# Optimizing Utility-Energy Efficiency for the Metaverse over Wireless
Networks under Physical Layer Security

Jun Zhao, Xinyu Zhou, Yang Li, Liangxin Qian  Nanyang Technological
University<EMAIL_ADDRESS>{xinyu003, yang048,
<EMAIL_ADDRESS>

(2023)

###### Abstract.

The Metaverse, an emerging digital space, is expected to offer various
services mirroring the real world. Wireless communications for mobile
Metaverse users should be tailored to meet the following user characteristics:
1) emphasizing application-specific perceptual utility instead of simply the
transmission rate, 2) concerned with energy efficiency due to the limited
device battery and energy intensiveness of some applications, and 3) caring
about security as the applications may involve sensitive personal data. To
this end, this paper incorporates application-specific utility, energy
efficiency, and physical-layer security (PLS) into the studied optimization in
a wireless network for the Metaverse. Specifically, after introducing utility-
energy efficiency (UEE) to represent each Metaverse user’s application-
specific objective under PLS, we formulate an optimization to maximize the
network’s weighted sum-UEE by deciding users’ transmission powers and
communication bandwidths. The formulated problem belongs to the sum-of-ratios
optimization, for which prior studies have demonstrated its difficulty.
Nevertheless, our proposed algorithm 1) obtains the global optimum for the
weighted sum-UEE optimization, via a transform to parametric convex
optimization problems, 2) applies to any utility function which is concave,
increasing, and twice differentiable, and 3) achieves a linear time complexity
in the number of users (the optimal complexity in the order sense).
Simulations confirm the superiority of our algorithm over other approaches. We
explain that our technique for solving the sum-of-ratios optimization is
applicable to other optimization problems in wireless networks and mobile
computing.

Wireless networks, Metaverse, physical-layer security, resource allocation,
utility-energy efficiency.

††copyright: acmcopyright††journalyear: 2023††doi: XXXXXXX.XXXXXXX††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

The Metaverse (wang2022survey, ) is regarded as the next generation of the
Internet, which consolidates technologies including extended reality (XR),
digital twin, and wireless communications. In 2021, Facebook changed its name
to Meta, raising public interest in the Metaverse.

Mobile users typically access the Metaverse via wireless communications. It is
important to optimize wireless networks to meet the attributes of Metaverse
users, which we present next.

Characteristics of Metaverse users. We identify the following traits for
mobile users of the Metaverse.

  * ①

Users aim to maximize application-specific perceptual utility rather than
simply the transmission rate. Traditional network optimization considers the
Quality of Service (QoS), such as the transmission rate, which quantifies the
objective performance of the system. For the Metaverse, humans are the main
players, so the Quality of Experience (QoE) capturing the perceptual
experience of users is a better metric than QoS. To this end, our utility
model should be adjusted accordingly.

  * ②

Users care about energy efficiency due to the limited battery of mobile
devices and energy intensiveness of some applications. For instance, Meta
Quest 2 with a fully charged battery can last for just 2 hours for gaming or 3
hours for video watching (chung2022xr, ).

  * ③

Users are concerned with security since certain Metaverse applications may
involve personal (e.g., biometric and health) data. Researchers at UC Berkeley
have shown in (nair2022exploring, ) that many existing Metaverse applications
are vulnerable to privacy breaches by an attacker who tries to infer users’
sensitive information.

The Metaverse over wireless networks: Utility-energy efficiency optimization
under physical-layer security. Since mobile users accessing the Metaverse are
constrained by wireless communication resources, it is vital to tailor
wireless networks to match the above characteristics of Metaverse users. We
formalize an optimization problem about the utility-energy efficiency (UEE)
under physical-layer security for the motivation discussed below, where UEE
for each user is defined as the application-specific perceptual utility over
energy consumption.

Energy efficiency (EE) plays a vital role in both the economy and the
environment. A faster transmission rate providing a higher quality of
experience for users will also increase energy consumption. Therefore, it is
essential to build an energy-efficient Metaverse system. Nevertheless, it is
not viable to emphasize energy saving overwhelmingly. The Metaverse will
provide many digital services, and lower transmission speeds will affect
users’ access to profits and high-quality experiences. Hence, how to allocate
the resources (e.g., the transmission power and bandwidth) in the network to
maximize the weighted sum of all users’ UEE is worth investigating, where each
user’s weight represents its priority in the optimization. The weighted sum-
UEE optimization aims to save energy and improve the utilities for users,
addressing “①” and “②” above.

