Approximations for generalized unsplittable flow on paths with application to power systems optimization

Abstract

The Unsplittable Flow on a Path (UFP) problem has garnered considerable attention as a challenging combinatorial optimization problem with notable practical implications. Steered by its pivotal applications in power engineering, the present work formulates a novel generalization of UFP, wherein demands and capacities in the input instance are monotone step functions over the set of edges. As an initial step towards tackling this generalization, we draw on and extend ideas from prior research to devise a quasi-polynomial time approximation scheme under the premise that the demands and capacities lie in a quasi-polynomial range. Second, retaining the same assumption, an efficient logarithmic approximation is introduced for the single-source variant of the problem. Finally, we round up the contributions by designing a (kind of) black-box reduction that, under some mild conditions, allows to translate LP-based approximation algorithms for the studied problem into their counterparts for the Alternating Current Optimal Power Flow problem—a fundamental workflow in operation and control of power systems.

Appendices

Appendix

A Proof of Lemma 2

Proof

For $r\in [d]$, consider the graph of the fractional profile $\sum _{k\in \mathcal S^q}f_k^r(e)\tilde{x}_k$ illustrated in Fig. 3. For $p\in [P_r]$, slice the region between the horizontal axis and horizontal line at height $h^{q,p,r}$ with $\frac{1}{\epsilon }+1$ horizontal lines, with inter-distance $\epsilon h^{q,p,r}$. The intersections of the optimal profile with these lines define a monotone function $g^{q,r}$, as pictured in Fig. 3, with $g^{q,r}(e)\in \{l\epsilon h^{p,r}:~l\in \{0,1\ldots ,1/\epsilon \},~p\in [P_r]\}$, for all $e\in \mathcal {E}$. We adopt a greedy procedure, explained in Algorithm 3 below, to remove a set of demands from $\mathcal S^q$ in each interval $\mathcal {E} _p^r$ such that the remaining set of demands fractionally fits below $g^{q,r}$ (see lines 2-9). The algorithm proceeds by removing the “left-most" set of demands that minimally ensures that the remaining ones in $\mathcal S^q$ can be packed under capacity $g^{q,r}$. This defines an intermediate fractional vector $\bar{{\bar{x}}}$ for separable d-USFP-R[$\mathcal S^q,g^q$], where $g^q=(g^{q,r})_{r\in [d]}$, which can be converted to a basic feasible solution (BFS) with the same or better objective value. Lastly, the fractional components of $\bar{{\bar{x}}}$ are rounded down yielding an integral solution ${\hat{x}}$.

Fig. 3

A profile and its $(h,\epsilon )$-restriction

We first show that condition (i) holds when ${\hat{x}}$ is replaced by ${\bar{x}}$. For $r\in [d]$, let $\mathcal J^{r}(e_i)$ be the set of demands $k\in \mathcal S^q$ for which ${\bar{x}}_k$ was set to 0 in step 7 when considering edge $e_i\in \mathcal {E}$. Consider an edge $e\in \mathcal {E} _p^r$ such that $\sum _{k\in \mathcal S^q}f_k^r(e){\bar{x}}_k>0$. Note that $0\le \sum _{k\in \mathcal S^q}f_k^r(e){\tilde{x}}_k-g^{q,r}(e)\le \epsilon h^{q,p,r}$ by (12) and the definition of $g^{q,r}$. By the monotonicity of $f^r_k(\cdot )$ and the condition of the while-loop in step 5 we have

$$\begin{aligned} \sum _{k\in \mathcal S^q}f_k^r(e){\bar{x}}_k&=\sum _{k\in \mathcal S^q}f_k^r(e){\tilde{x}}_k-\sum _{i:~e_i\le e}\sum _{k\in \mathcal J^{r}(e_i)}f_k^r(e){\tilde{x}}_k\\&\le \sum _{k\in \mathcal S^q}f_k^r(e){\tilde{x}}_k-\sum _{i:~e_i\le e}\sum _{k\in \mathcal J^{r}(e_i)}f_k^r(e_i){\tilde{x}}_k\\&\le \sum _{k\in \mathcal S^q}f_k^r(e){\tilde{x}}_k-\epsilon h^{q,p,r}\\&\le g^{q,r}(e). \end{aligned}$$

