* If $ \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(0,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),0,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})  > p_{\textnormal{max}}$, then the solution of $\beta$ to (<ref>)–(<ref>) cannot be $0$  (which along with (<ref>) means  $\beta>0$), since setting $\beta$ as $0$ violates (<ref>). We use $\beta>0$ in (<ref>) to get

∑_n ∈𝒩 p_n(ᾰ(β,T,ζ|$\bigstar$),β,ζ|$\bigstar$)  = p_max.

* If $ \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(0,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),0,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})  \leq p_{\textnormal{max}}$, then $0$ is a solution of $\beta$ to (<ref>)–(<ref>).

$\textnormal{Alg-Solve-}\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$.

if $ \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(0,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),0,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})  \leq p_{\textnormal{max}}$, then set $\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ as $0$;

else with function $C(\cdot)$ being set as $\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ (whose computation requires Algorithm 1.2.2.1 and Algorithm 1.2.1's Subprocedure 1), and $C_{\textnormal{target}}$ being set as $p_{\textnormal{max}} $, run $\textnormal{Bisection-Search-with-No-Known-Upper-Bound} \newline ([\boldsymbol{\zeta}, \textnormal{\small$\bigstar$}],\textnormal{function~}C(\cdot),C_{\textnormal{target}})$ based on Algorithm <ref> on Page algo:BisectionSearchwithNoUpperBound to obtain the desired $\beta$ satisfying Eq. (<ref>), and set this $\beta$ as   $\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$;

Algorithm $\text{2.1}$: Computing $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$ defined in Proposition 2.1 using $\mathcal{S}_{2.1}$ defined in (counterty).   Recall from (counterty) that $\mathcal{S}_{2.1}$ includes (<ref>),  (<ref>), and (<ref>e). Then for each $n \in \mathcal{N}$, Proposition 2.1 means  $\grave{{\zeta}}_n(T\mid \textnormal{\small$\bigstar$})$ is ${\zeta_n}$ which satisfies the following:

based on (<ref>), we ensure
ζ_n ·(t_n(b́_n(T,ζ|$\bigstar$),ṕ_n(T,ζ|$\bigstar$),s_n,f́_n^MS(T,ζ|$\bigstar$),
f́_n^VU(T,ζ|$\bigstar$)) - T) = 0;
based on (<ref>), we ensure
f́_n^VU(T,ζ|$\bigstar$)) ≤T ;
based on (<ref>e), we ensure ζ_n  ≥0.

Given $n \in \mathcal{N}$, we can discuss two cases for ${\zeta_n}$ as follows:

Case 1: If setting $\zeta_n$ to $0$ violates (<ref>), then $\zeta_n$ must be strictly positive, which is used in (<ref>) to show that the inequality in (<ref>) actually becomes equality in this case.

Case 2: If setting $\zeta_n$ to $0$ satisfies (<ref>), then we can just set $\zeta_n$ as $0$.

Summarizing the above two cases, we know that after defining

ζ^(n) := [ζ_1, ζ_2, …, ζ_n-1, 0,ζ_n+1, …, ζ_N],  and

h_n(ζ|T)  :=
-ζ_n, if t_n(b́_n(ζ^(n)| $\bigstar$),ṕ_n(ζ^(n)| $\bigstar$),s_n,
f́_n^MS(ζ^(n)|$\bigstar$),f́_n^VU(ζ^(n)| $\bigstar$)) ≤ T,   (<ref>a)
f́_n^VU(T,ζ|$\bigstar$)) - T,  otherwise,       (<ref>b)

setting $\boldsymbol{\zeta}$ as $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$ always ensures that $h_n(\boldsymbol{\zeta}\mid T)=0$. Letting $n$ iterate through $\mathcal{N}$ and defining

h(ζ|T) := [h_n(ζ|T)|_n ∈𝒩],

we know that setting $\boldsymbol{\zeta}$ as $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$   ensures that

h(ζ|T)  equals the   $N$-dimensional zero vector  0.

We define $\zeta_n^{\textnormal{upper}}$ such that when $\boldsymbol{\zeta}$ is $[0^{n-1},\zeta_n^{\textnormal{upper}},0^{N-n}]$,

h_n(ζ|T) = 0.

We will prove the following results and Lemma <ref>:

• $\grave{{\zeta}}_n(T\mid \textnormal{\small$\bigstar$}) \leq \zeta_n^{\textnormal{upper}}$, which along with $\zeta_n \geq 0$ means
$\zeta_n \in [0,\zeta_n^{\textnormal{upper}}]$;
• Given any $n \in \mathcal{N}$, for any $\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, \zeta_{n+1}, \ldots, \zeta_N$,

we have  $h_n([\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, \zeta_n^{\textnormal{upper}},\zeta_{n+1}, \ldots, \zeta_N]\mid T) \leq 0$;

• Given any $n \in \mathcal{N}$,  for any $\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, \zeta_{n+1}, \ldots, \zeta_N$,

we have  $h_n([\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, 0,\zeta_{n+1}, \ldots, \zeta_N]\mid T) \geq 0$;
i.e., $h_n(\mathring{\boldsymbol{\zeta}}^{(n)}) \geq 0$ for $\mathring{\boldsymbol{\zeta}}^{(n)}$ defined in (<ref>).

Given $\boldsymbol{\zeta} $ and $ T$, given  $n \in \mathcal{N}$, given $\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, \zeta_{n+1}, \ldots, \zeta_N$, then $h_n(\boldsymbol{\zeta}~\mid~T)$ is  as $\zeta_n$ increases.

We prove the above Result (<ref>). From Lemma <ref> and (<ref>), it holds that

h_n([0^n-1,ζ_n^upper,0^N-n]|T) = 0 = h_n(ζ|T)

≤h_n([0^n-1,ζ_n,0^N-n]|T) ≤0,

which with Lemma <ref> means $\grave{{\zeta}}_n(T\mid \textnormal{\small$\bigstar$}) \leq \zeta_n^{\textnormal{upper}}$.

The above Result (<ref>) clearly follows from Result (<ref>) and Lemma <ref>.

We prove the above Result (<ref>). From (<ref>), it holds that

h_n(ζ^(n))  := {
0, if t_n(b́_n(ζ^(n)|$\bigstar$),ṕ_n(ζ^(n)|$\bigstar$),

s_n,f́_n^MS(ζ^(n)|$\bigstar$),f́_n^VU(ζ^(n)|$\bigstar$)) ≤T,

s_n,f́_n^MS(ζ^(n)|$\bigstar$),f́_n^VU(ζ^(n)|$\bigstar$)) - T,
}  ≥0.
With the above Results (<ref>) (<ref>) (<ref>),
we apply the Poincaré–Miranda theorem [39]
and solve (<ref>) to obtain $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$ using the multivariate bisection algorithm of [40]. The pseudocode is given as Algorithm 2.1 below.  Readers may wonder why the multivariate bisection is not used to jointly solve $[\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},\alpha,\beta,\gamma,\boldsymbol{\delta}]$. The reason is that the conditions to use the multivariate bisection are quite strict; e.g., Results (<ref>) (<ref>) (<ref>) are for any $\zeta_1, \zeta_2, \ldots, \zeta_{n-1}, \zeta_{n+1}, \ldots, \zeta_N$ given any $n \in \mathcal{N}$. We do not have such strong conditions if we try to solve $[\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},\alpha,\beta,\gamma,\boldsymbol{\delta}]$ together.

$\textnormal{Alg-Solve-}\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$.

For $\boldsymbol{\zeta} :=[\zeta_1, \zeta_2, \ldots,  \zeta_N] \in \prod_{n \in \mathcal{N}} [0,\zeta_n^{\textnormal{upper}}]$ for $\zeta_n^{\textnormal{upper}}$ defined in (<ref>), given Results (<ref>) (<ref>) (<ref>), use the multivariate bisection algorithm proposed by [40]
to obtain $\boldsymbol{\zeta}$ which induces $\boldsymbol{h}(\boldsymbol{\zeta}\mid T)$ to be the $N$-dimensional zero vector $\boldsymbol{0}$, to achieve the tolerance level of  $\|\boldsymbol{h}(\boldsymbol{\zeta}\mid T)\|_2 \leq \epsilon_5$, where computing $\boldsymbol{h}(\boldsymbol{\zeta}\mid T)$ defined in (<ref>) and (<ref>) will call Algorithm 1.1 (resp., Algorithm 1.2) to compute $\acute{f}_n^{\textnormal{MS}}(\iffalse T,\fi\mathring{\boldsymbol{\zeta}}^{(n)}\mid \textnormal{\small$\bigstar$}) $ and $ \acute{f}_n^{\textnormal{VU}}(\iffalse T,\fi\mathring{\boldsymbol{\zeta}}^{(n)}\mid \textnormal{\small$\bigstar$})$ as well as $\acute{f}_n^{\textnormal{MS}}(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) $ and $ \acute{f}_n^{\textnormal{VU}}(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$
(resp., $\acute{b}_n(\iffalse T,\fi\mathring{\boldsymbol{\zeta}}^{(n)}\mid \textnormal{\small$\bigstar$})$ and $\acute{p}_n(\iffalse T,\fi\mathring{\boldsymbol{\zeta}}^{(n)}\mid \textnormal{\small$\bigstar$})$ as well as $\acute{b}_n(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ and $\acute{p}_n(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$) as inputs to the function $t_n()$ defined in (<ref>), where $\mathring{\boldsymbol{\zeta}}^{(n)}$ is  defined in (<ref>);

Return the obtained $\boldsymbol{\zeta}$ as $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$;

Algorithm $\text{2.2}$: Computing $T^{\#}(\textnormal{\small$\bigstar$})$ according to Proposition 2.2 using $\mathcal{S}_{2.2}$ defined in (countertt).  Recall from (countertt) that $\mathcal{S}_{2.2}$ includes (<ref>). Then Proposition 2.2 defines  $T^{\#}(\textnormal{\small$\bigstar$})$ as the solution of $T$ to

∑_n ∈𝒩 ζ̀_n(T|$\bigstar$) = y c_t.

Run $\textnormal{Bisection-Search-with-No-Known-Upper-Bound}$ $(\textnormal{\small$\bigstar$},\sum_{n \in \mathcal{N}} \grave{\zeta}_n(T\mid \textnormal{\small$\bigstar$}), y c_{\hspace{0.8pt}\textnormal{t}})$ based on Algorithm <ref> on Page algo:BisectionSearchwithNoUpperBound  to obtain the desired $T$ satisfying Eq. (<ref>), and set this $T$ as $T^{\#}(\textnormal{\small$\bigstar$})$, where computing $\sum_{n \in \mathcal{N}} \grave{\zeta}_n(T\mid \textnormal{\small$\bigstar$})$ will call Algorithm 2.1;

The bisection method is used repeatedly in the algorithms above. The pseudocodes are given below.

$\textnormal{Bisection-Search-with-No-Known-Upper-Bound}$ $(\boldsymbol{v},\textnormal{function~}C(u, \boldsymbol{v}),C_{\textnormal{target}})$, which returns $u \in [0,\infty)$ such that $C(u, \boldsymbol{v})$ equals (or is arbitrarily close to) $C_{\textnormal{target}}$ given $\boldsymbol{v}$, where $C(u, \boldsymbol{v})$ is  in $u$ given $\boldsymbol{v}$.

Randomly pick $u^{(0)}$ from $(0,\infty)$;

if $C(u^{(0)}, \boldsymbol{v})=C_{\textnormal{target}}$:  return $u^{(0)}$ as the desired $u$;

if $C(u^{(0)}, \boldsymbol{v})<C_{\textnormal{target}}$  //In this case, the solution $u$ is in $[0,u^{(0)})$

Use $\textnormal{Standard-Bisection-Search}(\boldsymbol{v},\textnormal{function~}C(u, \boldsymbol{v}),$ $C_{\textnormal{target}}, 0,u^{(0)})$ based on Algorithm <ref> to find the result, and return it as the desired $u$;

if $C(u^{(0)}, \boldsymbol{v})>C_{\textnormal{target}}$
//In this case, the solution $u$ is in $(u^{(0)},\infty)$

Find $i\geq 0$ such that $C(u^{(0)}\cdot 2^i, \boldsymbol{v})>C_{\textnormal{target}}$ but $C(u^{(0)}\cdot 2^{i+1}, \boldsymbol{v}) \leq C_{\textnormal{target}}$

if $C(u^{(0)}\cdot 2^{i+1}, \boldsymbol{v}) = C_{\textnormal{target}}$:  return $u^{(0)}\cdot 2^{i+1}$  as the desired $u$;

elsif $C(u^{(0)}\cdot 2^{i+1}, \boldsymbol{v}) < C_{\textnormal{target}}$

Use $\textnormal{Standard-Bisection-Search}(\boldsymbol{v},$$\textnormal{function~}C(u, \boldsymbol{v}),C_{\textnormal{target}}, u^{(0)}\cdot 2^i,u^{(0)}\cdot 2^{i+1})$  based on Algorithm <ref> to find the result,  and return it as the desired $u$;

$\textnormal{Standard-Bisection-Search}$ $(\boldsymbol{v},\textnormal{function~}C(u, \boldsymbol{v}),C_{\textnormal{target}}, \textnormal{lower bound}~L_0, \textnormal{upper bound}~U_0)$, which returns $u \in [L_0,U_0]$ such that $C(u, \boldsymbol{v})$ equals (or is arbitrarily close to) $C_{\textnormal{target}}$ given $\boldsymbol{v}$, where $C(u, \boldsymbol{v})$ is  in $u$ given $\boldsymbol{v}$.