For “③” above, the confidential data of Metaverse applications should be
accessible to only the intended users instead of eavesdroppers. To this end,
we aim to achieve physical-layer security to protect the information during
transmission. Secrecy capacity is an important metric in physical-layer
security. It refers to the communication rate that does not leak information
to an eavesdropper. In order to keep the information of users from the
eavesdroppers, we extend our Metaverse energy efficiency problem to physical-
layer security by considering the secrecy rate instead of the original rate.

Our contributions include problem formulation, a widely applicable
optimization technique, and an optimal algorithm in terms of the solution
quality and time-complexity order, as listed below.

  * •

We formulate the problem of maximizing the weighted sum of users’ utility-
energy efficiency (PLS) under physical-layer security for the Metaverse, by
deciding users’ transmission powers and bandwidth allocation. To the best of
our knowledge, this problem has not yet been studied in the literature, inside
and outside of Metaverse research.

  * •

The formulated problem belongs to the sum-of-ratios optimization, which is
non-convex. We explain that the problem is difficult to solve even using the
pseudoconcavity notion.

  * •

Despite the challenges, we solve the problem and develop an algorithm, via the
technique of transforming the sum-of-ratios to parametric convex optimization
problems.

  * •

Our proposed algorithm

    * •

obtains the global optimum,

    * •

applies to any utility function which is concave, increasing, and twice
differentiable, and

    * •

allows heterogeneous utility-function types among the users,

    * •

runs in linear time with respect to the number of users, which means the
optimal complexity in the order sense.

  * •

Simulations demonstrate the superiority of our algorithm over other approaches
in terms of the solution quality and time complexity. The utility functions
used in the simulations are based on real-world datasets.

  * •

We explain that our technique can go beyond our problem to handle functions of
product or quotient terms in general mathematical optimization. We illustrate
this by discussing example problems in wireless networks and mobile computing.
Researchers can use our technique to solve difficult problems.

Roadmap. The rest of the paper is organized as follows. Section 2 provides
related studies. In Section 3, we formulate the studied optimization problem.
Section 4 presents the challenges in solving the problem. Section 5 elaborates
on our algorithm which finds a global optimum of the problem. In Section 6, we
discuss the application of our optimization technique to other problems.
Simulation results are reported in Section 7. Section 8 concludes the paper.

##  2\. Related Work

We survey related research: energy efficiency and physical-layer security in
Section 2.1, and wireless Metaverse in Section 2.2.

###  2.1. Energy efficiency optimization and physical-layer security in
wireless networks

In wireless networks, the traditional notion of energy efficiency (EE) for a
user is defined as the ratio of data rate over power consumption (i.e., the
ratio of transmitted data size over energy consumption). Maximizing the
weighted sum of EE (WSEE) is addressed in (zamani2020optimizing, ; wu2016user,
). Different from WSEE, the system EE in (du2022weighted, ) is defined as the
ratio of all users’ sum rates over all users’ sum power consumption.

EE, WSEE, and system EE above do not examine specific application
requirements. Accommodating various applications requires the concept of
utility-energy efficiency (UEE), which for a user is the ratio of the
application-specific rate-dependent utility over power consumption. UEE in our
paper has also been investigated in (meshkati2009energy, ), which adopts game
theory to model an interference-constrained wireless network, where each user
maximizes its own UEE by deciding its transmission power. Different from UEE
defined for individual users, the system UEE in (huang2018utility, ) results
from dividing the sum of all users’ rate-dependent utilities by the sum of all
users’ power consumption. This system UEE optimization in (huang2018utility, )
deals with just one ratio, which is much easier than the sum-of-ratios
optimization in our paper. Moreover, the optimization method of
(huang2018utility, ) is applicable to only the specific utility function
$\kappa_{n}\ln r_{n}$ for data rate $r_{n}$ and constant $\kappa_{n}$. Even
just changing the utility function to $\kappa_{n}\ln(1+r_{n})$ will make
(huang2018utility, )’s approach invalid; in particular, (18a) in
(huang2018utility, ) will be non-concave and non-convex after the above
change. In contrast, our work applies to any utility function that is concave,
increasing, and twice differentiable. Besides the above major differences,
(huang2018utility, ) considers interference-constrained wireless networks and
optimizes only the transmission powers, while we adopt FDMA and jointly
optimize the transmission powers and bandwidth allocation.

Next, we discuss the incorporation of physical-layer security (PLS) into EE
optimization. Because the WSEE as the sum of ratios is more difficult to
analyze than the system EE, existing studies incorporating PLS into EE
typically investigate the system EE instead of WSEE, after replacing the
achievable rates with secrecy rates, as shown in (jiang2023secrecy, ;
zappone2019secrecy, ). Despite the above work on EE optimization under PLS, we
are unable to find any prior work on UEE optimization under PLS and hence the
problem of our paper is new.