Since ${\bar{x}}$ is feasible for d-USFP-R[$\mathcal S^q,g^q$], one can obtain a BFS $\bar{{\bar{x}}}$ for the same linear program with $\sum _{k\in \mathcal S^q}u_k\bar{{\bar{x}}}_k\ge \sum _{k\in \mathcal S}u_k{\bar{x}}_k$ as in step 10 of procedure Modify. Then, round down the fractional components in $\bar{\bar{x}}$ to obtain an integral solution ${\hat{x}}$. Note that, for all $e\in \mathcal {E}$,

$$\begin{aligned} \sum _{k\in \mathcal S^q}f_k^r(e){\hat{x}}_k\le \sum _{k\in \mathcal S^q}f_k^r(e)\bar{{\bar{x}}}_k\le g^{q,r}(e)\le \sum _{k\in \mathcal S^q}f_k^r(e)\tilde{x}_k, \end{aligned}$$

and hence (i) holds.

Note that the total fractional utility of demands removed by Algorithm 3 in steps 2-9 is

$$\begin{aligned} \sum _{r\in [d],~e\in \mathcal {E} }\sum _{k\in \mathcal J^r(e)}u_k{\tilde{x}}_k&= \sum _{r\in [d]:~H^{P_r,r}>0}\sum _{p=1}^{P_r}\sum _{e\in \mathcal {E} _p^r}\sum _{k\in \mathcal J^r(e)}u_k{\tilde{x}}_k\\&\le \sum _{r\in \mathcal H^{q}}\sum _{p=1}^{P_r}\sum _{e\in \mathcal {E} _p^r}\sum _{k\in \mathcal J^r(e)} \frac{1}{H^{q,p,r}} \sum _{t=1}^{T_r}a_k^{r,t}b^{r,t}(e_{{\overline{i}}(p,r)}){\tilde{x}}_k\\&=\sum _{r\in \mathcal H^{q}}\sum _{p=1}^{P_r}\frac{1}{H^{q,p,r}} \sum _{e\in \mathcal {E} _p^r}\sum _{k\in \mathcal J^r(e)} f_k^r(e_{{\overline{i}}(p,r)}){\tilde{x}}_k\\&\le \sum _{r\in \mathcal H^{q}}\sum _{p=1}^{P_r}\frac{C_r}{H^{q,p,r}} \left( \sum _{e\in \mathcal {E} _p^r}\sum _{k\in \mathcal J^r(e)} f_k^r(e){\tilde{x}}_k\right) \\&\le \sum _{r\in \mathcal H^{q}0}\sum _{p=1}^{P_r}\frac{C_r}{H^{q,p,r}}(\epsilon h^{q,p,r}+B^{q,p,r}), \end{aligned}$$

where we use the fact that $k\in \mathcal I^q$ in the first inequality, property (4) in the second inequality, and $\underline{f}_k^{p,r}\le B^{q,p,r}$ and the condition of the while-loop in step 5 in the last inequality. (Note that we sum above over $r\in [d]$ such that in $H^{q,P_r,r}>0$ since $k\in \mathcal J^r(e)$ implies that $f_k^r(e)>0$, which in turn implies by (11) that $H^{q,P_r,r}>0$.)