Initialize $B_{\textnormal{lower}} \leftarrow L_0$,  $B_{\textnormal{upper}} \leftarrow U_0$;

$B_{\textnormal{upper}} -B_{\textnormal{lower}}$ is no greater than $\epsilon_4$ for a small positive number $\epsilon_4$

$u \leftarrow \frac{B_{\textnormal{lower}} + B_{\textnormal{upper}}}{2} $;

if $C(u, \boldsymbol{v})=C_{\textnormal{target}}$:
return $u$;

if $C(u, \boldsymbol{v})>C_{\textnormal{target}}$:
$B_{\textnormal{lower}} \leftarrow  u$;

$B_{\textnormal{upper}} \leftarrow  u$;

To better understand bisection search in our algorithms above,
we prove in the Appendix of our full paper [36] that the left-hand side of (<ref>) (resp., (<ref>), (<ref>), (<ref>), (<ref>)) is  with respect to $\gamma$ (resp., $p_n$, $\alpha$, $\beta$, $T$). In simulations, the above often decreases so that there is a unique solution.

∑_n ∈𝒩 b_n = b_max;
β·( ∑_n ∈𝒩 p_n - p_max) = 0;
γ·(∑_n ∈𝒩 f_n^MS - f_max^MS)  = 0;
δ_n ·(f_n^VU -f_n,max^VU )= 0;
ζ_n ·(t_n(b_n,p_n,s_n,f_n^MS,f_n^VU) - T) = 0.

∀n ∈𝒩:
∂L/∂b_n = 0,  meaning  ∂U_n/∂r_n ·∂r_n/∂b_n  + y/2r_n^3z_n ∂r_n/∂b_n  + ζ_n
s_n/r_n^2∂r_n/∂b_n = α;
∀n ∈𝒩:∂L/∂p_n = 0,  meaning  ∂U_n/∂r_n ·∂r_n/∂p_n - 2yz_ns_n^2(p_n +p_n^cir)   + y/2r_n^3z_n ∂r_n/∂p_n  + ζ_n  s_n/r_n^2∂r_n/∂p_n = β;

Given $\beta$, if $p_n$ strictly decreases or does not change, then if $\alpha$ strictly increases, $\psi_n(p_n,\alpha,\beta \mid \boldsymbol{z},y, \boldsymbol{s})$ strictly increases, then $b_n$ strictly decreases. Contradiction. Hence, $p_n$ should strictly increase. given $\beta$, $\widetilde{p}_n(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \boldsymbol{z},y, \boldsymbol{s})$ strictly increases if $\alpha$ strictly increases.

Given $\alpha$, if $p_n$ strictly increases or does not change, then if $\beta$ strictly increases, $\psi_n(p_n,\alpha,\beta \mid \boldsymbol{z},y, \boldsymbol{s})$ strictly decreases, then $b_n$ strictly increases. Contradiction. Hence, $p_n$ should strictly decrease.  given $\alpha$, $\widetilde{p}_n(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \boldsymbol{z},y, \boldsymbol{s})$ strictly decreases if $\beta$ strictly increases.

b_n =   g_np_n/σ_n^2ϑ_n =  g_np_n/σ_n^2ψ_n(p_n,α,β|z,y, s)

r_n = b_n log_2(1+ϑ_n)=g_np_n/σ_n^2ψ_n(p_n,α,β|z,y, s)  log_2(1+ ψ_n(p_n,α,β|z,y, s))

∂U_n(r_n, s_n)/∂r_n   + y/2z_nr_n^3  + ζ_n   s_n/r_n^2 =  α/  ∂r_n/∂b_n   =  α/

=  α/
log_2(1+ψ_n(p_n,α,β|z,y, s))
-ψ_n(p_n,α,β|z,y, s)/(1+ψ_n(p_n,α,β|z,y, s))ln2

§ OUR ALGORITHM TO SOLVE PROBLEM $\MATHBB{P}_{1}$

Algorithm <ref> has been presented on Page algo:MN  to solve the system UCR optimization  $\mathbb{P}_{1}$. In Section <ref>, we have also explained how the different building blocks in Sections <ref>, <ref>, <ref>, and <ref> are combined together to produce Algorithm <ref>'s pseudocode. We now discuss the performance of Algorithm <ref>.

Solution quality and convergence. Algorithm <ref> comprises three levels of iterations, with the innermost iteration from Line 17 containing Algorithm <ref> in Line 20. Algorithm <ref> obtains a global optimum for Problem $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$. The outermost iteration from Line 4 is based on Dinkelbach's transform and does not lose optimality. However, the mid-level iteration from Line 11 is based on alternating optimization and cannot guarantee local/global optimality. Hence, Algorithm <ref> cannot guarantee local/global optimality for $\mathbb{P}_{1}$. Yet, using the terminology of stationary points in [12] for constrained optimization, Algorithm <ref> finds a stationary point for $\mathbb{P}_{1}$. The convergence of Algorithm <ref> is also clear from the above analysis.

Time Complexity. In Algorithm <ref> and its subroutine Algorithm <ref>, the bisection search is repeatedly used. Then the  complexity of Algorithm <ref> is polylogarithmic in the error tolerance's reciprocal in various calls of the bisection search. Below we analyze the  complexities of Algorithms <ref> and <ref> with respect to the number $N$ of users.

Line 1 of Algorithm <ref> calls Algorithm 2.2, which calls Algorithm 2.1. Algorithm 2.1 called in the above and in Line 2 of Algorithm <ref>  calls Algorithms 1.1 and 1.2. Algorithm 1.2 called in the above and in Line 3 of Algorithm <ref> calls Algorithms 1.2.2.2, 1.2.2.1, and 1.2.1. For the multivariate bisection search in Algorithm 2.1, from Theorem 2.3 of [40], to achieve the tolerance level of  $\|\boldsymbol{h}(\boldsymbol{\zeta}\mid T)\|_2 \leq \epsilon_5$, the number of iterations required is $\log_2 \frac{\sum_{n \in \mathcal{N}}\zeta_n^{\textnormal{upper}}}{\epsilon_5}$ for $\zeta_n^{\textnormal{upper}}$ define in (<ref>); i.e., logarithmic in $N$.
Computing $\widetilde{\boldsymbol{b}}(\alpha,\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ and $\widetilde{\boldsymbol{p}}(\alpha,\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ takes $\mathcal{O}(N)$. Then calculating $\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ costs $\mathcal{O}(N)$. Thus, obtaining $\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ takes $\mathcal{O}(N)$. Finally, computing $\boldsymbol{h}(\boldsymbol{\zeta}\mid T)$ costs $\mathcal{O}(N)$. Line 1 of Alg. <ref> takes $\mathcal{O}(N\log N)$. We analyze other lines of Alg. <ref> similarly. Then Alg. <ref> and each innermost iteration of Alg. <ref> cost $\mathcal{O}(N\log N)$. In the outermost and mid-level iterations, each computation of the utility $\mathcal{U}()$ in (<ref>) and the cost in (<ref>) requires $\mathcal{O}(N)$. To summarize, Algorithm <ref> takes $\mathcal{O}(N^2\log N)$.

§ MODELING THE HUMAN-CENTRIC UTILITY FROM REAL DATA

We now model users' human-centric utilities in the Metaverse over wireless communications using two datasets [10, 31] explained below, which are both based on real experiments of humans assessing videos.

SSV360 dataset. This dataset of [10] captures users' evaluation of 360 videos when wearing HTC Vive Pro Virtual Reality (VR) headsets. Each data point exhibits  a user's perceptual quality assessment of a 360 scene of a given bitrate and a given video resolution, under standing or seated viewing (SSV).

Netflix dataset. This dataset is a part of Netflix's Emmy Award-winning VMAF project [31]. Each data point represents users' mean opinion score for a video at a given bitrate and a given resolution.

The wireless data rate needs to be large enough for users' smooth watching experience at the given video bitrate [41]. We consider the  bitrate as a constant fraction (say $\theta$) of the wireless rate. Then substituting the bitrate $r_{\text{bitrate}}$ with the wireless rate $r_{\text{wireless}}$ just involves replacing $r_{\text{bitrate}}$ with $r_{\text{wireless}}/\theta$. Hence, for both datasets above, we perform curve-fitting with the bitrate and the resolution to obtain the utility functions.

Modeling human-centric utilities. Based on the two datasets above, the human-centric utility of each user $n$, denoted by $U_n(r_n,s_n)$, is modeled as a function of the bitrate $r_n$ and the video resolution $s_n$. We adopt the logarithmic utility function, which is used in [32, 33, 34] for various communication/network systems. The logarithmic function reflects users' diminishing marginal gain as the bitrate and the resolution  increase. Formally, we have $U_n(r_n,s_n)=\kappa_n \ln (1+ l^s_ns_n+l^r_nr_n)$ for coefficients $\kappa_n, l^s_n,l^r_n$, which are decided by fitting data. Since using a three-dimensional plot to show the two-variable function $U_n(r_n,s_n)$ is difficult for visual interpretation, we use the following transform to obtain a two-dimensional plot. Let $r_n ^{\max}$ (resp., $s_n^{\max}$) be the maximum  $r_n $ (resp.,  $s_n $) from the dataset. After defining $\alpha_n:=l^r_n r_n ^{\max}+l^s_n s_n^{\max}$, we  let $\frac{l^r_n}{\alpha_n}r_n+\frac{l^s_n}{\alpha_n}s_n$ be the $x$-axis coordinate, and plot $\kappa_n \ln (1+ \alpha_n x)$ as the $y$-axis coordinate, since it holds that $U_n(r_n,s_n)=\kappa_n \ln (1+ \alpha_n \cdot (\frac{l^r_n}{\alpha_n}r_n+\frac{l^s_n}{\alpha_n}s_n))=\kappa_n \ln (1+ \alpha_n x)$. With the above transformation, each data point's $x$-coordinate is between $0$ and $1$.

This function reflects the diminishing return when increasing the resolution and bitrate for VU $n$, and $\kappa_n$ is the user-specific parameters for better fitting their real dataset. Since it is unlikely to quantify the influences of these two factors on the utility separately, we fine-tune the utility function parameters $\kappa_n$, and normalization parameters $l^s_n,l^r_n$ to approximate the data from SSV360.

In Fig. <ref>(a) for the SSV360 dataset, the data and curves are about two users watching a 360 video (“Alcatraz” or “FormationPace” [10]) under seated or standing view. The score is an integer from $1$ to $5$ based on the well-known Absolute Category Rating. In Fig. <ref>(b) for the Netflix dataset, the data and curves present users' average assessment (from $0$ to $100$) of  different videos (BirdsInCage, BigBuckBunny, ElFuente1, or CrowdRun [31]). Both subfigures demonstrate that the curves of the logarithmic human-centric utility functions  fit the data. The specific expressions of the functions are provided in the

The issue of designing the human-perceived utility function of QoE is always the core. In order to precisely quantify the user's perceptual video quality, we build the function $U_n(\cdot)$ upon the subjective dataset, SSV360 [10], which is an inclusive dataset collected by real VUs from four representative $360^\circ$ video scenarios. This dataset includes comprehensive factors such as resolution, bitrate, locomotion, exposure time, framerate, and VUs' corresponding experience scores. It is user-centric and adequate for our formulated system. The VU rating (i.e., user experience score) is used as the utility value (y value) since it is the most user-centric subjective metric. Under the specific user, scenario, and locomotion (standing or seated), the rating is most influenced by the resolution and bitrate [10]. Therefore, the sum of these two is chosen to form the control variable: $l^s_ns_n+l^r_nr_n$, where $l^s_n,l^r_n$ are normalization parameters of resolution and bitrate for VU $n$ and the units of $s_n$ and $r_n$ are 100M pixels and 100M bps. Besides, we use $U_n(r_n,s_n)=\kappa_n \ln (1+ l^s_ns_n+l^r_nr_n)$ as the utility function??., which is used in [32, 33, 34] for various communication/network systems.

Use the data to learn $\kappa_n,l^s_n, l^r_n$ such that $U_n(r_n,s_n)=\kappa_n \ln (1+l^r_nr_n+ l^s_ns_n)$

Define $\alpha_n:=l^r_n r_n ^{\max}+l^s_n s_n^{\max}$

Then $U_n(r_n,s_n)=\kappa_n \ln (1+ \alpha_n \cdot (\frac{l^r_n}{\alpha_n}r_n+\frac{l^s_n}{\alpha_n}s_n))$

The $x$-axis is $(\frac{l^r_n}{\alpha_n}r_n+\frac{l^s_n}{\alpha_n}s_n)$, and the $y$-axis is $\kappa_n \ln (1+ \alpha_n x)$.

Each data point's $x$-coordinate is between $0$ and $1$.

This function reflects the diminishing return when increasing the resolution and bitrate for VU $n$, and $\kappa_n$ is the user-specific parameters for better fitting their real dataset. Since it is unlikely to quantify the influences of these two factors on the utility separately, we fine-tune the utility function parameters $\kappa_n$, and normalization parameters $l^s_n,l^r_n$ to approximate the data from SSV360. We assume that the bitrate is equivalent to the transmission rate, as the dominant factor contributing to the total delay in this wireless system is the transmission delay. To better demonstrate the subjective scoring of different videos by users, we also use the dataset in Netflix [31]??.