###  2.2. Metaverse over wireless networks

Calibrating wireless networks for mobile users accessing the Metaverse is an
emerging research topic. Recently, a number of papers on the topic have
appeared in different venues: (yu2022asynchronous, ) in JSAC co-authored by
the first author of the current paper, (meng2022sampling, ) in JSAC,
(wang2023semantic, ) in TWC, (jiang2022reliable, ; ren2022quantum, ) in TVT,
and a survey paper (wang2022survey, ) in COMST, where the meanings of the
abbreviations can be found in the references.

Among the technical papers above, (yu2022asynchronous, ; meng2022sampling, ;
ren2022quantum, ) adopt reinforcement learning to optimize wireless
performance for the Metaverse, while (wang2023semantic, ; jiang2022reliable, )
utilize economic theories to incentivize users for improving the usage of
semantic-aware sensing and coded distributed computing for wireless Metaverse.
The current paper’s co-authors have recently optimized wireless federated
learning in (zhou2022resource, ) for the Metaverse via alternating
optimization, which achieves neither local nor global optimum. In contrast,
our technique of the current paper goes beyond UEE optimization under PLS.
Using it in (zhou2022resource, ) will obtain a global optimum, based on our
Section 6 later.

##  3\. Problem Formulation

In this section, we will present the system model and formalize the
optimization problem.

###  3.1. System model and metrics

In our studied system, a base station acts as the Metaverse server for $N$
legitimate users $U_{n}|_{n=1,\ldots,N}$. There are also $N$ eavesdroppers
$E_{n}|_{n=1,\ldots,N}$, where $E_{n}$ tries to intercept the communication
between $U_{n}$ and the server. Fig. 1 illustrates our system.

Figure 1. Our system: A server provides Metaverse services for $N$ legitimate
users $U_{n}|_{n=1,\ldots,N}$, while the $n$th eavesdropper $E_{n}$ tries to
intercept the communication between user $U_{n}$ and the server. The studied
problem is to maximize the weighted sum of all users’ utility-energy
efficiency by deciding the bandwidth allocation and transmission powers.

Our problem is applicable to downlink and uplink communications between all
legitimate users and the Metaverse server. Suppose the communications follow
frequency division multiple access (FDMA), where different legitimate users’
signals will not interfere with each other. For each legitimate user $U_{n}$,
let $B_{n}$ be the bandwidth, and $p_{n}$ be its transmission power in the
case of uplink communication, or the transmission power of the server used to
communicate with $U_{n}$ in the case of downlink communication. For
simplicity, below we use uplink communication to introduce the problem.
Throughout this paper, the $n$th dimension of an $N$-dimensional vector
$\bm{x}$ is denoted by $x_{n}$ (unless stated otherwise). Hence, we have
$\bm{p}:=[p_{1},p_{2},\ldots,p_{N}]$ and $\bm{B}:=[B_{1},B_{2},\ldots,B_{N}]$.

Transmission rate. According to the Shannon formula, the transmission rate
$r_{n}(p_{n},B_{n})$ of legitimate user $U_{n}$ is

(1) $\displaystyle
r_{n}(p_{n},B_{n})=B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}}),$

where ${\sigma_{n}}^{2}$ is the power spectral density of Gaussian noise,
$g_{n}$ is the channel attenuation from $U_{n}$ to the server. The function
notation is used in this paper; e.g., $r_{n}(p_{n},B_{n})$ is a function of
$p_{n}$ and $B_{n}$.

Secrecy rate. Eavesdropper $E_{n}$ aims to intercept the communication between
legitimate user $U_{n}$ and the server. Let $r_{n,\textnormal{e}}$ be the
eavesdropping rate of $E_{n}$. We consider $r_{n,\textnormal{e}}$ as a
constant depending on only $n$. Then the secrecy rate of $U_{n}$ is given by

(2) $\displaystyle
r_{n,\textnormal{s}}(p_{n},B_{n}):=r_{n}(p_{n},B_{n})-r_{n,\textnormal{e}}.$