It follows that

$$\begin{aligned} \sum _{k \in \mathcal S^q}u_k{\bar{x}}_k&\ge \sum _{k \in \mathcal S^q}u_k{\tilde{x}}_k-\sum _{r,e}\sum _{k\in \mathcal J^r(e)}u_k{\tilde{x}}_k \nonumber \\&\ge \sum _{k \in \mathcal S^q}u_k\tilde{x}_k-\sum _{r\in \mathcal H^{q}}\sum _{p=1}^{P_r}\frac{C_r}{H^{q,p,r}}(\epsilon h^{q,p,r}+B^{q,p,r}). \end{aligned}$$

(58)

By the monotonicity of the functions $f^r_k(\cdot )$, d-USFP-R[$\mathcal S,g^q$] has only $\frac{1}{\epsilon }\sum _{r=1}^dP_r$ non-redundant packing inequalities of the form (1). It follows that the BFS $\bar{{\bar{x}}}$ computed in step 10 has at most $\frac{1}{\epsilon }\sum _{r=1}^dP_r$ fractional components $\bar{{\bar{x}}}\in (0,1)$. Thus,

$$\begin{aligned} \sum _{k \in \mathcal S^q}u_k{{\hat{x}}}_k&=\sum _{k \in \mathcal S^q}u_k\bar{{\bar{x}}}_k-\sum _{k \in \mathcal S^q:~\bar{{\bar{x}}}_k\in (0,1)}u_k\bar{{\bar{x}}}_k \nonumber \\&\ge \sum _{k \in \mathcal S^q}u_k{\bar{x}}_k-\frac{1}{\epsilon }\sum _{r=1}^dP_r\cdot \max _ku_k\bar{{\bar{x}}}_k \nonumber \\&\ge \sum _{k \in \mathcal S^q}u_k\bar{ x}_k-\frac{1}{\epsilon }\frac{\sum _{r=1}^dP_r}{\sum _{r\in \mathcal H^{q}}P_r}\sum _{r\in \mathcal H^{q}}\frac{P_rB^{q,P_r,r}}{H^{q,P_r,r}}, \end{aligned}$$

(59)

where we use in the last inequality that $\bar{{\bar{x}}}_k\le 1$ and

$$\begin{aligned} u_k&\le \frac{\sum _{r\in \mathcal H^{q}}P_r\sum _{t=1}^{P_r}a_k^{r,t}b^{r,t}(e_{n})/H^{q,P_r,r}}{\sum _{r\in \mathcal H^{q}}P_r}=\frac{\sum _{r\in \mathcal H^{q}}P_rf_k^r(e_n)/H^{q,P_r,r}}{\sum _{r\in \mathcal H^{q}}P_r}\\&\le \frac{\sum _{r\in \mathcal H^{q}}P_rB^{q,P_r,r}/H^{q,P_r,r}}{\sum _{r\in \mathcal H^{q}}P_r}, \end{aligned}$$

for $k\in \mathcal S^q$. Condition (ii) follows from (58) and (59). $\square$

B Proof of Lemma 4

Proof

The analysis follows the same lines as in Gan et al. (2015), Low (2014b), Huang et al. (2017) and is sketched here for completeness. Let $F''=(s_0'', x', v'',\ell '',S'')$ be an optimal solution of cOPF$[x']$, which can be found (to within any desired accuracy) in polynomial time, by solving a convex program. Consider the following problem.

$$\begin{aligned} \textsc {(cOPF}'[x']) \quad&\min _{\begin{array}{c} s_0, x, v,\ell , S \;\; \end{array}}\sum _{e\in \mathcal {E} }\ell _e,&\nonumber \\ \text {s.t.} \ {}&(29)-(34), (36), (37)\nonumber&\\&x=x'&\end{aligned}$$

(60)

$$\begin{aligned}&f_{\textsc {OPF}}(s_0, x)\ge f_{\textsc {OPF}}(s_0'',x').&\end{aligned}$$

(61)

Clearly, cOPF $'[x']$ is feasible as $F''$ satisfies all its constraints. Hence, it has an optimal solution $F'=(s_0', x', v',\ell ',S')$, which we claim satisfies the statement of the lemma. Suppose, for the sake of contradiction, that there exists an edge (h, t) such that $\ell _{h,t}' > \tfrac{|S_{h,t}'|^2}{v'_h}$. In the sequel, we construct a feasible solution ${\tilde{F}}=({\tilde{s}}_0, x', {\tilde{v}}, {\tilde{\ell }}, \tilde{S})$ for cOPF $'[x']$ such that $\sum _{e\in \mathcal {E} }{\tilde{\ell }}_e<\sum _{e\in \mathcal {E} }\ell _e'$, leading to a contradiction.