In Fig. <ref> (a) and (b), we present eight typical curves of utility functions of the user's subjective experience from measured data in SSV360 [10] and Netflix [31]. These four curves in Fig. <ref> (a) correspond to the four scenarios of user 1 sitting and watching the video FormationPace, user 1 standing and watching the video FormationPace, user 1 sitting and watching the video Alcatraz, and user 2 sitting and watching the video FormationPace. Another four curves in Fig. <ref> (b) correspond to video BigBuckBunny, ElFuente1, BirdInCage, and CrowdRun, respectively. In the datasets, SSV and Netflix, mean opinion scores (ranging from 0 to 5) and differential mean opinion scores (ranging from 0 to 100) are used to represent the user's subjective experience of the video, respectively. From these curves, different videos, postures, and subjective factors of the user themselves could affect the user experience.

Parameter table of utility functions in Fig. <ref>

2*dataset     2*user scenario     3l|parameter
1l|$\kappa$     1l|$l^s$     $l^r$
4*SSV360 [10]         user 1 seated - FormationPace video                                   1l|1.7031      1l|57.2059      0.0131
user 1 standing - FormationPace video                                   1l|1.7134      1l|37.0186      0.0085
user 1 seated - Alcatraz video                                   1l|1.0210      1l|232.9612      0.0532
User 2 seated - FormationPace video                                   1l|1.0210      1l|232.9612      0.0532
4*Netflix [31]     BigBuckBunny video                                   1l|31.9725      1l|3745.8765      218.4595
ElFuente1 video                                   1l|33.1874      1l|2210.8196      128.935
BirdInCage video                                   1l|27.6723      1l|6692.7040      390.3185
CrowdRun video                                   1l|33.1931      1l|808.8521      67.2965

SSV360 dataset.

Netflix dataset.

Modeling the logarithmic human-centric utility functions from the SSV360 and
Netflix datasets.

§ SIMULATIONS

In this section on simulations,
we first describe the default settings  and then report various results.

Default settings.
We consider a macro-cell wireless channel model for urban areas. With $d_n$ denoting the distance between the Metaverse server (MS) and a virtual-reality user (VU) indexed by $n$, the path loss between them is $128.1+37.6\log d_n$ along with 8 decibels (dB) for the standard deviation of shadow fading [6], where the unit of $d_n$ is kilometer. The power spectral density of Gaussian noise ${\sigma_n}^2$ is $-174$ dBm/Hz (i.e., the thermal noise amount at 20 C room temperature). VUs are randomly located in a circle of radius 500m centered at the MU.
The default total bandwidth $b_{\textnormal{max}}$ is 20GHz, and the total transmission power $p_{\textnormal{max}}$ is 30W. The effective switched capacitance $\kappa^{\textnormal{MS}}$ and $\kappa^{\textnormal{VU}}$ are set as $10^{-27}$. The number $\mu_n$ of bits per pixel is 16, and the compression rate $\nu_n$ is 100. The maximum CPU frequencies at the MS and VUs, $f_{\textnormal{max}}^{\textnormal{MS}}$ and $f_{n,\textnormal{max}}^{\textnormal{VU}}$, are 300GHz and 50GHz, respectively. The default  weights for  energy and delay are $c_e = 0.5$ and $c_t = 0.5$. The default VU number is 5. Based on measurements,
Section V of [41] quantifies the computational complexity of processing a video frame of resolution $s_n$ as   $w(s_n) = (7 \times 10^{-10} \times {s_n}^{3/2} +  0.083) $ tera (i.e., trillion) floating-point operations (FLOPs). From Fig. <ref> in Section <ref>, we know that $\mathcal{A}_n(s_n,\Lambda_n) $ (resp., $  \mathcal{B}_n(s_n,\Lambda_n)$) of Page defineAB for delay is less than $\mathcal{F}_n(s_n,\Lambda_n) $ (resp., $  \mathcal{G}_n(s_n,\Lambda_n)$) of Page defineFG for energy. In the simulations, we set both $\mathcal{A}_n(s_n,\Lambda_n) $ and $ \mathcal{B}_n(s_n,\Lambda_n)$  as
$\Lambda_n w(s_n) /30$, and set both $\mathcal{F}_n(s_n,\Lambda_n) $ and $ \mathcal{G}_n(s_n,\Lambda_n)$ as
$\Lambda_n w(s_n)$, which make all of them convex in $s_n$. For all VU $n$, we let $\Lambda_n$ be the same $\Lambda$.
Then the optimization objective becomes a multiple of $\Lambda$, which thus has no impact. The above avoids considering the impact of heterogeneous $\Lambda_n$ for simplicity. Possible values for the resolution $s_n$ are $4096 \times 2160$, $3072 \times 1620$, $2048 \times 1080$, $1920 \times 1080$, and $1280 \times 720$ pixels, which are also referred to  as 4k, 3k, 2k, 1080p, and 720p. The SSV360 dataset [10]  in Section <ref>  includes the perceptual assessment of  users watching VR videos. We use those data for curve-fitting different logarithmic utility functions, and assign the functions to users in the simulations: one function for one user.

Comparison with baselines. For the simulation results, we first compare our algorithm with baselines:

* average allocation, which sets each $b_n$ as $\frac{b_{\textnormal{max}}}{N}$, each $p_n$ as $\frac{p_{\textnormal{max}}}{N}$, each $s_n$ as $2048 \times 1080$ (i.e., 2k resolution), each $f_n^{\textnormal{MS}}$ as $\frac{f_{\textnormal{max}}^{\textnormal{MS}}}{N}$, and each $f_n^{\textnormal{VU}}$ as $f_{n,\textnormal{max}}^{\textnormal{VU}}$;

* optimize $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$ only, while setting each $f_n^{\textnormal{MS}}$ as $\frac{f_{\textnormal{max}}^{\textnormal{MS}}}{N}$ and each $f_n^{\textnormal{VU}}$ as $f_{n,\textnormal{max}}^{\textnormal{VU}}$,

* optimize $\boldsymbol{f^{\textnormal{MS}}}$ and $\boldsymbol{f^{\textnormal{VU}}}$ only, while setting each $b_n$ as $\frac{b_{\textnormal{max}}}{N}$, each $p_n$ as $\frac{p_{\textnormal{max}}}{N}$, and each $s_n$ as $2048 \times 1080$.

Various simulation results are plotted in the subfigures of Fig. <ref> for a detailed comparison and examining the impact of different parameters on the system utility-cost ratio (UCR). We discuss the results below.

* UCR versus the total bandwidth.

Here we vary the total bandwidth from 1GHz to 20GHz. In Fig. <ref>(a), larger bandwidth induces higher data rates, which reduce latency  and energy consumption, thus increasing the system UCR. In addition, the difference in UCR  between the proposed algorithm and the average allocation baseline rises from 522.3% to 630.4% as the total bandwidth increases.

* UCR versus minimum resolution.
We fix the maximum resolution as $4096 \times 2160$ (4k) and change the minimum resolution from $1280\times720$ (720P) to $4096 \times 2160$. From Fig. <ref>(b), the UCR performance of all the algorithms improves as the minimum resolution decreases, with our proposed algorithm showing a significant improvement. The reason for this is that the high data volumes associated with high resolution can lead to higher energy consumption and system delay.

The UCR of the proposed algorithm gradually plateaus when the minimum resolution reaches below $1920\times1080$ (1080p).

* UCR versus transmission power.
Here we configure the maximum downlink transmission power from 0.03W to 100W. From Fig. <ref>(c), the UCR of all algorithms increases as the transmission power grows, since raising the transmission power   widens the search space for the optimization. When the transmission power is very small (e.g., 0.03W or 0.3W), the UCR of the proposed algorithm is slightly higher than other algorithms, but as the transmission power increases, the performance of the proposed algorithm far exceeds others. The UCR of all methods plateaus when the transmission power reaches 50W.

* UCR versus computation resource.

We vary the maximum server CPU frequency from 0.5GHz to 60GHz and all VU's maximum  CPU frequencies from 10MHz to 10GHz. In Fig. <ref>(d) and Fig. <ref>(e), the system UCR increases as the maximum CPU frequency grows, since the optimization problem has a wider search space.

The proposed algorithm outperforms the average allocation, reaching a difference of 660.9% and 653.2% for server CPU frequency at 60GHz and VU CPU frequencies at 10GHz, respectively.

* Impact of user number on UCR.
We now amplify the bandwidth, transmission power, and server's CPU frequency by 10 times and fix other parameters to see the impact of user number $N$ on UCR.

In Fig. <ref>, as the user number increases from 10 to 160, the average UCR decreases. This is because the server allocates fewer resources to each VU and induces decreasing utility. In general,  the comfortable frame rate for VR applications is at least 90 [42], i.e., at most 11ms for one frame. Note that the system delay in Fig. <ref> is to complete each user's all frames. In all user scenarios shown in Fig. <ref>, the delay for one frame (i.e., $\frac{\text{delay}}{\text{frame number}}$) is less than 7 ms, which satisfies the comfortable frame rate requirement.

All the above simulations compare our algorithm with the baselines. Below we provide additional simulation results to show the impact of other settings to our algorithm.

The system utility-cost ratio (UCR) versus various parameters.

Impact of cost weights on UCR.
We configure different cost weights of energy and delay $(c_e, c_t)$ to see the effect on the system UCR, where we enforce $c_e + c_t=1$.

In Fig. <ref>(f), as $c_t$ rises to 0.8, the system UCR also increases, reflecting the importance of delay optimization for the whole system. However, as $c_t$ increases to 0.9, the system UCR instead drops significantly, since emphasizing the latency overwhelmingly while undervaluing the energy may enlarge the system cost.

Metrics with respect to the number of users.

Allocated parameters for each user.

Comparing individual users' UCR.

Reviewing the allocated results of different users and the impact of user scenarios on UCR.

Reviewing different users' allocated results.

In Fig. <ref>(a), we present the optimized bandwidth $b_n$, transmit power $p_n$, resolution $s_n$, MS resource allocation $f^{\textnormal{MS}}_n$, and user CPU frequency $f^{\textnormal{VU}}_n$ of five users to visualize the resource allocation. Fig. <ref>(b) shows the impact of user scenarios on individual UCRs. Different user preferences and physical states affect subjective scores. For example, users who prefer high-quality videos may give stricter subjective scores than those who do not require high video quality. Generally, many users found sitting to provide more comfort than standing, as indicated by a higher UCR in most seated scenarios. However, it's important to note that individual differences were observed, and this trend did not hold true for all participants.

To summarize, extensive simulation results above confirm the effectiveness of our proposed

Here we compare the proposed algorithm with optimize $\boldsymbol{f^{\textnormal{MS}}}$ and $\boldsymbol{f^{\textnormal{VU}}}$ only, and optimize $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$ only. For all algorithms, the initial resource setting is exactly the initialization setting of Algorithm <ref>. The other parameters are set consistently with the default setting in <ref>. For the two algorithms that only optimize parts of the resources, the difference between them and the proposed method is that they only optimize $\boldsymbol{f^{\textnormal{MS}}}$, $\boldsymbol{f^{\textnormal{VU}}}$ or $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$. From Fig. <ref> (b), it can be inferred that optimizing only $\boldsymbol{f^{\textnormal{MS}}}$, $\boldsymbol{f^{\textnormal{VU}}}$ or $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$ can't lead to an optimal solution to the optimization problem, as reflected in the fact that energy consumption and delay are not effectively improved. The proposed algorithm consistently achieves the best performance, in terms of energy consumption, system delay, and total UCR. Note that the optimization of $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$ only also outperforms the optimization of $\boldsymbol{f^{\textnormal{MS}}}$ and $\boldsymbol{f^{\textnormal{VU}}}$ only, as optimizing $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$ results in higher transmission rate and moderate resolution, which both enhance the user experience and reduce costs to a certain extent. Furthermore, the utility function is directly related to $\boldsymbol{b}$, $\boldsymbol{p}$, and $\boldsymbol{s}$.

§ CONCLUSION

In this paper, we optimize the system utility-cost ratio (UCR) for the Metaverse over wireless networks. The optimization variables include the allocation of both communication and computation resources as well as the resolutions of virtual reality (VR) videos. Our human-centric utility measure represents users' subjective assessment of the VR video quality, and is supported by real datasets.

We tackle the non-convex system UCR optimization by proposing a novel technique for fractional programming. Our computationally efficient algorithm for the system UCR optimization is validated by extensive simulations. Three future directions are as follows. Firstly, since the current paper solves the optimization problem via alternating optimization (AO) of video frame resolution and other variables, a future task is to see whether we can optimize all variables simultaneously to obtain the globally optimal solution. Secondly, we may incorporate the priorities of different users into computing the system utility (e.g., using a weighted sum with weights representing users' priorities), and investigate the impact of such formulation on the optimization. Thirdly, while the current paper contains extensive simulation results to support the analysis, we can implement real-world systems to evaluate the performance of our proposed algorithm in practice.

§ ACKNOWLEDGEMENT

This research is partly supported by the Singapore Ministry of Education Academic Research Fund under Grant Tier 1 RG90/22, Grant Tier 1 RG97/20, Grant Tier 1 RG24/20 and Grant Tier 2 MOE2019-T2-1-176; and partly by the Nanyang Technological University (NTU)-Wallenberg AI, Autonomous Systems and Software Program (WASP) Joint Project.