Utility. We regard user $U_{n}$’s application-specific perceptual utility rate
as a function of the secrecy rate $r_{n,\textnormal{s}}(p_{n},B_{n})$ to
emphasize physical-layer security (PLS). Specifically, using111We require
$f_{n}(x)$ to be defined for any $x>0$. We do not require $f_{n}(x)$ to be
defined for $x=0$, but if $\lim_{x\to 0^{+}}f_{n}(x)$ exists and is finite, we
can just use it to define $f_{n}(0)$. We also do not enforce any condition on
whether $f_{n}(x)$ is non-negative or not. Additional conditions of $f_{n}(x)$
are discussed in Section 4.2. $f_{n}(\cdot):(0,\infty)\to(-\infty,\infty)$ to
denote the utility rate function, user $U_{n}$’s utility rate is given by
$f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))$. Consider a small time interval
$[t,t+\Delta t]$, where $\Delta t$ is small enough such that
$r_{n,\textnormal{s}}(p_{n},B_{n})$ can be seen as invariant during
$[t,t+\Delta t]$. Then the utility of user $U_{n}$ over the time interval
$[t,t+\Delta t]$ is $\mathcal{U}_{n}^{[t,t+\Delta
t]}:=f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))\Delta t$.

Power & energy consumption. The same as (huang2018utility, ; xu2013throughput,
), the power consumed by user $U_{n}$ includes not just the transmission power
$p_{n}$, but also the circuit power $p_{n}^{\textnormal{cir}}$, which is a
constant given $n$. During the time interval $[t,t+\Delta t]$, user $U_{n}$’s
energy consumption is given by $\mathcal{E}_{n}^{[t,t+\Delta
t]}:=(p_{n}+p_{n}^{\textnormal{cir}})\Delta t$.

Utility-energy efficiency. For user $U_{n}$, we define its utility-energy
efficiency (UEE) $\varphi_{n}(p_{n},B_{n})$ under PLS as the ratio of
$\frac{\mathcal{U}_{n}^{[t,t+\Delta t]}}{\mathcal{E}_{n}^{[t,t+\Delta t]}}$
for small enough $\Delta t$, which induces

(3)
$\displaystyle\varphi_{n}(p_{n},B_{n}):=\frac{f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))}{p_{n}+p_{n}^{\textnormal{cir}}}=\frac{f_{n}(r_{n}(p_{n},B_{n})-r_{n,\textnormal{e}})}{p_{n}+p_{n}^{\textnormal{cir}}}.$

When $f_{n}(\cdot)$ becomes the identity function (i.e., $f_{n}(x)=x$),
$\varphi_{n}(p_{n},B_{n})$ becomes
$\frac{r_{n,\textnormal{s}}(p_{n},B_{n})}{p_{n}+p_{n}^{\textnormal{cir}}}$
(i.e., $\frac{\text{secrecy rate}}{\text{power consumption}}$), which is just
the traditional notion of energy efficiency under PLS (jiang2018secrecy, ).

###  3.2. Utility-energy efficiency (UEE) optimization

Our goal is to maximize the weighted sum of all users’ UEE under PLS. This
optimization problem is formalized as follows:

(4) Problem $\mathbb{P}_{1}$:
$\displaystyle\max_{\bm{p},\bm{B}}~{}\sum_{n\in\mathcal{N}}c_{n}\varphi_{n}(p_{n},B_{n})$
(4a) subject to: $\displaystyle\sum_{n\in\mathcal{N}}B_{n}\leq
B_{\textnormal{total}},$ (4b) $\displaystyle r_{n}(p_{n},B_{n})\geq
r_{n}^{\min},\text{ for all }n\in\mathcal{N}:=\\{1,\cdots,N\\},$

where $c_{n}>0$ represents the priority of user $U_{n}$ in the optimization.
Larger $c_{n}$ means higher priority. Constraints (4a) sets the total
bandwidth for FDMA. Constraint (4b) ensures that the transmission rate
$r_{n}(p_{n},B_{n})$ of user $U_{n}$ should be at least a constant
$r_{n}^{\min}$ ($r_{n}^{\min}$ can vary for different $n$). Condition 1 below
is about minimum legitimate rates $r_{n}^{\min}|_{n\in\mathcal{N}}$ and
eavesdropping rates $r_{n,\textnormal{e}}|_{n\in\mathcal{N}}$.

###### Condition 1.

For all $n\in\mathcal{N}$, we have $r_{n}^{\min}\geq
r_{n,\textnormal{e}},r_{n}^{\min}>0,r_{n,\textnormal{e}}\geq 0$.

We have the following remarks about Condition 1.

###### Remark 1.

Condition 1 with (4b) ensures $r_{n}(p_{n},B_{n})\geq r_{n,\textnormal{e}}$;
i.e., each user $U_{n}$’s secrecy rate $r_{n,\textnormal{s}}(p_{n},B_{n})$ is
non-negative.