Apply the forward-backward sweep algorithm, illustrated in Alg. 4, on the solution $F'$ to obtain a feasible solution ${\tilde{F}}$.

We show the feasibility of the solution ${\tilde{F}}$. By Steps 6, 7 and 10 of Alg. 4, all equality constraints of cOPF $'[x']$ are satisfied. By Step 5 and the feasibility of $F'$, we also have

$$\begin{aligned} {\tilde{\ell }}_e \le \ell _e'\le {\overline{\ell }}_e\text { for all } e \in \mathcal {E} . \end{aligned}$$

(62)

Next, by rewriting ${\tilde{S}}_{i,j}$, recursively substituting from the leaves, we get

$$\begin{aligned} {\tilde{S}}_{i,j}=\sum _{k \in \mathcal N_j} s_k {\tilde{x}}_k + \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}}z_e {\tilde{\ell }}_{e}. \end{aligned}$$

(63)

Write $\Delta \ell _e \triangleq {\tilde{\ell }}_e - \ell _e'\le 0$, $\Delta S_e \triangleq {\tilde{S}}_e - S_e'$, and $\Delta |S_e|^2 \triangleq |{\tilde{S}}_e|^2 - |S_e'|^2$, for $e\in \mathcal {E}$. Let ${\widehat{S}}_{j} \triangleq \sum _{k \in \mathcal N_j} s_k x_k'$, $\tilde{L}_{i,j} \triangleq \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}}z_e \tilde{\ell }_e$, and $L'_{i,j} \triangleq \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}}z_e \ell _e'$. Note by (63) that $\tilde{S}_{i,j} = {\widehat{S}}_{j} + {\tilde{L}}_{i,j}$ and, similarly, $S_{i,j} = {\widehat{S}}_{j} + L_{i,j}'$. It follows that, for all $(i,j)\in \mathcal {E}$,

$$\begin{aligned} \Delta S_{i,j}&= {\tilde{L}}_{i,j}-L_{i,j}'=\sum _{e \in \mathcal {E} _j\cup \{(i,j)\}}z_{e} \Delta \ell _{e} \le 0, \end{aligned}$$

(64)

where the inequality follows by assumption A1. In particular, for $(i,j)=(0,1)$, we obtain

$$\begin{aligned} {\tilde{s}}_0^{\textrm{R}}&= -{\tilde{S}}_{0,1}^{\textrm{R}} \ge - S_{0,1}'^{\textrm{R}} = s_0'^{\textrm{R}}, \end{aligned}$$

(65)

implying by A0 that $f_0({\tilde{s}}_0^{\textrm{R}}) \ge f_0(s_0'^{\textrm{R}})$ and hence (61) is satisfied.

Furthermore,

$$\begin{aligned} \Delta |S_{i,j}|^2&= |{\tilde{S}}_{i,j}|^2 - |S_{i,j}'|^2 \end{aligned}$$

(66)

$$\begin{aligned}&= ({\tilde{S}}_{i,j}^{\textrm{R}})^2 - (S_{i,j}'^{\textrm{R}})^2 +({\tilde{S}}_{i,j}^{\textrm{I}})^2 - (S_{i,j}'^{\textrm{I}})^2 \end{aligned}$$

(67)

$$\begin{aligned}&=\Delta S_{i,j}^{\textrm{R}} ({\tilde{S}}_{i,j}^{\textrm{R}} + S_{i,j}'^{\textrm{R}}) + \Delta S_{i,j}^{\textrm{I}} ({\tilde{S}}_{i,j}^{\textrm{I}} + S_{i,j}'^{\textrm{I}})\end{aligned}$$