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Jun Zhao
(Member, IEEE) received a bachelor’s degree in Information Engineering from Shanghai Jiao Tong University, China, in July 2010, and a Ph.D. degree in Electrical and Computer Engineering from Carnegie Mellon University (CMU), USA, in May 2015, (advisors: Virgil Gligor and Osman Yagan; collaborator: Adrian Perrig), affiliating with CMU’s renowned CyLab Security & Privacy Institute. He is currently an Assistant Professor in the School of Computer Science and Engineering (SCSE), Nanyang Technological University (NTU), Singapore. Before joining NTU as a Faculty Member, he was a Postdoctoral Researcher under the supervision of Xiaokui Xiao at NTU. Before that, he was a Postdoctoral Researcher at Arizona State University as an Arizona Computing Post-Doctoral Researcher Best Practices Fellow (advisors: Junshan Zhang and Vincent Poor).

Liangxin Qian received bachelor's and master's degrees in communication engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2019 and 2022, respectively. He is currently working toward his Ph.D. at the School of Computer Science and Engineering, Nanyang Technological University, Singapore. His research interests include Metaverse, mobile edge computing, and communication theory.

Wenhan Yu
(Student Member, IEEE) received his B.S. degree in Computer Science and Technology from Sichuan University, Sichuan, China in 2021. He is currently pursuing a Ph.D. degree in the School of Computer Science and Engineering, Nanyang Technological University (NTU), Singapore. His research interests cover wireless communications, deep reinforcement learning, optimization, and Metaverse.

§.§ Proving the relationship between  in (<ref>) and minimizing $W(\boldsymbol{x},\boldsymbol{y}):=G(\boldsymbol{x}) + \sum_{n=1}^N K_n(\boldsymbol{x},y_n)$ subject to $\boldsymbol{x} \in \mathcal{S}$ and $y_n \in \mathbb{R}^+$, stated in Section <ref> on Pages opt-sum-of-ratios and opt-sum-of-ratios2

Recall that  in (<ref>) means minimizing $H(\boldsymbol{x}) : = G(\boldsymbol{x}) + \sum_{n=1}^N \frac{A_n(\boldsymbol{x})}{B_n(\boldsymbol{x})}$ subject to $\boldsymbol{x}$ in a convex set $\mathcal{S}$, for convex  $A_n(\boldsymbol{x})$ and concave $B_n(\boldsymbol{x})$.

We now consider alternating optimization (AO) of $\boldsymbol{x}$ and $\boldsymbol{y}$ to minimize $W(\boldsymbol{x},\boldsymbol{y}):=G(\boldsymbol{x}) + \sum_{n=1}^N K_n(\boldsymbol{x},y_n)$ subject to $\boldsymbol{x} \in \mathcal{S}$ and $\boldsymbol{y}:=[y_1,\ldots,y_N] \in (\mathbb{R}^+)^N$, where $K_n(\boldsymbol{x},y_n):=[A_n(\boldsymbol{x})]^2y_n + \frac{1}{4[B_n(\boldsymbol{x})]^2y_n} $, and $\mathbb{R}^+$ denotes the set of positive numbers. We will show that

\begin{align}
\begin{array}{l}
\text{if the above AO process of minimizing $W(\boldsymbol{x},\boldsymbol{y})$ converges}\\
\text{to $(\boldsymbol{x}^*,\boldsymbol{y}^*)$, $\boldsymbol{x}^*$ is a stationary point for \mbox{\texttt{FP-minimization}}} \\
\text{in~(\ref{FP-minimization}).}
\end{array}     \label{eqpartialKn0}
\end{align}

The AO process is as follows: We start with a randomly initialized $\boldsymbol{x}^{(0)} \in \mathcal{S}$. Then we optimize $\boldsymbol{y}$ with $\boldsymbol{x}$ being $\boldsymbol{x}^{(0)}$ to  minimize $W(\boldsymbol{x},\boldsymbol{y})$, and denote the obtained  $\boldsymbol{y} \in (\mathbb{R}^+)^N$ as $\boldsymbol{y}^{(1)}$. Given $\boldsymbol{y}$ as $\boldsymbol{y}^{(1)}$, we optimize $\boldsymbol{x}$ to  minimize $W(\boldsymbol{x},\boldsymbol{y})$, and denote the obtained  $\boldsymbol{x} \in \mathcal{S}$ as $\boldsymbol{x}^{(1)}$. Given $\boldsymbol{x}$ as $\boldsymbol{x}^{(1)}$, we optimize $\boldsymbol{y}$ to  minimize $W(\boldsymbol{x},\boldsymbol{y})$, and denote the obtained  $\boldsymbol{y} \in (\mathbb{R}^+)^N$ as $\boldsymbol{y}^{(2)}$. The above process continues iteratively. The $i$-th iteration includes the following two steps:

* Given $\boldsymbol{x}$ as $\boldsymbol{x}^{(i-1)}$, we optimize $\boldsymbol{y}$ to  minimize $W(\boldsymbol{x},\boldsymbol{y})$, and denote the obtained  $\boldsymbol{y} \in (\mathbb{R}^+)^N$ as $\boldsymbol{y}^{(i)}$.

* Given $\boldsymbol{y}$ as $\boldsymbol{y}^{(i)}$, we optimize $\boldsymbol{x}$ to  minimize $W(\boldsymbol{x},\boldsymbol{y})$, and denote the obtained  $\boldsymbol{x} \in \mathcal{S}$ as $\boldsymbol{x}^{(i)}$.

The above AO process converges when the relative difference between $W(\boldsymbol{x}^{(i-1)},\boldsymbol{y}^{(i-1)})$ and $W(\boldsymbol{x}^{(i)},\boldsymbol{y}^{(i)})$ is smaller than a predefined small error tolerance.

To examine the AO process of minimizing $W(\boldsymbol{x},\boldsymbol{y})$, we will analyze 1) optimizing $\boldsymbol{y}$ given $\boldsymbol{x}$, and 2) optimizing $\boldsymbol{x}$ given $\boldsymbol{y}$. Optimizing $\boldsymbol{y}$ given $\boldsymbol{x}$ means minimizing $K_n(\boldsymbol{x},y_n):=[A_n(\boldsymbol{x})]^2y_n + \frac{1}{4[B_n(\boldsymbol{x})]^2y_n} $ with respect to  $y_n$ for each $n=1,2,\ldots,N$; i.e., letting $y_n $ be $y_n^{\#}(\boldsymbol{x}):=\frac{1}{2A_n(\boldsymbol{x})B_n(\boldsymbol{x})} $. If such $\boldsymbol{y}$ is substituted back to $K_n(\boldsymbol{x},y_n)$, then $K_n(\boldsymbol{x},y_n)$ will become the desired $\frac{A_n(\boldsymbol{x})}{B_n(\boldsymbol{x})}$. Moreover, we now show that the partial derivative of $W(\boldsymbol{x},\boldsymbol{y})$ with respect to $\boldsymbol{x}$ at $\boldsymbol{y}$ being $\boldsymbol{y}^{\#}(\boldsymbol{x})$ is the same as the  derivative of $H(\boldsymbol{x})$ with respect to $\boldsymbol{x}$. In fact, we have

\begin{align}
\frac{\partial K_n(\boldsymbol{x},y_n)}{\partial \boldsymbol{x}}   & =   2A_n(\boldsymbol{x})y_n \frac{\partial A_n(\boldsymbol{x})}{\partial \boldsymbol{x}}  - \frac{1}{2[B_n(\boldsymbol{x})]^3y_n} \cdot \frac{\partial B_n(\boldsymbol{x})}{\partial \boldsymbol{x}} , \label{eqpartialKn}
\end{align}

\begin{align}
\frac{\partial \big(\frac{A_n(\boldsymbol{x})}{B_n(\boldsymbol{x})}\big)}{\partial \boldsymbol{x}}   & =   \frac{\frac{\partial A_n(\boldsymbol{x})}{\partial \boldsymbol{x}}\cdot B_n(\boldsymbol{x}) - \frac{\partial B_n(\boldsymbol{x})}{\partial \boldsymbol{x}}\cdot A_n(\boldsymbol{x})}{(B_n(\boldsymbol{x}))^2} .\label{eqpartialAnBn}
\end{align}

From (<ref>) and (<ref>), it holds that

\begin{align}
\bigg(  \frac{\partial K_n(\boldsymbol{x},y_n)}{\partial \boldsymbol{x}} \bigg)|_{y_n=\frac{1}{2A_n(\boldsymbol{x})B_n(\boldsymbol{x})} }  & =    \frac{\partial \big(\frac{A_n(\boldsymbol{x})}{B_n(\boldsymbol{x})}\big)}{\partial \boldsymbol{x}},\label{eqpartialKn2}
\end{align}

which further implies

\begin{align}
\bigg(  \frac{\partial W(\boldsymbol{x},\boldsymbol{y})}{\partial \boldsymbol{x}} \bigg)|_{\boldsymbol{y} = \boldsymbol{y}^{\#}(\boldsymbol{x})}  & =    \frac{\partial H(\boldsymbol{x})}{\partial \boldsymbol{x}}.\label{eqpartialKn3}
\end{align}

Moreover, as explained, we have

\begin{align}
W(\boldsymbol{x},\boldsymbol{y})|_{\boldsymbol{y} = \boldsymbol{y}^{\#}(\boldsymbol{x})}  & =    H(\boldsymbol{x}).\label{eqpartialKn4}
\end{align}

Using (<ref>) and (<ref>), we now show (<ref>). The AO process of minimizing $W(\boldsymbol{x},\boldsymbol{y})$ is . Specifically, we have $W(\boldsymbol{x}^{(i)},\boldsymbol{y}^{(i)}) \leq W(\boldsymbol{x}^{(i-1)},\boldsymbol{y}^{(i)}) \leq W(\boldsymbol{x}^{(i-1)},\boldsymbol{y}^{(i-1)})$. For  $W(\boldsymbol{x},\boldsymbol{y})$, we know $W(\boldsymbol{x}^{(i)},\boldsymbol{y}^{(i)})$ converges as the iteration number $i \to \infty$. Supposing the variable solution of the AO process converges to  $(\boldsymbol{x}^*,\boldsymbol{y}^*)$, we know that

172 $\boldsymbol{y}^*$ is the optimal $\boldsymbol{y}$ for minimizing $W(\boldsymbol{x},\boldsymbol{y})$ given $\boldsymbol{x}$ as $\boldsymbol{x}^*$ (i.e., $y_n^*:=\frac{1}{2A_n(\boldsymbol{x}^*)B_n(\boldsymbol{x}^*)} $), and

173 $\boldsymbol{x}^*$ is the optimal $\boldsymbol{x}$ for minimizing $W(\boldsymbol{x},\boldsymbol{y})$ given $\boldsymbol{y}$ as $\boldsymbol{y}^*$.

Result “173” means that $\boldsymbol{x}^*$ satisfies the KKT conditions for optimizing $\boldsymbol{x}$ to minimize $W(\boldsymbol{x},\boldsymbol{y}^*)$ subject to $\boldsymbol{x} \in \mathcal{S}$. Suppose $\boldsymbol{x}$ is $M$-dimensional, and $\boldsymbol{x} \in \mathcal{S}$ means

\begin{align}
\begin{cases}
\mathcal{Q}_q(\boldsymbol{x}) \leq 0, & q=1,2,\ldots,Q, \\
\mathcal{R}_r(\boldsymbol{x}) = 0, & r=1,2,\ldots,R,\\
\boldsymbol{x} \in \mathbb{R}^M.
\end{cases} \nonumber
\end{align}

Then with $\boldsymbol{\alpha} $ and $\boldsymbol{\beta}$ denoting  the multipliers,

the KKT conditions mean

\hspace{-8pt}~~~\frac{\partial }{\partial x_m} (W(\boldsymbol{x}^*,\boldsymbol{y}^*) \\\hspace{-8pt}+ \sum_{q=1}^Q  \alpha_q \mathcal{Q}_q(\boldsymbol{x}^*)
\\\hspace{-8pt}+ \sum_{r=1}^R \beta_r \mathcal{R}_r(\boldsymbol{x}^*) ) = 0,
\\ \hspace{-8pt}\text{for $m=1,2,\ldots,M$,}
\end{array}$
Primal feasibility:    $\mathcal{Q}_q(\boldsymbol{x}^*) \leq 0,$
for $ q=1,2,\ldots,Q,$
$\mathcal{R}_r(\boldsymbol{x}^*) = 0, $
for $ r=1,2,\ldots,R,$

Dual feasibility:    $\alpha_q  \geq 0$,
for $ q=1,2,\ldots,Q,$
Complementary slackness:    $\alpha_q \mathcal{Q}_q(\boldsymbol{x}^*) = 0$
for $ q=1,2,\ldots,Q.$

Using (<ref>) in (<ref>), we know that (<ref>)–(<ref>) are equivalent to

Stationarity:     $\begin{array}{l}\hspace{-2pt}\frac{\partial }{\partial x_m} (H(\boldsymbol{x}^*) \\
+ \sum_{q=1}^Q  \alpha_q \mathcal{Q}_q(\boldsymbol{x}^*) \\
+ \sum_{r=1}^R \beta_r \mathcal{R}_r(\boldsymbol{x}^*) ) = 0, \\
\text{for $m=1,2,\ldots,  M$,}\end{array}$

Primal feasibility:   $\mathcal{Q}_q(\boldsymbol{x}^*) \leq 0,$

for $ q=1,2,\ldots,Q,$

$\mathcal{R}_r(\boldsymbol{x}^*) = 0, $

for $ r=1,2,\ldots,R,$

Dual feasibility:   $\alpha_q  \geq 0$

for $ q=1,2,\ldots,Q,$
Complementary slackness:   $\alpha_q \mathcal{Q}_q(\boldsymbol{x}^*) = 0$

for $ q=1,2,\ldots,Q.$

The above (<ref>)–(<ref>) mean that $\boldsymbol{x}^*$ is a stationary point for optimizing $\boldsymbol{x}$ to minimize $H(\boldsymbol{x})$ subject to $\boldsymbol{x} \in \mathcal{S}$; i.e., $\boldsymbol{x}^*$ is a stationary point of  in (<ref>). Hence, we have proved (<ref>).