###### Remark 2.

Condition 1 covers the following special case where we do not impose physical-
layer security but still enforce a minimum transmission rate for each user:
$r_{n,\textnormal{e}}=0$ and $r_{n}^{\min}>0$ for all $n\in\mathcal{N}$.

###### Remark 3.

We enforce $r_{n}^{\min}>0$ in Condition 1 so that each user $U_{n}$ will
always be allocated with a strictly positive bandwidth; i.e., $B_{n}>0$ for
all $n\in\mathcal{N}$. This avoids analyzing the degenerate case where only a
subset of $N$ users share the total bandwidth $B_{\textnormal{total}}$.

We also comment on how Problem $\mathbb{P}_{1}$ is optimized in practice.

###### Remark 4.

Problem $\mathbb{P}_{1}$ will be solved using our Algorithm 1 in Section 5.2.
Then a question is which entity solves $\mathbb{P}_{1}$ in practice. We let
the Metaverse server perform the task, assuming that it has obtained the
values of $p_{n}^{\textnormal{cir}},r_{n}^{\min},r_{n,\textnormal{e}}$ for all
$n$ (e.g., these are shared with the server before the optimization stage).
After the server solves $\mathbb{P}_{1}$, it will notify each legitimate user
$U_{n}$ of the $p_{n}$ and $B_{n}$ values.

##  4\. Challenges in Solving Problem $\mathbb{P}_{1}$

We first state the optimization preliminaries and conditions of the function
$f_{n}(x)$, which are used to explain the difficulty in solving
$\mathbb{P}_{1}$.

###  4.1. Preliminaries of mathematical optimization

Let $f(\bm{x})$ be a function defined on a convex set $\mathcal{S}$, which is
a subset of a real vector space. Then we have the following from Definitions
1.3.1, 2.2.1, and 3.2.1 of the book (cambini2008generalized, ).

###### Definition 1 (Convexity).

$f$ is convex in $\bm{x}$ if and only if for any
$\bm{x}_{1},\bm{x}_{2}\in\mathcal{S}$ and $t\in[0,1]$, it holds that
$f(t\bm{x}_{1}+(1-t)\bm{x}_{2})\leq tf(\bm{x}_{1})+(1-t)f(\bm{x}_{2})$.

###### Definition 2 (Pseudoconvexity).

$f$ is pseudoconvex in $\bm{x}$ if and only if for any
$\bm{x}_{1},\bm{x}_{2}\in\mathcal{S}$, $f(\bm{x}_{1})>f(\bm{x}_{2})$ implies
$\nabla f(\bm{x}_{1})\cdot(\bm{x}_{2}-\bm{x}_{1})<0$, where $\nabla f$ denotes
the gradient of $f$.

###### Definition 3 (Quasiconvexity).

$f$ is quasiconvex in $\bm{x}$ if and only if for any
$\bm{x}_{1},\bm{x}_{2}\in\mathcal{S}$ and $t\in[0,1]$, it holds that
$f(t\bm{x}_{1}+(1-t)\bm{x}_{2})\leq\max\\{f(\bm{x}_{1}),f(\bm{x}_{2})\\}$.

With convexity above, Lemma 4.1 helps us understand concavity.

###### Lemma 4.1 (Convexity versus Concavity).

A function $f$ is said to be concave (resp., pseudoconcave, quasiconcave) if
and only if $-f$ is convex (resp., pseudoconvex, quasiconvex).

For the reasoning behind Lemma 4.1, interested readers can refer to Section 3
of the book (cambini2008generalized, ). Lemma 4.2 below presents the
relationships between the definitions discussed above.

###### Lemma 4.2 (Relationships between notions).

With “$\Rightarrow$” denoting “implies”, we have the following assuming
differentiability

(5)
$\displaystyle\text{Convexity}\Rightarrow\text{Pseudoconvexity}\Rightarrow\text{Quasiconvexity},\text{
and}$ (6)
$\displaystyle\text{Concavity}\Rightarrow\text{Pseudoconcavity}\Rightarrow\text{Quasiconcavity}.$

Lemma 4.2 follows from Fig. 2.2 and Fig. B.1 of the book
(cambini2008generalized, ).

For a minimization problem, if the objective function and222Note that the
constraint of a convex (resp., concave) function being at most (resp., least)
a constant is a convex constraint. the constraints are all convex, then we
have a convex optimization problem, for which the following lemma holds.

###### Lemma 4.3 (Chapters 3 and 4 of (boyd2004convex, )).

For convex optimization, the Karush–Kuhn–Tucker (KKT) conditions are

  * •

sufficient for optimality, and

  * •

are necessary for optimality if Slater’s condition holds (i.e., if the
feasible set contains at least one interior point).

