to the box-constrained injection region P ∩ B. The box B(ˆpn,p˜n) is contained inP ∩ B.

Proof: The proof follows immediately from Theorem 1.

The rest of this subsection is dedicated to the proof of Theorem 1, which is based on a series of definitions and lemmas.

Definition 4. DefineB_d as the box containing all vectorsp_d introduced in (5.2c) satisfying the condition pij ∈[p^min_ij , p^max_ij ] for every(i, j)∈E.~

Definition 5. Given two arbitrary points p¯_d,p˜_d∈ B_d, defineM(¯p_d,p˜_d) as follows:

• Let M(¯p_d,p˜_d) be a matrix with |N | rows indexed by the vertices of G and with |E|~ columns indexed by the edges in E.~

• For every vertexk∈ N and edge(i, j)∈E~, set the(k,(i, j))^th entry ofM(¯pd,p˜d)(the

one in the intersection of rowk and column (i, j)) as



















1 if k=i

fij( ¯pij)−fij( ˜pij)

¯

pij−˜pij if k=j and p¯ij 6= ˜pij

f_ij⁰ (¯p_ij) if k=j and p¯_ij = ˜p_ij

0 otherwise

(5.11)

where f_ij⁰ (¯p_ij) denotes the right derivative of f_ij(¯p_ij) if p¯_ij < p^max_ij and the left derivative of fij(¯pij) if p¯ij =p^max_ij .

To illustrate Definition 5, consider the three-node graph G~ depicted in Figure 5.4(a).

The matrixM(¯pd,p˜d) associated with this graph has the structure shown in Figure 5.4(b), where the “*” entries depend on the specific values of ¯p_dand ˜p_d. Consider an edge (i, j)∈E.~ The (j,(i, j))^th entry ofM(¯pd,p˜d) is equal to

fij(¯pij)−fij(˜pij)

¯ pij−p˜ij

, (5.12)

provided ¯pij 6= ˜pij. As can be seen in Figure 5.4(c), this is equal to the slope of the line connecting the point (¯p_ij,p¯_ji) to the point (˜p_ij,p˜_ji) on the parameterized curve (p_ij, p_ji), where pji = fij(pij). Moreover, f_ij⁰ (¯pij) is the limit of this slope as the point (˜pij,p˜ji) approaches (¯p_ij,p¯_ji). It is also interesting to note that M(¯p_d,p˜_d) has one positive entry, one negative entry and m−2 zero entries in each column (note that the slope of the line connecting (¯p_ij,p¯_ji) to (˜p_ij,p˜_ji) is always negative). The next lemma explains how the matrixM(¯p_d,p˜_d) can be used to relate the semi-flow vector to the injection vector.

Lemma 1. Consider two arbitrary injection vectors p¯_n and p˜_n in P, associated with the semi-flow vectors p¯_d and p˜_d (defined in (5.2)). The relation

¯

p_n−p˜_n =M(¯p_d,p˜_d)×( ¯P_d−P˜_d) (5.13)

holds.

Proof: One can write

¯

pi−p˜i = X

j∈N(i)

(¯pij −p˜ij), ∀i∈ N. (5.14)

By using the relations

¯

pji =fij(¯pij), p˜ji=fij(˜pij), ∀(i, j)∈E,~ (5.15) it is straightforward to verify that (5.13) and (5.14) are equivalent.

The next lemma investigates an important property of the matrixM(¯pd,p˜d).

Lemma 2. Given two arbitrary points p¯d,p˜d ∈ B_d, assume that there exists a nonzero vector x ∈ R^m such that x^TM(¯p_d,p˜_d) ≥ 0. If x has at least one strictly positive entry, then there exists a nonzero vector y∈ R^m₊ such thaty^TM(¯pd,p˜d)≥0.