§.§ Proving that the left-hand side of (<ref>) is  with respect to $\gamma$

With $v(x)$ defined by $\zeta_n \frac{\mathcal{A}_n(s_n,\Lambda_n)}{x^2} - y c_{\hspace{0.5pt}\textnormal{e}} \cdot2 \kappa^{\textnormal{MS}} \mathcal{F}_n(s_n,\Lambda_n)x$, clearly $v(x)$ is  with respect to $x$. Then $\text{PositiveRoot}(v(x)= \gamma)$, denoting the positive root satisfying $v(x)= \gamma$, is  with respect to $\gamma$. Thus, the left-hand side of (<ref>) is  with respect to $\gamma$.

§.§ Proving that the left-hand side of (<ref>) is  with respect to $p_n$

From (<ref>), we know (i) $\psi_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})$ is positive and decreasing in $p_n$, where “$\textnormal{\small$\bigstar$}$” denotes “$\boldsymbol{z},y, \boldsymbol{s}$”. In addition, (ii) the function $\log_2(1+x) - \frac{x}{(1+x) \ln 2}$ is increasing and positive   for $x>0$ since the derivative $\frac{x}{(1+x)^2 \ln 2}$ is positive for $x>0$, and $\log_2(1+x) - \frac{x}{(1+x) \ln 2}$ at $x=0$ equals $0$. From the above Results “(i)” and “(ii)”, we obtain (iii) $\left[
\log_2\big(1+\psi_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})\big)
-\frac{\psi_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})}{(1+\psi_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$}))\ln2}\right]$ is positive and decreasing  in $p_n$.

We also have (iv) the function $\frac{\log_2(1+x)}{x}$ is decreasing for $x>0$ since the derivative $\frac{1}{x^2}[\frac{x}{(1+x) \ln 2} - \log_2(1+x)]  $ is negative due to Result “(ii)” above. From the above Results “(i)” and “(iv)”, we obtain (v) $\overline{r}_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})$ defined in (<ref>) is increasing in $p_n$. Since the utility function $U_n(r_n, s_n)$ is concave and  in $r_n$, $\frac{\partial U_n(r_n, s_n)}{\partial r_n}$ is  and  in $r_n$. Then (vi) $ \frac{\partial U_n(r_n, s_n)}{\partial r_n}   + \frac{yc_{\hspace{0.5pt}\textnormal{e}}}{2z_n{r_n}^3{{\nu_n}^2}}  + \frac{\zeta_n s_n\textcolor{black}{\mu_n}  {\color{black}\Lambda_n}}{{r_n}^2{\nu_n}} $ is positive and  in $r_n$.
From the above Results “(v)” and “(vi)”, we obtain (vii) $     \big(\frac{\partial U_n(r_n, s_n)}{\partial r_n}\big)|_{r_n=\overline{r}_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})}   + \frac{y c_{\hspace{0.5pt}\textnormal{e}}}{2z_n \cdot [\overline{r}_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})]^3\textcolor{black}{{\nu_n}^2}}   + \frac{\zeta_n s_n\textcolor{black}{\mu_n}  {\color{black}\Lambda_n}}{ [\overline{r}_n(p_n,\alpha,\beta \mid \textnormal{\small$\bigstar$})]^2\textcolor{black}{\nu_n}}
$ is positive and  in $p_n$.

From the above Results “(iii)” and “(vii)”, the left hand side of (<ref>) is  as $p_n$ increases.

§.§ Further explanations of the step-by-step analysis in (<ref>) on Page roadmap for Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$, which will be useful for Appendices <ref>, <ref>, and <ref>

In this part, we provide further explanations of the step-by-step analysis in (<ref>) on Page roadmap for Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$, which will be useful to prove the results in Appendices <ref>, <ref>, and <ref> later.

We recall Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$ (i.e., $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$) from (<ref>):

\begin{align}
\textnormal{$\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$}:\hspace{-10pt}\max\limits_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T} \hspace{-5pt} & F(\boldsymbol{b},\hspace{-2pt}\boldsymbol{p},\hspace{-2pt}\boldsymbol{f}^{\textnormal{MS}},\hspace{-2pt}\boldsymbol{f}^{\textnormal{VU}},\hspace{-2pt}T  \mid \boldsymbol{s}, \hspace{-2pt}y) \nonumber \\
&- yc_{\hspace{0.5pt}\textnormal{e}} \hspace{-2pt}\cdot \hspace{-2pt}\sum_{n \in \mathcal{N}} \big\{[(p_n +p_n^{\textnormal{cir}})  s_n\textcolor{black}{\mu_n}{\color{black}\Lambda_n}]^2z_n \hspace{-2pt} \nonumber \\
&+\hspace{-2pt} \frac{1}{4(r_n(b_n,p_n)\textcolor{black}{\nu_n})^2z_n}\big\} \nonumber \\
\textrm{s.t.} \quad &  \textnormal{(\ref{constraintbn}): } \sum_{n \in \mathcal{N}} b_n \leq b_{\textnormal{max}},\nonumber \\  &\textnormal{(\ref{constraintpn}): }\sum_{n \in \mathcal{N}} p_n \leq p_{\textnormal{max}} , \nonumber \\ &\textnormal{(\ref{constraintfMS}): }\sum_{n \in \mathcal{N}} f_n^{\textnormal{MS}} \leq f_{\textnormal{max}}^{\textnormal{MS}},  \nonumber \\ &\textnormal{(\ref{constraintfn}): }f_n^{\textnormal{VU}} \leq f_{n,\textnormal{max}}^{\textnormal{VU}},~\forall n \in \mathcal{N},  \nonumber \\ &\textnormal{(\ref{constraintTtau}): } t_n(b_n,p_n,s_n,f_n^{\textnormal{MS}},f_n^{\textnormal{VU}}) \leq T,~\forall n \in \mathcal{N}.  \nonumber
\end{align}

As shown at the beginning of Section <ref> on Page kktproblem5, $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$ belongs to convex optimization, and $\alpha, \beta, \gamma, \boldsymbol{\delta}, \boldsymbol{\zeta}$ denote  the Lagrange multipliers for (<ref>), (<ref>), (<ref>), (<ref>), and  (<ref>), respectively.

Suppose we already know $\boldsymbol{\zeta}$. We move $\boldsymbol{\zeta}$ and (<ref>) to the objective function, and construct the following problem (recall that for an optimization problem $\mathbb{P}_{i}$, we use  $H_{\mathbb{P}_{i}}$ to denote its objective function):

\begin{align}
\textnormal{$\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$}:\hspace{-10pt}\min\limits_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T} \hspace{-5pt} & - H_{\mathbb{P}_{5}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T \mid \textnormal{\small$\bigstar$}) \nonumber \\
&+  \sum_{n \in \mathcal{N}} \big[\zeta_n \cdot (t_n(b_n,p_n,s_n,f_n^{\textnormal{MS}},f_n^{\textnormal{VU}}) \hspace{-2pt}-\hspace{-2pt} T)\big]    \nonumber \\
\textrm{s.t.} \quad &  \textnormal{(\ref{constraintbn}): } \sum_{n \in \mathcal{N}} b_n \leq b_{\textnormal{max}},\nonumber \\  &\textnormal{(\ref{constraintpn}): }\sum_{n \in \mathcal{N}} p_n \leq p_{\textnormal{max}} , \nonumber \\ &\textnormal{(\ref{constraintfMS}): }\sum_{n \in \mathcal{N}} f_n^{\textnormal{MS}} \leq f_{\textnormal{max}}^{\textnormal{MS}},  \nonumber \\ &\textnormal{(\ref{constraintfn}): }f_n^{\textnormal{VU}} \leq f_{n,\textnormal{max}}^{\textnormal{VU}},~\forall n \in \mathcal{N} .  \nonumber
\end{align}

We define

* Statement $\mathcal{V}_{\mathbb{P}_{5}}$:

$\left\{ \begin{array}{l}
[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\textnormal{\small$\bigstar$}), \\
T^{\#}(\textnormal{\small$\bigstar$}),\alpha^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}),\gamma^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{\delta}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$})] \\
\textnormal{is a solution to the KKT conditions $\mathcal{S}_{KKT}$ in (\ref{KKTP5})} \\ \textnormal{for Problem $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$.}
\end{array}  \right\}$, and

* Statement $\mathcal{V}_{\mathbb{P}_{7}}$:

$\left\{ \begin{array}{l}
T^{\#}(\textnormal{\small$\bigstar$}), \alpha^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}),\gamma^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{\delta}^{\#}(\textnormal{\small$\bigstar$})] \\ \textnormal{is a solution to the KKT conditions of} \\
\textnormal{Problem $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}))$.}
\end{array}  \right\} $.

By checking the KKT conditions of $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$ and $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}))$, we build the following relationship between
$\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$ and $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$:

\begin{align}
\text{Statement $\mathcal{V}_{\mathbb{P}_{5}}$}   & \hspace{-2pt}\Leftrightarrow\hspace{-2pt} \left\{ \begin{array}{l}  \textnormal{Statement $\mathcal{V}_{\mathbb{P}_{7}}$ holds,}  \\
\text{and $[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\textnormal{\small$\bigstar$}),$}\\
\text{$(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\textnormal{\small$\bigstar$}),T^{\#}(\textnormal{\small$\bigstar$})  ,\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$})]$ satisfies}  \\
\text{$\mathcal{S}_{2.1} \cup \mathcal{S}_{2.2}:=\big\{\textnormal{(\ref{partialLpartialT}), (\ref{Complementarytime}),  (\ref{constraintTtau}), (\ref{Dualfeasibility}e)}\big\}$.}
\end{array}  \hspace{-5pt}\right\}. \label{statev5v7}
\end{align}

In $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$, the optimizations of $[\boldsymbol{b},\boldsymbol{p}] $, $[\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}}]$ and $T$ are independent and thus separable. This independence holds because $\boldsymbol{\zeta}$ is already given for $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$. We do not have such independence in optimizing $\mathbb{P}_{5}(\textnormal{\small$\bigstar$})$ where $\boldsymbol{\zeta}$ is not decided yet. Hence, $\mathbb{P}_{7}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ is equivalent to the combination of $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$, $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$, and $\mathbb{P}_{10}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ defined below:

\begin{align}
&\textnormal{Problem $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$}: \hspace{-1pt}\min_{T}   \quad y c_{\hspace{0.8pt}\textnormal{t}} T - T \sum_{n \in \mathcal{N}} \zeta_n   \nonumber \\
&\textnormal{Problem $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$}: \hspace{-1pt}\min_{\boldsymbol{b},\boldsymbol{p}}   \hspace{-1pt} \left\{\hspace{-5pt}  \begin{array}{l} -\mathcal{U}(\boldsymbol{b},\hspace{-2pt}\boldsymbol{p},\hspace{-2pt}\boldsymbol{s})   \\ +yc_{\hspace{0.5pt}\textnormal{e}} \hspace{-2pt}\cdot \hspace{-2pt}\sum\limits_{n \in \mathcal{N}} \big\{[(p_n +p_n^{\textnormal{cir}})  s_n\textcolor{black}{\mu_n}{\color{black}\Lambda_n}]^2z_n \hspace{-2pt}\\
+\hspace{-2pt} \frac{1}{4(r_n(b_n,p_n)\textcolor{black}{\nu_n})^2z_n}\big\} \\ +   \sum\limits_{n \in \mathcal{N}} (\zeta_n \cdot t_n^{\textnormal{Tx}}(b_n,p_n, s_n)) \end{array}  \hspace{-5pt} \right\} \label{eqP9} \\
&~~~~~~\textrm{s.t.} \quad   \textnormal{(\ref{constraintbn})}, \textnormal{(\ref{constraintpn})},  \nonumber
\\
&\textnormal{Problem $\mathbb{P}_{10}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$}: \hspace{-1pt}\min_{\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}}}   \hspace{-1pt} \left\{\hspace{-5pt}
\begin{array}{l} y \hspace{-2pt}\cdot\hspace{-2pt} [c_{\hspace{0.5pt}\textnormal{e}}\hspace{-2pt} \cdot \hspace{-2pt}(\sum_{n\in\mathcal{N}}E_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\hspace{-2pt}f_n^{\textnormal{MS}})\hspace{-2pt}\\
+\hspace{-2pt}\sum_{n \in \mathcal{N}}E_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\hspace{-2pt}f_n^{\textnormal{VU}}))\hspace{-1pt}]  \\  +   \sum_{n \in \mathcal{N}} (\zeta_n \cdot [t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,f_n^{\textnormal{MS}})\\
+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,f_n^{\textnormal{VU}})] \end{array}  \hspace{-5pt}
\right\} \label{eqP10} \\
&~~~~~~\textrm{s.t.} \quad   \textnormal{(\ref{constraintfMS})}, \textnormal{(\ref{constraintfn})},  \nonumber
\end{align}