Readers unfamiliar with the KKT conditions can refer to Theorem 4.2.3 of
(cambini2008generalized, ), and can also look into (13a)–(13l) to be presented
on Page 13a of the current paper, where we will use the KKT conditions.

Lemma 4.4 below broadens problems under which KKT conditions are sufficient
for optimality, to go beyond convex optimization.

###### Lemma 4.4 (Theorem 4.4.1 of (cambini2008generalized, )).

For a minimization problem with all constraints being inequalities, if the
objective function is pseudoconvex, and all constraints are quasiconvex and
differentiable, then a feasible point satisfying the KKT conditions is
globally optimal.

###  4.2. Conditions of the utility function $f_{n}(x)$

The requirements of the utility rate function $f_{n}(x)$ for any
$n\in\mathcal{N}$ are formally presented as Condition 2 below (we will just
call $f_{n}(x)$ as the utility function hereafter for simplicity).

###### Condition 2.

The utility function $f_{n}(x)$ for any $n\in\mathcal{N}$ is concave,
increasing, and twice differentiable, with respect to $x>0$; i.e.,
$f_{n}^{\prime\prime}(x)\leq 0$ and $f_{n}^{\prime}(x)>0$ for $x>0$.

In Condition 2, the concavity of $f_{n}(x)$ means diminishing marginal return,
which holds in various practical applications (yang2012crowdsourcing, ;
liu2018edge, ; mo2000fair, ; xiong2020reward, ).

We also remark that $f_{n}(x)$, $f_{n}^{\prime}(x)$ and
$f_{n}^{\prime\prime}(x)$ are defined for any $x>0$. Similar to Footnote 1,
for any of $f_{n}(x)$, $f_{n}^{\prime}(x)$ and $f_{n}^{\prime\prime}(x)$, we
do not require it to be defined for $x=0$. If it has a finite (resp., no)
limit as $x\to 0^{+}$, we can just use the limit to define the corresponding
value at $x=0$ (resp., do not define any value at $x=0$).

We can even allow heterogeneous types of utility functions among the users.
For the specific expressions of the utility function $f_{n}(x)$ used in
simulations, we will discuss three types in Section 7.1.

We now discuss the properties of functions in our studied system.

###### Lemma 4.5 (Lemma 1 of (zhou2022resource, )).

$r_{n}(p_{n},B_{n})$ is jointly concave with333For a function $f(\bm{x})$,
“being convex (resp., concave) in $\bm{x}$” has the same meaning as “jointly
convex (resp., concave) in all dimensions of the vector $\bm{x}$”. respect to
$p_{n}$ and $B_{n}$.

###### Lemma 4.6.

Under Condition 2, we have:

  * •

$f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))$ is jointly concave with respect to
$p_{n}$ and $B_{n}$;

  * •

$\varphi_{n}(p_{n},B_{n})$ is jointly pseudoconcave with respect to $p_{n}$
and $B_{n}$.

###### Proof.

From Lemma 4.5 and Eq. (2), $r_{n,\textnormal{s}}(p_{n},B_{n})$ is jointly
concave in $p_{n}$ and $B_{n}$. According to the composition rule in Eq.
(3.11) of (boyd2004convex, ), for concave $f_{n}(\cdot)$, since
$\widetilde{f_{n}}(\cdot)$ defined as $f_{n}(\cdot)$ (resp., $-\infty$) for
points inside (resp., outside) of the domain of $f_{n}$ is non-decreasing, the
function $f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))$ is jointly concave in
$p_{n}$ and $B_{n}$.

From Page 245 (the book’s internal page number, not the pdf page number) of
the book (cambini2008generalized, ), for a ratio, if the numerator is non-
negative, concave and differentiable, and the denominator is positive, convex
and differentiable, then the ratio is pseudoconcave. Based on the above, we
have proved the pseudoconcavity of $\varphi_{n}(p_{n},B_{n})$ with respect to
$p_{n}$ and $B_{n}$. ∎

###  4.3. Challenges of solving Problem $\mathbb{P}_{1}$

Based on the proof of Lemma 4.6, we now call
$\small\frac{f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))}{p_{n}+p_{n}^{\textnormal{cir}}}$
(i.e., $\varphi_{n}(p_{n},B_{n})$) a concave-convex ratio: a ratio having a
concave function as the numerator and a convex function as the denominator.
Then $\mathbb{P}_{1}$ is maximizing the sum of concave-convex ratios. Such
sum-of-ratios optimization is non-convex and difficult to solve
(jong2012efficient, ; shen2018fractional, ).