(68)

$$\begin{aligned}&=\sum _{e \in \mathcal {E} _j\cup \{(i,j)\}} z_e^{\textrm{R}} \Delta \ell _e \big ( 2 {\widehat{S}}^{\textrm{R}}_{j} + {\tilde{L}}_{i,j}^{\textrm{R}} + L_{i,j}'^{\textrm{R}} \big ) \nonumber \\&\quad + \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}} z_e^{\textrm{I}} \Delta \ell _e \big ( 2 {\widehat{S}}^{\textrm{I}}_{j} + {\tilde{L}}_{i,j}^{\textrm{I}} + L_{i,j}'^{\textrm{I}} \big ) \end{aligned}$$

(69)

$$\begin{aligned}&= \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}} 2\Delta \ell _e \textrm{Re}(z_e^*{\widehat{S}}_j)+ \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}} \Delta \ell _e \textrm{Re}(z_e^*{\tilde{L}}_{i,j})\nonumber \\&\quad +\sum _{e \in \mathcal {E} _j\cup \{(i,j)\}} \Delta \ell _e\textrm{Re}(z_e^*L'_{i,j})\le 0, \end{aligned}$$

(70)

where Eqn. (70) follows by A1, A3 (or A4 $'$) and $\Delta \ell _e \le 0$. Therefore, by the feasibility of $S_e$,

$$\begin{aligned} |\tilde{S}_{e}| \le |S_{e}'| \le {\overline{S}}_{e} \quad \text {for all } e \in \mathcal {E} . \end{aligned}$$

(71)

Note that, by A1, the inequalities in  (71) also imply that the reverse power constraint in (33) is satisfied for ${\tilde{S}}$.

Rewrite Cons. (31) by recursively substituting $\tilde{v}_j$, for j moving away from the root, and then substituting for ${\tilde{S}}_{h,t}$ using (63):

$$\begin{aligned} {\tilde{v}}_j&= v_0 -2\sum _{ (h,t) \in \mathcal {P}_{j} } \textrm{Re}(z_{h,t}^*{\tilde{S}}_{h,t}) + \sum _{( h,t) \in \mathcal {P}_{j} } |z_{h,t}|^2 {\tilde{\ell }}_{h,t}\nonumber \\&=v_0 -2 \sum _{ (h,t) \in \mathcal {P}_{j} } \textrm{Re}\Big (z^*_{h,t}\big (\sum _{k\in \mathcal N_t} s_k {\tilde{x}}_k + \sum _{e \in \mathcal {E} _t\cup \{(h,t)\}} z_{e} {\tilde{\ell }}_{e} \big )\Big ) + \sum _{ (h,t) \in \mathcal {P}_{j} } |z_{h,t}|^2 {\tilde{\ell }}_{h,t},\nonumber \\&=v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) {\tilde{x}}_k - 2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\Big ( z_{h,t}^* \sum _{e \in \mathcal {E} _t} z_{e} {\tilde{\ell }}_{e}\Big ) \nonumber \\&\quad - 2\sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 {\tilde{\ell }}_{h,t} +\sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 {\tilde{\ell }}_{h,t}, \end{aligned}$$

(72)

where the last statement follows from exchanging the summation operators, and $z^*_{e} z_{e} = |z_{e}|^2$. Thus,

$$\begin{aligned} {\tilde{v}}_j&=v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) {\tilde{x}}_k \nonumber \\&\quad - \Big (2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\big ( z_{h,t}^* \sum _{e \in \mathcal {E} _t} z_{e} {\tilde{\ell }}_{e}\big ) + \sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 {\tilde{\ell }}_{h,t}\Big ) \nonumber \\&\le v_0 <{\overline{v}}_j, \end{aligned}$$

(73)

where the first inequality follows by A1 and A3, and the last inequality follows by A2. Since ${\tilde{\ell }}_e \le \ell '_e$ and ${\tilde{x}} = x'$, we get by A1 and the feasibility of $F'$,

$$\begin{aligned} {\tilde{v}}_j&\ge v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) x_k' \nonumber \\&\quad - \Big (2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\big ( z_{h,t} \sum _{e \in \mathcal {E} _t} z_{e} \ell _{e}\big ) + \sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 \ell _{h,t}\Big ) \nonumber \\&=v_j'\ge {\underline{v}}_j. \end{aligned}$$

(74)

By Ineqs. (71) and (74), ${\tilde{\ell }}_{i,j} = \frac{| S_{i,j} '|^2}{v_i'} \ge \frac{| {\tilde{S}}_{i,j} |^2}{{\tilde{v}}_i}$, hence, ${\tilde{F}}$ is feasible.