Then after defining

* Statement $\mathcal{V}_{\mathbb{P}_{8}}$:

$\left\{ \begin{array}{l}
\textnormal{$T^{\#}(\textnormal{\small$\bigstar$})$ is a solution to Problem $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}))$.}
\end{array}  \right\} $,

* Statement $\mathcal{V}_{\mathbb{P}_{9}}$:

$\left\{ \begin{array}{l}
\textnormal{$[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),\alpha^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$})]$} \\ \textnormal{is a solution to the KKT conditions of} \\
\textnormal{Problem $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}))$.}
\end{array}  \right\} $, and

* Statement $\mathcal{V}_{\mathbb{P}_{10}}$:

$\left\{ \begin{array}{l}
\textnormal{$(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\textnormal{\small$\bigstar$}), \gamma^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{\delta}^{\#}(\textnormal{\small$\bigstar$})]$} \\ \textnormal{is a solution to the KKT conditions of}\\
\textnormal{Problem $\mathbb{P}_{10}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}))$.}
\end{array}  \right\} $

we obtain

\begin{align}
&\text{Statement $\mathcal{V}_{\mathbb{P}_{5}}$}  \nonumber\\
& \Leftrightarrow \left\{ \begin{array}{l}  \textnormal{Statements $\mathcal{V}_{\mathbb{P}_{8}}$,  $\mathcal{V}_{\mathbb{P}_{9}}$, and  $\mathcal{V}_{\mathbb{P}_{10}}$ hold,}  \\
\text{and $[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\textnormal{\small$\bigstar$}),$}\\
\text{$T^{\#}(\textnormal{\small$\bigstar$})  ,\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$})]$ satisfies}  \\
\text{$\mathcal{S}_{2.1} \cup \mathcal{S}_{2.2}:=\big\{\textnormal{(\ref{partialLpartialT}), (\ref{Complementarytime}),  (\ref{constraintTtau}), (\ref{Dualfeasibility}e)}\big\}$.}
\end{array}  \right\}, \nonumber \\  & \Leftrightarrow \left\{ \begin{array}{l}  \textnormal{Statements $\mathcal{V}_{\mathbb{P}_{9}}$ and  $\mathcal{V}_{\mathbb{P}_{10}}$ hold,}  \\
\text{and $[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\textnormal{\small$\bigstar$}),(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\textnormal{\small$\bigstar$}),$}\\
\text{$T^{\#}(\textnormal{\small$\bigstar$})  ,\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$})]$ satisfies}  \\
\text{$\mathcal{S}_{2.1} \cup \mathcal{S}_{2.2}:=\big\{\textnormal{(\ref{partialLpartialT}), (\ref{Complementarytime}),  (\ref{constraintTtau}), (\ref{Dualfeasibility}e)}\big\}$.}
\end{array}  \right\} \label{statev5v8910}
\end{align}

where the last step means $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ can be neglected since (<ref>) induces the objective function of $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ to be always $0$.

Now we analyze Problem $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$.
Note that $\beta$ is  the Lagrange multiplier for (<ref>). Suppose we already know $\beta$.  We move $\beta$ and (<ref>) to the objective function, and construct the following problem:

\begin{align}
&\textnormal{Problem $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$}: \nonumber\\
&\hspace{-1pt}\min_{\boldsymbol{b},\boldsymbol{p}}   \begin{array}{l} \hspace{-1pt} \left\{ H_{\mathbb{P}_{9}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})
+ \beta \cdot \big( \sum_{n \in \mathcal{N}} p_n - p_{\textnormal{max}}\big)  \right\} \end{array}\\
&~~~~~~~~\textrm{s.t.} \quad  \textnormal{(\ref{constraintbn})}, \label{eqP11}
\end{align}

where $H_{\mathbb{P}_{9}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ denotes the objective function of Problem $\mathbb{P}_{9}$.
Then after defining

* Statement $\mathcal{V}_{\mathbb{P}_{11}}$:

$\left\{ \begin{array}{l}
\textnormal{$[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$}),\alpha^{\#}(\textnormal{\small$\bigstar$})]$} \\ \textnormal{is a solution to the KKT conditions of}\\
\textnormal{Problem $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}))$.}
\end{array}  \right\} $

and checking the KKT conditions of $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$})$ and $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}))$, we have

\begin{align}
\text{Statement $\mathcal{V}_{\mathbb{P}_{9}}$}   & \Leftrightarrow \left\{ \begin{array}{l}  \textnormal{Statement $\mathcal{V}_{\mathbb{P}_{11}}$ holds,}  \\
\text{and $[\beta^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$})]$ satisfies}\\
\text{$\mathcal{S}_{1.2.2.2}:=\big\{\textnormal{(\ref{Complementarybeta})}, \textnormal{(\ref{constraintpn})}, \textnormal{(\ref{Dualfeasibility}b)}\big\}$.}
\end{array}  \right\} \label{statev8v10}
\end{align}

Now we analyze Problem $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$.
Note that $\alpha$ is  the Lagrange multiplier for (<ref>). Suppose we already know $\alpha$.  We move $\alpha$ and (<ref>) to the objective function, and construct the following problem:

\begin{align}
&\textnormal{Problem $\mathbb{P}_{12}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha)$}: \nonumber\\
&\hspace{-1pt}\min_{\boldsymbol{b},\boldsymbol{p}}   \hspace{-1pt} \left\{\hspace{-5pt}  \begin{array}{l} H_{\mathbb{P}_{11}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)  + \alpha \cdot \big( \sum_{n \in \mathcal{N}} b_n - b_{\textnormal{max}}\big)   \end{array}  \hspace{-5pt} \right\},  \label{eqP12}
\end{align}

where $H_{\mathbb{P}_{11}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$ denotes the objective function of Problem $\mathbb{P}_{11}$.
Then after defining

* Statement $\mathcal{V}_{\mathbb{P}_{12}}$:

$[\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{p}^{\#}(\textnormal{\small$\bigstar$})]$ is a globally optimal solution to Problem $\mathbb{P}_{12}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}),\alpha^{\#}(\textnormal{\small$\bigstar$}))$,

and checking the KKT conditions of $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}))$ and $\mathbb{P}_{12}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{\#}(\textnormal{\small$\bigstar$}),\beta^{\#}(\textnormal{\small$\bigstar$}),\alpha^{\#}(\textnormal{\small$\bigstar$}))$, we get

\begin{align} \text{Statement $\mathcal{V}_{\mathbb{P}_{11}}$}   & \Leftrightarrow \left\{ \begin{array}{l}  \textnormal{Statement $\mathcal{V}_{\mathbb{P}_{12}}$ holds,}  \\
\text{and $[\alpha^{\#}(\textnormal{\small$\bigstar$}),\boldsymbol{b}^{\#}(\textnormal{\small$\bigstar$})]$ satisfies }\\
\text{$\mathcal{S}_{1.2.2.1}:=\big\{\textnormal{(\ref{Stationaritypn3alpha}), (\ref{Complementaryalpha2})}\big\}$.}
\end{array}  \right\} \label{statev10v11}
\end{align}

\begin{align}
& \stackrel{\text{\ding{203}}}{\Longleftrightarrow}  \left\{ \begin{array}{l}
\text{$??$ is the solution to ? Problem ??,}  \\
\text{and ?? satisfies ??}
\end{array}  \right\} \nonumber \\  & \stackrel{\text{\ding{204}}}{\Longleftrightarrow}  \left\{ \begin{array}{l}
\text{$??$ is the solution to ? Problem ??,}  \\
\text{and ?? satisfies ??}
\end{array}  \right\} \nonumber \\  & \stackrel{\text{\ding{205}}}{\Longleftrightarrow}  \left\{ \begin{array}{l}
\text{$??$ is the solution to ? Problem ??,}  \\
\text{and ?? satisfies ??}
\end{array}  \right\} \nonumber
\end{align}

\begin{align}
-H_{\mathbb{P}_{5}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T \mid \textnormal{\small$\bigstar$}) +  \sum_{n \in \mathcal{N}} \big[\zeta_n \cdot (t_n(b_n,p_n,s_n,f_n^{\textnormal{MS}},f_n^{\textnormal{VU}}) - T)\big]
\end{align}

\begin{align}
\textnormal{Problem $\mathbb{P}_{8}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$}: \hspace{-10pt}\min_{\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}}}  & \hspace{-5pt} \left\{\hspace{-5pt}  \begin{array}{l} -\mathcal{U}(\boldsymbol{b},\hspace{-2pt}\boldsymbol{p},\hspace{-2pt}\boldsymbol{s}) \hspace{-2pt}+\hspace{-2pt} y \hspace{-2pt}\cdot\hspace{-2pt} [c_{\hspace{0.5pt}\textnormal{e}}\hspace{-2pt} \cdot \hspace{-2pt}(\sum_{n\in\mathcal{N}}E_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\hspace{-2pt}f_n^{\textnormal{MS}})\hspace{-2pt}+\hspace{-2pt}\sum_{n \in \mathcal{N}}E_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\hspace{-2pt}f_n^{\textnormal{VU}}))\hspace{-1pt}]  \\ +yc_{\hspace{0.5pt}\textnormal{e}} \hspace{-2pt}\cdot \hspace{-2pt}\sum_{n \in \mathcal{N}} \left\{[(p_n +p_n^{\textnormal{cir}})  s_n\textcolor{black}{\mu_n}{\color{black}\Lambda_n}]^2z_n \hspace{-2pt}+\hspace{-2pt} \frac{1}{4(r_n(b_n,p_n)\textcolor{black}{\nu_n})^2z_n}\right\} \\  +   \sum_{n \in \mathcal{N}} (\zeta_n \cdot (t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,f_n^{\textnormal{MS}})+t_n^{\textnormal{Tx}}(b_n,p_n, s_n)+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,f_n^{\textnormal{VU}})) \end{array}  \hspace{-5pt} \right\}  \\
\textrm{s.t.} \quad &  \textnormal{(\ref{constraintbn})}, \textnormal{(\ref{constraintpn})}, \textnormal{(\ref{constraintfMS})}, \textnormal{(\ref{constraintfn})},  \nonumber
\end{align}

§.§ Proving that the left-hand side of (<ref>) is  with respect to $\beta$

From the conditions of Proposition 1.2.1 and Proposition 1.2.2.1, setting $[\boldsymbol{b},\boldsymbol{p},\alpha]$ as
$[\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$ satisfies $\mathcal{S}_{1.2.1} \cup \mathcal{S}_{1.2.2.1}=\big\{\textnormal{(\ref{Stationaritybn}),
(\ref{Stationaritypn}), (\ref{Stationaritypn3alpha}), (\ref{Complementaryalpha2})}\big\}$; i.e., the KKT conditions

of convex optimization $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$.

\begin{align}
\begin{array}{l}
\text{$[\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$} \\  \text{is a globally optimal solution to $\mathbb{P}_{11}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$.}
\end{array} \label{P6result}
\end{align}

To prove the desired result, we consider the case where $\beta$ equals $\beta_1$, and the case where $\beta$ equals $\beta_2$, respectively, for arbitrarily chosen $\beta_1$ and $\beta_2$. Due to Result (<ref>) above, for $H_{\mathbb{P}_{11}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)$ denoting the objective function of Problem $\mathbb{P}_{11}$, we obtain

\begin{align}
&H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$} \hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_1)) \nonumber \\ & \leq \nonumber \\
&H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$} \hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_1)  , \label{eqbeta1beta21}
\end{align}

\begin{align}
&H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$} \hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_2) \nonumber \\ & \leq \nonumber\\
&H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$} \hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_2) .\label{eqbeta1beta22}
\end{align}

From (<ref>) and (<ref>), it follows that

\begin{align}
& \left[ \begin{array}{l}
H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\\
\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_1))   \\   - H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\\
\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_2))
\end{array} \right] \nonumber \\ & + \left[ \begin{array}{l}
H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\\
\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_2))   \\   - H_{\mathbb{P}_{11}}(\widetilde{\boldsymbol{b}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\\
\widetilde{\boldsymbol{p}}(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta_1)
\end{array} \right] \leq 0.  \label{eqbeta1beta2}
\end{align}

Since $H_{\mathbb{P}_{11}}(\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T \mid \beta,\iffalse T,\fi\boldsymbol{\zeta},\boldsymbol{z},y, \boldsymbol{s})$ equals $H_{\mathbb{P}_{9}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})  + \beta \cdot \big( \sum_{n \in \mathcal{N}} p_n - p_{\textnormal{max}}\big)  $ from (<ref>), the term inside the first “$[\cdot]$” of (<ref>) equals $(\beta_1 - \beta_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - p_{\textnormal{max}}\big)$, and the term inside the second “$[\cdot]$” of (<ref>) equals $(\beta_2 - \beta_1) \cdot \big( \sum_{n \in \mathcal{N}}\widetilde{p}_n(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - p_{\textnormal{max}}\big)$. Then we obtain

\begin{align}
&\left\{ \hspace{-5pt}\begin{array}{l}
(\beta_1  - \beta_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - p_{\textnormal{max}}\big) \\
+ (\beta_2 - \beta_1) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - p_{\textnormal{max}}\big)
\end{array}\hspace{-5pt} \right\} \nonumber \\
& \leq 0;
\end{align}

i.e., $(\beta_1 - \beta_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta_1,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_1,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta_2,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta_2,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})  \big) \leq 0$. Hence, $\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) $, i.e., the left-hand side of (<ref>), is  as $\beta$ increases.