Lemma 4.6 also shows that $\varphi_{n}(p_{n},B_{n})$ for each $n$ is
pseudoconcave, unfortunately the sum of pseudoconcave functions may not be
pseudoconcave. Even if we manage to prove the pseudoconcavity of
$\sum_{n\in\mathcal{N}}c_{n}\varphi_{n}(p_{n},B_{n})$ (the objective function
of $\mathbb{P}_{1}$), which is very difficult (e.g., just analyzing the
pseudoconvexity of the sum of two linear fractional functions is already
challenging, as shown in (cambini2005pseudoconvexity, )), then we can in
principle use the KKT conditions of Problem $\mathbb{P}_{1}$, as explained
in444For $\mathbb{P}_{1}$, all constraints are differentiable and convex (and
hence quasiconvex) with Lemma 4.5 and Foonote 2. Hence, if we can prove that
the objective function is pseudoconcave, Lemma 4.4 means that KKT conditions
can solve $\mathbb{P}_{1}$. Nonetheless, even if we can do the above, the KKT
conditions of $\mathbb{P}_{1}$ are intractable to get a solution.  Footnote 4,
but those conditions involve taking derivatives of the ratios, inducing quite
complex expressions, and the corresponding analysis becomes intractable. In
this paper, instead of analyzing the pseudoconcavity of
$\sum_{n\in\mathcal{N}}c_{n}\varphi_{n}(p_{n},B_{n})$ and being trapped in the
intractable analysis, we will present an elegant approach (to be detailed in
Section 5.1) for solving Problem $\mathbb{P}_{1}$.

Recently, Shen and Yu (shen2018fractional, ; shen2018fractional2, ) proposed a
novel technique to solve the sum-of-ratios optimization (referred to as
fractional programming in their papers). However, since their technique relies
on block coordinate ascent (i.e., alternating optimization), applying their
technique to our Problem $\mathbb{P}_{1}$ will find a point which has no local
or global optimality guarantee. In contrast, our approach will find a globally
optimal solution of $\mathbb{P}_{1}$.

##  5\. Algorithm to Find A Global Optimum

In this section, we will discuss how to transform $\mathbb{P}_{1}$ into a
sequence of convex optimization problems, and then use the transform to
propose an algorithm that finds a global optimum of $\mathbb{P}_{1}$.

###  5.1. Transforming Problem $\mathbb{P}_{1}$ into parametric convex
optimization problems

Firstly, we introduce an auxiliary variable $\beta_{n}$ to transform Problem
$\mathbb{P}_{1}$ into the epigraph form. Let
$\frac{c_{n}f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))}{p_{n}+p_{n}^{\textnormal{cir}}}\geq\beta_{n}$
and $\mathbb{P}_{1}$ can be transformed to the following equivalent form as
$\mathbb{P}_{2}$:

(7) Problem $\mathbb{P}_{2}$:
$\displaystyle\max_{\bm{p},\bm{B},\bm{\beta}}~{}\sum_{n\in\mathcal{N}}\beta_{n},\text{
where }\mathcal{N}:=\\{1,\cdots,N\\},$ (7a) subject to:
$\displaystyle\text{(\ref{constra:bandwidth}), (\ref{constra:rate})},$ (7b)
$\displaystyle
F_{n}(p_{n},B_{n})-\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})\geq
0,\textnormal{ for all $n\in\mathcal{N}$},$

where we use $F_{n}(p_{n},B_{n})$ to simplify the representation:

(8) $\displaystyle
F_{n}(p_{n},B_{n}):=c_{n}f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n})).$

Problem $\mathbb{P}_{2}$ is not convex optimization since
$\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$ in (7b) is not jointly convex
(actually also not jointly concave) in $\beta_{n}$ and $p_{n}$, since the
Hessian matrix for $\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$ is
$\tiny\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ which is not positive semifinite
(actually also not negative semidefinite).

We have explained in Section 4.3 that Problem $\mathbb{P}_{1}$ belongs to the
following kind of problems: maximizing the sum of concave-convex ratios. Such
problems have at least one global maximum according to (jong2012efficient, ;
shen2018fractional, ). Hence, $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ have at
least one global maximum.