Proof: Consider an indexi0∈ N such thatxi0 >0. DefineV(i₀) as the set of all vertices i∈ N from which there exists a directed path to vertex i₀ in the graphG. Note that~ V(i₀) includes vertexi0 itself. The first goal is to show that

xi ≥0, ∀i∈ V(i0). (5.16)

To this end, consider an arbitrary set of vertices i₁, ..., i_k in V(i₀) such that {i₀, i₁..., i_k} forms a direct path inG~ as

i_k→ik−1→ · · ·i₁→i₀. (5.17) To prove (5.16), it suffices to show that x_i1, ..., x_ik ≥0. For this purpose, one can expand the product x^TM(¯pd,p˜d) and use the fact that each column of M(¯pd,p˜d) hasm−2 zero entries to conclude that

x_i1 +f_i1i0(¯p_i1i0)−f_i1i0(˜p_i1i0)

¯

p_i1i0−p˜_i1i0 x_i0 ≥0. (5.18) Sincex_i0 is positive andf_i1i0(·) is a decreasing function, x_i1 turns out to be positive. Now, repeating the above argument for i1 instead of i0 yields that xi2 ≥ 0. Continuing this reasoning leads to x_i1, ..., x_ik ≥0. Hence, inequality (5.16) holds. Now, definey as

yi =







x_i if i∈ V(i₀) 0 otherwise

, ∀i∈ N. (5.19)

In light of (5.16),yis a nonzero vector inR^m₊. To complete the proof, it suffices to show that y^TM(¯p_d,p˜_d) ≥ 0. Similar to the indexing procedure used for the columns of the matrix

M(¯p_d,p˜_d), we index the entries of the|E|~ dimensional vectory^TM(¯p_d,p˜_d) according to the edges ofG. Now, given an arbitrary edge (α, β)~ ∈E, the following statements hold true:~

• If α, β ∈ V(i₀), then the (α, β)^th entries of y^TM(¯p_d,p˜_d) and x^TM(¯p_d,p˜_d) (i.e., the entries corresponding to the edge (α, β)) are identical.

• Ifα∈ V(i0) and β 6∈ V(i₀), then the (α, β)^th entry of y^TM(¯pd,p˜d) is equal to yα.

• Ifα6∈ V(i₀) and β 6∈ V(i₀), then the (α, β)^th entry of y^TM(¯p_d,p˜_d) is equal to zero.

Note that the case α 6∈ V(i₀) and β ∈ V(i₀) cannot happen, because if β ∈ V(i₀) and (α, β)∈E, then~ α∈ V(i₀) by the definition of V(i₀). It follows from the above results and the inequalityx^TM(¯p_d,p˜_d)≥0 thaty^TM(¯p_d,p˜_d)≥0.

Definition 6. Consider the graph G and an arbitrary flow vector p_e. Given a subgraph G_s of the graph G, define pe(G_s) as the flow vector associated with the edges of G_s, which has been induced by p_e. Definep_d(G_s),p_n(G_s) andp_i(G_s) as the semi-flow vector, injection vector and injection at node i∈ G_s corresponding tope(G_s), respectively. Define also P(G_s) as the injection region associated with G_s.

The next lemma studies the injection region P in the case whenfij(·)’s are all piecewise linear.

Lemma 3. Assume that the functionf_ij(·)is piecewise linear for every(i, j)∈E. Consider~ two arbitrary points pˆn,p¯n ∈ P and a vector ∆¯pn ∈ R^m satisfying the relations

ˆ

pn≤p¯n−∆¯pn ≤p¯n. (5.20) There exists a strictly positive number^max with the property

¯

p_n−ε∆¯p_n ∈ P, ∀ε∈[0, ^max]. (5.21) Proof: In light of (5.20), we have ∆¯p_n ≥ 0. If ∆¯p_n = 0, then the lemma becomes trivial asε can take any arbitrary value. So, assume that ∆¯pn 6= 0. Let ˆpe and ¯pe denote two flow vectors associated with the injection vectors ˆp_n and ¯p_n, respectively. Denote the corresponding semi-flow vectors as ˆp_d and ¯p_d. Given an edge (i, j)∈E, the curve~

(pij, fij(pij))|pij ∈[p^min_ij , p^max_ij ] (5.22)

is a Pareto set in R² due to fij(·) being monotonically decreasing. Since (ˆpij,pˆji) and (¯p_ij,p¯_ji) both lie on the above curve, one of the following cases occurs:

• Case 1: pˆ_ij ≥p¯_ij and ˆp_ji≤p¯_ji.

• Case 2: pˆ_ij ≤p¯_ij and ˆp_ji≥p¯_ji.

This fact can be observed in Figure 5.4(c) for the points (¯p_ij,p¯_ji) and (˜p_ij,p˜_ji) instead of (ˆpij,pˆji) and (¯pij,p¯ji). With no loss of generality, it can be assumed that Case (1) occurs.