§.§ Proving that the left-hand side of (<ref>) is  with respect to $\alpha$

From Proposition 1.2.1's condition, setting $[\boldsymbol{b},\boldsymbol{p}]$ as
$[\widetilde{\boldsymbol{b}}(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$})]$ satisfies $\mathcal{S}_{1.2.1}=\big\{\textnormal{(\ref{Stationaritybn}),
(\ref{Stationaritypn})}\big\}$; i.e., the KKT conditions

of convex optimization $\mathbb{P}_{12}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha)$.

\begin{align}
\begin{array}{l}
\text{$[\widetilde{\boldsymbol{b}}(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$ is a globally optimal solution } \\
\text{to $\mathbb{P}_{12}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha)$.}
\end{array} \label{P7result}
\end{align}

To prove the desired result, we consider the case where $\alpha$ equals $\alpha_1$, and the case where $\alpha$ equals $\alpha_2$, respectively, for arbitrarily chosen $\alpha_1$ and $\alpha_2$. Due to Result (<ref>) above, for $H_{\mathbb{P}_{12}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha)$ denoting the objective function of Problem $\mathbb{P}_{12}$, we obtain

\begin{align}
& H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_1) \nonumber \leq\\
&H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_2, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_2, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_1)
, \label{alphaeqalpha1alpha21}
\end{align}

\begin{align}
& H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_2, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_2, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_2) \leq \nonumber \\
&H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_2)
. \label{alphaeqalpha1alpha22}
\end{align}

From (<ref>) and (<ref>), it follows that

\begin{align}
& [ H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_1) \nonumber\\
&- H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_1, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha_2)] \nonumber \\ & +\hspace{-2pt}  [ H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_2,\hspace{-.5pt}  \beta,\hspace{-.5pt} \iffalse T,\fi\boldsymbol{\zeta}\hspace{-.5pt} \mid \hspace{-.5pt} \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_2, \hspace{-.5pt} \beta,\hspace{-.5pt} \iffalse T,\fi\boldsymbol{\zeta}\hspace{-.5pt} \mid\hspace{-.5pt}  \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\hspace{-.5pt} \boldsymbol{\zeta},\hspace{-.5pt} \beta,\hspace{-.5pt} \alpha_2)  \nonumber \\
&- H_{\mathbb{P}_{12}}(\widetilde{\boldsymbol{b}}(\alpha_2,\hspace{-.5pt} \beta,\hspace{-.5pt} \iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\widetilde{\boldsymbol{p}}(\alpha_2, \beta,\iffalse T,\fi\boldsymbol{\zeta}\hspace{-.5pt} \mid\hspace{-.5pt}  \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\hspace{-.5pt} \boldsymbol{\zeta},\hspace{-.5pt} \beta,\hspace{-.5pt} \alpha_1) \hspace{-.5pt} ] \nonumber \\ & \leq 0.  \label{alphaeqalpha1alpha2}
\end{align}

Since $H_{\mathbb{P}_{12}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta,\alpha)$ equals $H_{\mathbb{P}_{11}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta},\beta)  + \alpha \cdot \big( \sum_{n \in \mathcal{N}} b_n - b_{\textnormal{max}}\big)   $ from (<ref>),  the term inside the first “$[\cdot]$” of (<ref>) equals $(\alpha_1 - \alpha_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_1,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - b_{\textnormal{max}}\big)$,
and the term inside the second “$[\cdot]$” of (<ref>) equals $(\alpha_2 - \alpha_1) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_2,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - b_{\textnormal{max}}\big)$. Then we obtain

\begin{align}
&(\alpha_1 - \alpha_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_1,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - b_{\textnormal{max}}\big) \nonumber \\
+& (\alpha_2 - \alpha_1) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_2,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - b_{\textnormal{max}}\big) \leq 0;
\end{align}

i.e., $(\alpha_1 - \alpha_2) \cdot \big( \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_1,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) - \sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha_2,\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) \big) \leq 0$. Hence, $\sum_{n \in \mathcal{N}} \widetilde{b}_n(\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) $, i.e., the left-hand side of (<ref>), is  as $\alpha$ increases.

§.§ Proving Page lemsolvepngivenalphaandbetazeta's Lemma <ref>

From Lemma <ref> to be presented in  Appendix <ref>, $t_n(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$ is  as $\zeta_n$ increases. For $h_n(\boldsymbol{\zeta}\mid T)$ defined in (<ref>), given $[\zeta_1, \ldots, \zeta_{n-1}, \zeta_{n+1}\ldots, \zeta_N]$ and $T$, we either have (<ref>a) or (<ref>b). In either case, “$t_n(\acute{b}_n(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),s_n,\acute{f}_n^{\textnormal{MS}}(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{f}_n^{\textnormal{VU}}(\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})) - T$” or “$-\zeta_n$”  defined for $h_n(\boldsymbol{\zeta}\mid T)$ is  as $\zeta_n$ increases. Hence, $h_n(\boldsymbol{\zeta}\mid T)$ is  as $\zeta_n$ increases, given $[\zeta_1, \ldots, \zeta_{n-1}, \zeta_{n+1}\ldots, \zeta_N]$ and $T$.

§.§ Proving that the left-hand side of (<ref>) is  with respect to $T$

We recall from (<ref>) and (<ref>) that

setting $\boldsymbol{\zeta}$ as $\grave{\boldsymbol{\zeta}}(T\mid \textnormal{\small$\bigstar$})$   ensures $h_n(\boldsymbol{\zeta}\mid T)=0$ for any $n \in \mathcal{N}$,

where $h_n(\boldsymbol{\zeta}\mid T)=0$ is defined in (<ref>).

Lemma <ref> to be presented in Appendix <ref> below, we can prove that $\sum_{n \in \mathcal{N}} \grave{\zeta}_n(T\mid \textnormal{\small$\bigstar$})$; i.e., the left-hand side of (<ref>) is  with respect to $T$. The proof is similar to those in Appendices <ref> and <ref>.

To prove the desired result, we consider the case where $T$ equals $T_1$, and the case where $T$ equals $T_2$, respectively, for arbitrarily chosen $T_1 < T_2$. Below we will prove

∑_n ∈𝒩 ζ̀_n(T_1|$\bigstar$) ≥∑_n ∈𝒩 ζ̀_n(T_2|$\bigstar$).

We prove (<ref>) by contradiction. Suppose  (<ref>) does not hold; i.e., suppose we have $\sum_{n \in \mathcal{N}} \grave{\zeta}_n(T_1\mid \textnormal{\small$\bigstar$}) < \sum_{n \in \mathcal{N}} \grave{\zeta}_n(T_2\mid \textnormal{\small$\bigstar$})$. Then there exists at least one $k_1 \in \mathcal{N}$ such that $\grave{\zeta}_{k_1}(T_1\mid \textnormal{\small$\bigstar$}) < \grave{\zeta}_{k_1}(T_2\mid \textnormal{\small$\bigstar$})$.

\begin{align}
&h_n([\grave{\zeta}_1(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_{k_1-1}(T_1\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{k_1}(T_2\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{k_1+1}(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_1\mid \textnormal{\small$\bigstar$})]\mid T_1) \nonumber \\ &\leq   h_n([\grave{\zeta}_1(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_1\mid \textnormal{\small$\bigstar$})]\mid T_1)  \nonumber \\ & = 0 \nonumber
\end{align}

Suppose for $n \in \mathcal{N} \setminus \{k_1\}$ such that $\grave{\zeta}_{n}(T_1\mid \textnormal{\small$\bigstar$}) \geq \grave{\zeta}_{n}(T_2\mid \textnormal{\small$\bigstar$})$.

\begin{align}
&h_n([\grave{\zeta}_1(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_{k_1-1}(T_1\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{k_1}(T_2\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{k_1+1}(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_1\mid \textnormal{\small$\bigstar$})]\mid T_1) \nonumber \\ &\leq   h_n([\grave{\zeta}_1(T_2\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_2\mid \textnormal{\small$\bigstar$})]\mid T_1)  \nonumber
\end{align}

\begin{align}
&h_n([\grave{\zeta}_1(T_2\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_2\mid \textnormal{\small$\bigstar$})]\mid T_2) \geq 0
\end{align}

Based on the definition of
$h_n(\boldsymbol{\zeta}\mid T)$ in (<ref>), we consider the following three cases, respectively.

ζ̀^(n) := [ζ_1, ζ_2, …, ζ_n-1, 0,ζ_n+1, …, ζ_N],

$T_1 \leq T_2$

$h_n(\boldsymbol{\grave{\zeta}}(T_1\mid \textnormal{\small$\bigstar$})\mid T_1) =0$

need to prove $h_n([\grave{\zeta}_1(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_{n-1}(T_1\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_n(T_2\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{n+1}(T_1\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_1\mid \textnormal{\small$\bigstar$})]\mid T_1) \geq 0$

need to prove $h_n([\grave{\zeta}_1(T_2\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_{n-1}(T_2\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_n(T_1\mid \textnormal{\small$\bigstar$}),\grave{\zeta}_{n+1}(T_2\mid \textnormal{\small$\bigstar$}),\ldots,\grave{\zeta}_N(T_2\mid \textnormal{\small$\bigstar$})]\mid T_2) \leq 0$

$h_n(\boldsymbol{\grave{\zeta}}(T_2\mid \textnormal{\small$\bigstar$})\mid T_2) =0$

$\grave{\zeta}_n(T_2\mid \textnormal{\small$\bigstar$})$

Then from the above, we know that $\grave{\zeta}_n(T\mid \textnormal{\small$\bigstar$})$ is  as $T$ increases.?

§.§ Lemma <ref> and its proof

Given “$\textnormal{\small$\bigstar$}$” (i.e., $[\boldsymbol{z},y, \boldsymbol{s}]$) and $[\zeta_1, \ldots, \zeta_{n-1}, \zeta_{n+1}\ldots, \zeta_N]$, we have:

given $[\zeta_1, \ldots, \zeta_{n-1}, \zeta_{n+1}\ldots, \zeta_N]$, and “$\textnormal{\small$\bigstar$}$” (i.e., “$\boldsymbol{z},y, \boldsymbol{s}$”), then as $\zeta_n$ increases,

i)  $t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}), s_n)$ is ;

ii)  $t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$ is ; and

iii)  $t_n(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$ is .

Proof of Lemma <ref>:

Below we prove Results “i)”,  “ii)”, and  “iii)”, respectively.

Proving Lemma <ref>'s Result “i)”:

From Proposition 1.2's condition, setting $[\boldsymbol{b},\boldsymbol{p},\alpha,\beta]$ as
$[\acute{\boldsymbol{b}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\alpha}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$ satisfies $\mathcal{S}_{1.2.1} \cup \mathcal{S}_{1.2.2.1} \cup \mathcal{S}_{1.2.2.2}=\big\{\textnormal{(\ref{Stationaritybn}), (\ref{Stationaritypn}),  (\ref{Complementarybeta}), \textnormal{(\ref{constraintpn})}, (\ref{Dualfeasibility}b), (\ref{Stationaritypn3alpha}), (\ref{Complementaryalpha2})}\big\}$; i.e., the KKT conditions

of convex optimization $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$.

\begin{align}
\begin{array}{l}
\text{$[\acute{\boldsymbol{b}}(\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}\hspace{-2pt}\mid\hspace{-2pt} \textnormal{\small$\bigstar$})]$ is a globally optimal solution to $\mathbb{P}_{9}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$.}
\end{array} \label{P7resultlem3i}
\end{align}

To prove the desired result, we consider the case where $\zeta_n$ equals $\zeta_n^{(1)}$, and the case where $\zeta_n$ equals $\zeta_n^{(2)}$, respectively, for arbitrarily chosen $\zeta_n^{(1)}$ and $\zeta_n^{(2)}$. Due to Result (<ref>) above, after

\begin{align}
\boldsymbol{\zeta}^{(n,1)}&:=  [\zeta_1,\ldots,\zeta_{n-1},\zeta_n^{(1)},\zeta_{n+1},\ldots,\zeta_{N}] , \nonumber \\ \boldsymbol{\zeta}^{(n,2)}&:=  [\zeta_1,\ldots,\zeta_{n-1},\zeta_n^{(2)},\zeta_{n+1},\ldots,\zeta_{N}] ,\nonumber
\end{align}

then with $H_{\mathbb{P}_{9}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ denoting the objective function of Problem $\mathbb{P}_{9}$, we obtain

\begin{align}
& H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)}) \nonumber \\
&\leq H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})
, \label{alphaeqalpha1alpha21lem3i}
\end{align}

\begin{align}
& H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)}) \nonumber \\
&\leq H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})
. \label{alphaeqalpha1alpha22lem3i}
\end{align}

From (<ref>) and (<ref>), it follows that

\begin{align}
& \big[H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})  \nonumber \\
&- H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})\big] \nonumber \\ & +\hspace{-2pt}  \big[ H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})  \nonumber\\
& - H_{\mathbb{P}_{9}}(\acute{\boldsymbol{b}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{p}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})\big] \nonumber \\ & \leq 0.  \label{alphaeqalpha1alpha2lem3i}
\end{align}

Note that $H_{\mathbb{P}_{9}}(\boldsymbol{b},\boldsymbol{p} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ is given by (<ref>). Then the term inside the first “$[\cdot]$” of (<ref>) equals $(\zeta_n^{(1)} - \zeta_n^{(2)}) \cdot t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}), s_n) $, and the term inside the second “$[\cdot]$” of (<ref>) equals $(\zeta_n^{(2)} - \zeta_n^{(1)}) \cdot t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}), s_n) $. Then we obtain

\begin{align}
&(\zeta_n^{(1)} - \zeta_n^{(2)}) \cdot t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}), s_n) \nonumber \\
+ &(\zeta_n^{(2)} - \zeta_n^{(1)}) \cdot t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}), s_n) \leq 0;
\end{align}

i.e., $(\zeta_n^{(1)} - \zeta_n^{(2)}) \cdot \big( t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}), s_n) - t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}), s_n)\big) \leq 0$. Hence, $t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}), s_n) $ is  as $\zeta_n$ increases.