Finally, by the first inequality in (62) and the fact that $\ell _{h,t} '> \tfrac{|S_{h,t}|^2}{v_h}={\tilde{\ell }}_{h,t}$, we have $\sum _{e\in \mathcal {E} }{\tilde{\ell }}_{e}<\sum _{e\in \mathcal {E} } \ell _{e}'$, contradicting the optimality of $F'$ for cOPF $'[x']$.

C Proof of Lemma 5

Proof

The argument is similar to that in Lemma 4. We apply a slightly modified version of Alg. 4 on the solution $F'$ to obtain a feasible solution ${\tilde{F}}$. Replace steps 1 and 5 in Alg. 4, respectively, by:

$$\begin{aligned} \text {1: }{\tilde{x}} \leftarrow {\bar{x}}; {\tilde{v}}_0 \leftarrow v_0,\hbox { and 5: }{\tilde{\ell }}_{i,j} \leftarrow \ell '_{i,j}. \end{aligned}$$

(75)

By Steps 6, 7 and 10 of the (modified) algorithm, all equality constraints of (rcOPF $[{\bar{x}}]$) are satisfied. By (modified) Step 5 and the feasibility of $F'$, we also have

$$\begin{aligned} {\tilde{\ell }}_e =\ell _e'\le {\overline{\ell }}_e\text { for all }e \in \mathcal {E} . \end{aligned}$$

(76)

Write $\Delta S_e \triangleq {\tilde{S}}_e - S_e'$, and $\Delta |S_e|^2 \triangleq |{\tilde{S}}_e|^2 - |S_e'|^2$, for $e\in \mathcal {E}$. Let $S_{j}' \triangleq \sum _{k \in \mathcal N_j} s_k x_k'$, $\tilde{S}_{j}\triangleq \sum _{k \in \mathcal N_j} s_k {\tilde{x}}_k$, and $\tilde{L}_{i,j} \triangleq \sum _{e \in \mathcal {E} _j\cup \{(i,j)\}}z_e \tilde{\ell }_e$. Note by (63) that ${\tilde{S}}_{i,j} = {\tilde{S}}_{j} + {\tilde{L}}_{i,j}$ and, $S_{i,j}' = S_{j}' + {\tilde{L}}_{i,j}$. It follows that, for all $(i,j)\in \mathcal {E}$,

$$\begin{aligned} \Delta S_{i,j}&= {\tilde{S}}_{j}-S_{j}'=\sum _{k \in \mathcal N_j} s_k {\bar{x}}_k-\sum _{k \in \mathcal N_j} s_k x_k' \le 0, \end{aligned}$$

(77)

where the inequality follows from (40) and (41). In particular, for $(i,j)=(0,1)$, we obtain

$$\begin{aligned} {\tilde{s}}_0^{\textrm{R}}&= -{\tilde{S}}_{0,1}^{\textrm{R}} \ge - S_{0,1}'^{\textrm{R}} = s_0'^{\textrm{R}}, \end{aligned}$$

(78)

implying by A0 $'$ that $f_0({\tilde{s}}_0^{\textrm{R}} \cos \phi + {\tilde{s}}_0^{\textrm{I}} \sin \phi ) \ge f_0(s_0'^{\textrm{R}} \cos \phi + s_0'^{\textrm{I}} \sin \phi ))$ and hence $f_{\textsc {OPF}}(\tilde{s}_0,{\tilde{x}})\ge (1-\varepsilon )f_{\textsc {OPF}}(s_0',x')$ follows from (38) and (42).