Indeed, if Case (2) happens, it suffices to make two changes:

• Change the orientation of the edge (i, j) in the graph G~ so that (j, i) ∈ E~ instead of (i, j)∈E.~

• Replace the constraint p_ji = f_ij(p_ij) in (5.4c) with p_ij = f_ij⁻¹(p_ji), where the exis- tence and monotonicity of the inverse functionf_ij⁻¹(·) is guaranteed by the decreasing property off_ij(·).

Therefore, suppose that ˆ

p_ij ≥p¯_ij, pˆ_ji≤p¯_ji, ∀(i, j)∈E~ (5.23) or

ˆ

p_d≥p¯_d. (5.24)

First, consider the case ˆp_d > p¯_d. In light of Lemma 1, the assumption ˆp_n ≤ p¯_n can be expressed as

M(ˆp_d,p¯_d)×(ˆp_d−p¯_d) = ˆp_n−p¯_n≤0. (5.25) In order to guarantee the relation ¯p_n−ε∆¯p_n ∈ P, it suffices to seek a vector ∆¯p_d ∈ R^|E|~ satisfying

¯

p_d−ε∆¯p_d∈ B_d (5.26)

and

M(¯pd,p¯d−ε∆¯pd)×(¯pd−(¯pd−ε∆¯pd)) = ¯pn−(¯pn−ε∆¯pn) (5.27)

(see the proof of Lemma 1), or equivalently

¯

p_d−ε∆¯p_d∈ B_d (5.28a)

M(¯p_d,p¯_d−ε∆¯p_d)×∆¯p_d= ∆¯pn. (5.28b) Consider an arbitrary vector ∆¯p_d ∈ R^|E|~ with all negative entries. In light of Definition 5, the inequality ˆpd > p¯d and the piecewise linear property of the fij(·)’s, there exists a positive numberε^max such that

¯

p_d−ε∆¯p_d∈ B_d (5.29a)

M(¯p_d,p¯_d−ε∆¯p_d) =M(¯p_d,p¯_d) (5.29b) for every ε ∈ [0, ε^max]. To prove the lemma, it follows from (5.28) and (5.29) that it is enough to show the existence of a negative vector ∆¯p_d satisfying the relation

M(¯p_d,p¯_d)×∆¯p_d = ∆¯pn (5.30) in whichεdoes not appear. To prove this by contradiction, assume that the above equation does not have a solution. By Farkas’ Lemma, there exists a vectorx∈ R^m such that

x^TM(¯pd,p¯d)≥0, x^T∆¯pn>0. (5.31) Since ∆¯pn is nonnegative, the inequality x^T∆¯pn > 0 does not hold unless x has at least one strictly positive entry. Now, it follows from x^TM(¯p_d,p¯_d)≥0 and Lemma 2 that there exists a nonzero vector y∈ R^m such that

y^TM(¯p_d,p¯_d)≥0, y≥0. (5.32) On the other hand, given an edge (i, j) ∈ E, since ˆ~ p_ij ≥ p¯_ij (due to (5.23)), the slope of the line connecting the points (ˆpij,pˆji) and (¯pij,p¯ji) is more than or equal tof⁰(¯pij). This yields that

M(¯pd,p¯d)≤M(ˆpd,p¯d). (5.33)

Now, it follows from (5.24), (5.25), (5.32), and (5.33) that

0≥y^TM(ˆp_d,p¯_d)×(ˆp_d−p¯_d)≥y^TM(¯p_d,p¯_d)×(ˆp_d−p¯_d)≥0. (5.34) Thus,

0 =y^TM(ˆpd,p¯d)×(ˆpd−p¯d) =y^T(ˆpn−p¯n). (5.35) This is a contradiction because ˆpn −p¯n is strictly negative and the nonzero vector y is positive.