Proving Lemma <ref>'s Result “ii)”:

From Proposition 1.1's condition, setting $[\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},\gamma,\boldsymbol{\delta}]$ as
$[\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\gamma}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{\delta}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$ satisfies $\mathcal{S}_{1.1}:=\big\{\textnormal{(\ref{partialLpartialfMS}), (\ref{partialLpartialfVU}), (\ref{Complementarygamma}), (\ref{Complementarydelta}), \textnormal{(\ref{constraintfMS})}, \textnormal{(\ref{constraintfn})}, (\ref{Dualfeasibility}c), (\ref{Dualfeasibility}d)}\big\}$; i.e., the KKT conditions

of convex optimization $\mathbb{P}_{10}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$.

\begin{align}
\begin{array}{l}
\text{$[\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})]$ is a globally optimal solution} \\
\textnormal{to $\mathbb{P}_{10}(\textnormal{\small$\bigstar$},\boldsymbol{\zeta})$.}
\end{array} \label{P7resultlem3ii}
\end{align}

To prove the desired result, we consider the case where $\zeta_n$ equals $\zeta_n^{(1)}$, and the case where $\zeta_n$ equals $\zeta_n^{(2)}$, respectively, for arbitrarily chosen $\zeta_n^{(1)}$ and $\zeta_n^{(2)}$. Due to Result (<ref>) above, after

\begin{align}
\boldsymbol{\zeta}^{(n,1)}&:=  [\zeta_1,\ldots,\zeta_{n-1},\zeta_n^{(1)},\zeta_{n+1},\ldots,\zeta_{N}] , \nonumber \\ \boldsymbol{\zeta}^{(n,2)}&:=  [\zeta_1,\ldots,\zeta_{n-1},\zeta_n^{(2)},\zeta_{n+1},\ldots,\zeta_{N}] ,\nonumber
\end{align}

then with $H_{\mathbb{P}_{10}}(\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ denoting the objective function of Problem $\mathbb{P}_{10}$, we obtain

\begin{align}
& H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})\leq \nonumber \\
&H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})
, \label{alphaeqalpha1alpha21lem3ii}
\end{align}

\begin{align}
& H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})\leq \nonumber \\
&H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})
. \label{alphaeqalpha1alpha22lem3ii}
\end{align}

From (<ref>) and (<ref>), it follows that

\begin{align}
& \big[H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})  \nonumber \\
&- H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})\big] \nonumber \\ & +\hspace{-2pt}  \big[ H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,2)})   \nonumber \\
&- H_{\mathbb{P}_{10}}(\acute{\boldsymbol{f}}^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}),\acute{\boldsymbol{f}}^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}) \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta}^{(n,1)})\big] \nonumber \\ & \leq 0.  \label{alphaeqalpha1alpha2lem3ii}
\end{align}

Note that $H_{\mathbb{P}_{10}}(\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}} \mid \textnormal{\small$\bigstar$},\boldsymbol{\zeta})$ is given by (<ref>). Then the term inside the first “$[\cdot]$” of (<ref>) equals $(\zeta_n^{(1)} - \zeta_n^{(2)}) \cdot [t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))] $, and the term inside the second “$[\cdot]$” of (<ref>) equals $(\zeta_n^{(2)} - \zeta_n^{(1)}) \cdot [t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))]$. Then we obtain

\begin{align}
\left\{\begin{array}{l}
(\zeta_n^{(1)} - \zeta_n^{(2)}) \cdot [t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))\\
+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))]  \\
+ (\zeta_n^{(2)} - \zeta_n^{(1)}) \cdot [t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))\\
+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))]
\end{array}\right\}\leq 0;
\end{align}

namely, $(\zeta_n^{(1)} \hspace{-2pt}-\hspace{-2pt} \zeta_n^{(2)}) \cdot \left\{\begin{array}{l}
[t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))\\
+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,1)}\mid \textnormal{\small$\bigstar$}))] \\
-[t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))\\
+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}^{(n,2)}\mid \textnormal{\small$\bigstar$}))]
\end{array}\right\} \hspace{-2pt}\leq\hspace{-2pt} 0$. Therefore,  $t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$ is  as $\zeta_n$ increases.

Proving Lemma <ref>'s Result “iii)”: From (<ref>), $t_n(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$ is the sum of $t_n^{\textnormal{Tx}}(\acute{b}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\acute{p}_n(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}), s_n)$ and $t_n^{\textnormal{MS}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{MS}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))+t_n^{\textnormal{VU}:\textnormal{Pro}}(s_n,\acute{f}_n^{\textnormal{VU}}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}))$. Then the desired result clearly follows from Lemma <ref>'s Results “i)” and “ii)”.

We need to use the above to solve for $[\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T, \alpha,\beta,\gamma,\boldsymbol{\delta},\boldsymbol{\zeta}]$. Each boldfaced symbol above is an $N$-dimensional vector, so we actually need to solve for $(6N+4)$ variables. We also have $(6N+4)$ equalities, where each of (<ref>)–(<ref>),   (<ref>) and
(<ref>) has $N$ equalities, and each of  (<ref>), (<ref>), (<ref>) and  (<ref>) (i.e.,  (<ref>)) has one equality. In view of the equalities, since it is challenging to simultaneously solve for all the variables, we instead propose the following steps to compute the variables gradually (how we come up with them will be discussed soon):

Step 2.1: Suppose $\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}, \boldsymbol{z},y, \boldsymbol{s}$ are given. Based on (<ref>)–(<ref>) and (<ref>), (<ref>), (<ref>), (<ref>),   (<ref>c), (<ref>d), decide $\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},\gamma,\boldsymbol{\delta}$ as functions of $[\alpha, \beta,\iffalse T,\fi\boldsymbol{\zeta}, \boldsymbol{z},y, \boldsymbol{s}]$.

Step 2.2: On top of Step 2.1, decide $\alpha$ ensuring (<ref>), (<ref>) and (<ref>a) as a function of $[\beta,\iffalse T,\fi\boldsymbol{\zeta}, \boldsymbol{z},y, \boldsymbol{s}]$.

Step 2.3: On top of Step 2.2, decide $\beta$ ensuring (<ref>), (<ref>) and (<ref>b) as a function of $[\iffalse T,\fi\boldsymbol{\zeta}, \boldsymbol{z},y, \boldsymbol{s}]$.

Step 2.4: On top of Step 2.3, decide $\iffalse T,\fi\boldsymbol{\zeta}$ ensuring (<ref>), (<ref>), (<ref>) and (<ref>e) as functions of $[\boldsymbol{z},y,\boldsymbol{s}]$.

Step 2.5: Based on the above, we can express $[\boldsymbol{b},\boldsymbol{p},\boldsymbol{f}^{\textnormal{MS}},\boldsymbol{f}^{\textnormal{VU}},T, \alpha,\beta,\gamma,\boldsymbol{\delta},\boldsymbol{\zeta}]$ satisfying (<ref>)–(<ref>) as functions of  $[\boldsymbol{z},y,\boldsymbol{s}]$, thus solving Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$ globally.

In principle, the bisection method does not need the monotonicity of the function involved. Yet, for (<ref>)?, we still provide Lemma <ref>? below, to better understand the obtained solution, as explained below.

Theoretically, we can only show  of $\sum_{n \in \mathcal{N}} \breve{p}_n(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ as given in Lemma <ref> below since there may be degenerate cases (e.g., when all users have the same parameters). Yet, as shown in our simulation, for  cases, $\sum_{n \in \mathcal{N}} \breve{p}_n(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ is strictly decreasing as $\beta$ increases, which means the uniqueness of $\acute{\beta}(\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$})$ in practice, as the solution of $\beta$ to (<ref>)–(<ref>) given $[\boldsymbol{\zeta}, \textnormal{\small$\bigstar$}]$.

Given $[\iffalse T,\fi\boldsymbol{\zeta},\textnormal{\small$\bigstar$}]$, we have that $\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}),\beta,\boldsymbol{\zeta}\mid \textnormal{\small$\bigstar$}) $ is  as $\beta$ increases.

§.§ The bisection method to solve the KKT conditions <ref> of Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$

Based on ??, we use the bisection method several times to find a solution to the KKT conditions ?? of Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$. The details are shown in Algorithms <ref>, ??, and ??.
In principle, the bisection method does not require the monotonicity of the function to be evaluated. Yet, we still provide Propositions ??-?? below to obtain the following insight. Although  theoretically we show only “” instead of “” in Propositions ?? and ??, in practice, as long as the considered scenario is not degenerate, we actually have “” for Propositions ??-??. Since the bisection method for monotone functions has only one root, we get the insight that in  scenarios in practice, there is exactly one globally optimal solution $[\boldsymbol{b}^{\#}(\boldsymbol{z},y,\boldsymbol{s}),\boldsymbol{p}^{\#}(\boldsymbol{z},y,\boldsymbol{s}),(\boldsymbol{f}^{\textnormal{MS}})^{\#}(\boldsymbol{z},y,\boldsymbol{s}),(\boldsymbol{f}^{\textnormal{VU}})^{\#}(\boldsymbol{z},y,\boldsymbol{s}),T^{\#}(\boldsymbol{z},y,\boldsymbol{s})]$ to Problem $\mathbb{P}_{5}(\boldsymbol{z},y,\boldsymbol{s})$, which our Algorithm <ref> can find at convergence.

Given $[\alpha,\beta,\iffalse T,\fi\boldsymbol{\zeta},\boldsymbol{z},y, \boldsymbol{s}]$, the left hand side of (<ref>) is  as $\zeta_n$ increases.  ? $h_n(\boldsymbol{\zeta}\mid T)$

We define $T^{\#}(\boldsymbol{z},y,\boldsymbol{s})$ as the solution of $T$ such that

\begin{align}
\sum_{n \in \mathcal{N}} \max\{\grave{\zeta}_n(T\mid \boldsymbol{z},y, \boldsymbol{s}), 0\} = y c_{\hspace{0.8pt}\textnormal{t}}.
\end{align}

We define $\breve{\alpha}(\beta,\iffalse T,\fi\boldsymbol{\zeta}\mid \boldsymbol{z}, y, \boldsymbol{s})$ as a solution of $\alpha$ to ensuring

Then $\beta^{\#}(y, \boldsymbol{z})$ is defined as a solution of $\beta$ to ensuring

\begin{align}
\beta \cdot \bigg( \sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta\mid \boldsymbol{z}, y, \boldsymbol{s}),\beta \mid \boldsymbol{z},y, \boldsymbol{s}) - p_{\textnormal{max}}\bigg) = 0. \label{betahash2}
\end{align}

\begin{align}
\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta\mid \boldsymbol{z}, y, \boldsymbol{s}),\beta \mid \boldsymbol{z},y, \boldsymbol{s}) - p_{\textnormal{max}} = 0. \label{betahash}
\end{align}

With $\breve{\alpha}(\beta\mid \boldsymbol{z}, y, \boldsymbol{s})$ defined above, for convenience, we define

b̆_n(β,T,ζ|z,y, s): = b_n(ᾰ(β|z, y, s),β|y, z_n)  defined in ??,
p̆_n(β,T,ζ|z,y, s): = p_n(ᾰ(β|z, y, s),β|y, z_n)  defined in ??, .
f̆^MS_n(β,T,ζ|z,y, s): =
f̆^VU_n(β,T,ζ|z,y, s): =

$\alpha^*$ and $\beta^*$ satisfying ??, ?? and ?? are solutions of $\alpha$ and $\beta$ to

∑_n ∈𝒩 b_n(α,β|z,y)  = b_max ,
β·( ∑_n ∈𝒩 p_n(α, β,T,ζ|z,y, s) - p_max) = 0.

Case 1: If $\beta = 0$,

\begin{align}
\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(0\mid \boldsymbol{z},y, \boldsymbol{s}),0 \mid \boldsymbol{z},y, \boldsymbol{s}) \leq p_{\textnormal{max}} \label{conditionbeta0}
\end{align}

Case 2: If $\beta > 0$,

\begin{align}
\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(\beta\mid \boldsymbol{z}, y, \boldsymbol{s}),\beta \mid \boldsymbol{z},y, \boldsymbol{s}) = p_{\textnormal{max}}
\end{align}

We need to show

\begin{align}
\sum_{n \in \mathcal{N}} \widetilde{p}_n(\breve{\alpha}(0\mid y, \boldsymbol{z}),0 \mid \boldsymbol{z},y, \boldsymbol{s}) > p_{\textnormal{max}}
\end{align}

If ∑_n ∈𝒩 p_n(ᾰ(0|y, z),0 |z,y, s) ≤p_max, then  β(y, z) = 0,
α(y, z) = ᾰ(0|y, z),
b_n(y, z)= b_n(ᾰ(0|y, z),0 |y, z_n),