Furthermore,

$$\begin{aligned} \Delta |S_{i,j}|^2&= |{\tilde{S}}_{i,j}|^2 - |S_{i,j}'|^2\\&= ({\tilde{S}}_{i,j}^{\textrm{R}})^2 - (S_{i,j}'^{\textrm{R}})^2 +({\tilde{S}}_{i,j}^{\textrm{I}})^2 - (S_{i,j}'^{\textrm{I}})^2 \\&=\Delta S_{i,j}^{\textrm{R}} ({\tilde{S}}_{i,j}^{\textrm{R}} + S_{i,j}'^{\textrm{R}}) + \Delta S_{i,j}^{\textrm{I}} ({\tilde{S}}_{i,j}^{\textrm{I}} + S_{i,j}'^{\textrm{I}})\\&=\Delta S_{i,j}^{\textrm{R}}({\tilde{S}}_j^{\textrm{R}}+S_j'^{\textrm{R}}+2\tilde{L}_{i,j}^{\textrm{R}})+ \Delta S_{i,j}^{\textrm{I}}({\tilde{S}}_j^{\textrm{I}}+S_j'^{\textrm{I}}+2{\tilde{L}}_{i,j}^{\textrm{I}})\le 0, \end{aligned}$$

where the last inequality follows by A1, A4 $'$ and (77). Therefore,

$$\begin{aligned} |\tilde{S}_{i,j}| \le |S'_{i,j}| \le {\overline{S}}_{i,j}. \end{aligned}$$

(79)

Next, we show ${\underline{v}}_j \le {\tilde{v}}_j \le {\overline{v}}_j$. As in (72), rewrite Cons. (31) by recursively substituting $v_j'$, for j moving away from the root, and then substituting for ${\tilde{S}}_{h,t}$ using (63):

$$\begin{aligned} v_j'&=v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) x_k' \nonumber \\&\quad - \Big (2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\big ( z_{h,t}^* \sum _{e \in \mathcal {E} _t} z_{e} \ell '_{e}\big ) + \sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 \ell '_{h,t}\Big ) \end{aligned}$$

(80)

A similar equation can be derived for ${\tilde{v}}_j$, where $x'$ and $\ell '$ in (80) are replaced by ${\tilde{x}}$ and ${\tilde{\ell }}$, respectively. By assumptions A2 and A3, we have

$$\begin{aligned} {\tilde{v}}_j&=v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) {\tilde{x}}_k \nonumber \\&\quad - \Big (2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\big ( z_{h,t}^* \sum _{e \in \mathcal {E} _t} z_{e} {\tilde{\ell }}_{e}\big ) + \sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 {\tilde{\ell }}_{h,t}\Big ) \nonumber \\&\le v_0 <{\overline{v}}_j. \end{aligned}$$

Moreover, since ${\tilde{\ell }}_e = \ell '_e$ and ${\tilde{x}} = \bar{x}$ satisfies (39), we get by A1 and the feasibility of $F'$,

$$\begin{aligned} {\tilde{v}}_j&\ge v_0 -2 \sum _{k \in \mathcal N} \textrm{Re}\Big ( \sum _{(h,t)\in \mathcal {P}_k \cap \mathcal {P}_j} z^*_{h,t} s_k\Big ) x_k' \nonumber \\&\quad - \Big (2 \sum _{(h,t)\in \mathcal {P}_j} \textrm{Re}\big ( z_{h,t} \sum _{e \in \mathcal {E} _t} z_{e} \ell _{e}'\big ) + \sum _{ (h,t) \in \mathcal {P}_{j} }|z_{h,t}|^2 \ell _{h,t}'\Big ) \nonumber \\&=v_j'\ge {\underline{v}}_j. \end{aligned}$$

(81)

Finally, by inequalities (79) and (81), $\tilde{\ell }_{i,j} = \ell _{i,j}' = \frac{|S'_{i,j}|^2}{v'_i} \ge \frac{|{\tilde{S}}_{i,j}|^2}{{\tilde{v}}_i}$, hence $\tilde{\ell }_{i,j}$ satisfies Cons. (37). $\square$