So far, the lemma has been proven in the case when ˆpd >p¯d. To extend the proof to the case ˆp_d ≥p¯_d, defineE_r as the set of every edge (i, j)∈ E such that

ˆ

p_ij 6= ¯p_ij (5.36)

(note that ˆp_ij = ¯p_ij if and only if ˆp_ji = ¯p_ji). Define also G_r as the unique subgraph of G induced by the edge setE_r. LetN_r denote the vertex set ofG_r, which may be different from N. It is easy to verify that

pˆd(G_r)>p¯d(G_r), (5.37a)

¯

p_i−pˆ_i= ¯p_i(G_r)−pˆ_i(G_r), ∀i∈ N_r. (5.37b) Therefore,

ˆ

p_n(G_r)≤p¯_n(G_r)−∆¯p_n(G_r)≤p¯_n(G_r) (5.38) where the relationship between ∆¯p_n and the new vector ∆¯p_n(G_r) is as follows:

∆¯pi =







∆¯pi(G_r) if i∈ N_r 0 otherwise

∀i∈ N. (5.39)

In light of (5.37a) and (5.38), one can adopt the proof given earlier for the case ˆp_d >p¯_d to conclude the existence of a positive number ^max with the property

¯

p_n(G_r)−ε∆¯p_n(G_r)∈ P(G_r), ∀ε∈[0, ^max]. (5.40) Given an arbitrary numberε∈[0, ^max], we use the shorthand notationpn(G_r) for ¯pn(G_r)−

ε∆¯pn(G_r). Let pe(G_r) denote a flow vector corresponding to the injection vector pn(G_r).

One can expand the vector p_e(G_r) into a flow vector p_efor the graph G as follows:

• For every (i, j) ∈ E_r, the (i, j)^th entries of p_e and p_e(G_r) (the ones corresponding to the edge (i, j)) are identical.

• For every (i, j)∈ E\E_r, the (i, j)^th entry of pe is equal to ¯pij.

It is straightforward to show thatpn ∈ P, wherepn denotes the injection vector associated with the flow vector p_e. The proof is completed by noting thatp_n= ¯p_n−ε∆¯p_n.

The next lemma proves Theorem 1 in the case when fij(·)’s are all piecewise linear.

Lemma 4. Assume that the function f_ij(·) is piecewise linear for every (i, j)∈ E. Given~ any two arbitrary pointspˆn,p˜n ∈ P, the boxB(ˆpn,p˜n)is a subset of the injection regionP. Proof: With no loss of generality, assume that ˆpn ≤ p˜n (because otherwise B(ˆpn,p˜n) is empty). To prove the lemma by contradiction, suppose that there exists a point p_n ∈ B(ˆpn,p˜n) such that pn6∈ P. Consider the set

γ

γ ∈[0,1], p˜_n+γ(p_n−p˜_n)∈ P

(5.41) and denote its maximum as γ^max (the existence of this maximum number is guaranteed by the closedness and compactness ofP). Note that ˜pn+γ(pn−p˜n) is equal to pn atγ = 1.

Since p_n6∈ P by assumption, we haveγ^max<1. Denote ˜p_n+γ^max(p_n−p˜_n) as ¯p_n. Hence, p¯n ∈ P and ˆpn≤pn≤p¯n (recall that γ^max<1). Define ∆¯pn as ¯pn−pn. One can write:

ˆ

pn≤p¯n−∆¯pn≤p¯n, pˆn,p¯n∈ P. (5.42) By Lemma 3, there exists a strictly positive number^max with the property

¯

p_n−ε∆¯p_n∈ P, ∀ε∈[0, ^max] (5.43) or equivalently

˜

p_n+ (γ^max+ε(1−γ^max))(p_n−p˜_n)∈ P, ∀ε∈[0, ^max]. (5.44)

Notice that

γ^max+ε(1−γ^max)> γ^max, ∀ε >0. (5.45) Due to (5.44), this violates the assumption that γ^max is the maximum of the set given

in (5.41).

Lemma 4 will be deployed next to prove Theorem 1 in the general case.

Proof of Theorem 1: Consider an arbitrary approximation offij(·) by a piecewise linear function for every (i, j) ∈E. As a counterpart of~ P, let P_s denote the injection region in the piecewise-linear case. By Lemma 4, we have

B(ˆpn,p˜n)⊆ P_s. (5.46)

Since the piecewise linear approximation can be made in such a way that the sets P and P_s become arbitrarily close to each other, the above relation implies that the interior of B(ˆp_n,p˜_n) is a subset of P. On the other hand, P is a closed set. Hence, the boxB(ˆp_n,p˜_n)

must entirely belong toP.