To solve Problem $\mathbb{P}_{2}$, one initial idea is trying to use Lemma 4.4
and hence the KKT conditions directly. Yet, deciding the quasiconvexity of
$F_{n}(p_{n},B_{n})-\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$ in (7b) is
very difficult. Hence, instead of trying to use $\mathbb{P}_{2}$’s KKT
conditions directly, we take a step back and use the Fritz-John conditions
(viz., Remark 4.2.2 of (cambini2008generalized, ) and Lemma 2.1’s proof in
(jong2012efficient, )), which do not need the quasiconvexity of constraints.
The Fritz-John conditions provide the necessary conditions for a global
optimum. Basically, in the Fritz-John conditions, the Lagrange multiplier (say
$w$) on the gradient of the objective function can be zero or positive. Yet,
following the proof of Lemma 2.1 in (jong2012efficient, ), we obtain $w>0$.
Then as shown in (jong2012efficient, ), $w$ can be absorbed into other
multipliers and hence omitted, after which the Fritz-John conditions reduce to
the KKT conditions. Based on the above discussion, we have

(11) $\textstyle\begin{array}[]{l}\text{any global maximum of $\mathbb{P}_{2}$
needs to satisfy}\\\ \text{the KKT conditions
(\ref{P2KKTeqStationaritypn})--(\ref{P2KKTeqDualfeasibility})
below.}\end{array}$

For Problem $\mathbb{P}_{2}$, with $\bm{\nu}:=[\nu_{n}|_{n\in\mathcal{N}}]$,
$\bm{\tau}:=[\tau_{n}|_{n\in\mathcal{N}}]$ and $\lambda$ denoting the
multipliers, and the Lagrangian function given by

$\textstyle
L_{\mathbb{P}_{2}}(\bm{p},\bm{B},\bm{\beta},\bm{\nu},\bm{\tau},\lambda)$
$\textstyle=-\sum_{n\in\mathcal{N}}\beta_{n}+\sum_{n\in\mathcal{N}}\nu_{n}\cdot\big{(}\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})-F_{n}(p_{n},B_{n})\big{)}$
(12)
$\textstyle\quad\,\textstyle{+\sum_{n\in\mathcal{N}}\tau_{n}\cdot(r_{n}^{\min}-r_{n})+\lambda\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\text{total}})},$

the KKT conditions of Problem $\mathbb{P}_{2}$ are as follows, with
$L_{\mathbb{P}_{2}}$ short for
$L_{\mathbb{P}_{2}}(\bm{p},\bm{B},\bm{\beta},\bm{\nu},\bm{\tau},\lambda)$ (see
(cambini2008generalized, , Theorem 4.2.3) or (boyd2004convex, , Section 1.4.2)
for a formal introduction to the KKT conditions):

Stationarity: (13a) $\textstyle\frac{\partial L_{\mathbb{P}_{2}}}{\partial
p_{n}}=0,\text{ for all }n\in\mathcal{N},$ (13b) $\textstyle\frac{\partial
L_{\mathbb{P}_{2}}}{\partial B_{n}}=0,\text{ for all }n\in\mathcal{N},$ (13c)
$\textstyle\frac{\partial
L_{\mathbb{P}_{2}}}{\partial\beta_{n}}=-1+\nu_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})=0,\text{
for all }n\in\mathcal{N};$ Complementary slackness: (13d)
$\textstyle\nu_{n}\cdot\big{(}\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})-F_{n}(p_{n},B_{n})\big{)}=0,\text{
for all }n\in\mathcal{N},$ (13e)
$\textstyle\tau_{n}\cdot(r_{n}^{\min}-r_{n})=0,\text{ for all
}n\in\mathcal{N},$ (13f)
$\textstyle\lambda\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\text{total}})=0;$
Primal feasibility: (13g) $\textstyle
F_{n}(p_{n},B_{n})-\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})\geq
0,\textnormal{ for all $n\in\mathcal{N}$},$ (13h) $\textstyle
r_{n}(p_{n},B_{n})\geq r_{n}^{\min},\text{for all }n\in\mathcal{N},$ (13i)
$\textstyle\sum_{n\in\mathcal{N}}B_{n}\leq B_{\text{total}};$ Dual
feasibility: (13j) $\textstyle\nu_{n}\geq 0,\text{for all }n\in\mathcal{N},$
(13k) $\textstyle\tau_{n}\geq 0,\text{for all }n\in\mathcal{N}.$ (13l)
$\textstyle\lambda\geq 0.$

From (13c), it follows that

(14) $\textstyle\nu_{n}=\frac{1}{p_{n}+p_{n}^{\textnormal{cir}}},$

and (13j) holds.

Using (14) in (13d), we know

(15)
$\textstyle\beta_{n}=\frac{F_{n}(p_{n},B_{n})}{p_{n}+p_{n}^{\textnormal{cir}}},$

and (13g) holds (actually the equal sign in (13g) is taken).

Instead of solving $\mathbb{P}_{2}$’s KKT conditions (13a)–(13l) directly,
which is complex, we will connect them to a series of parametric convex
optimization problems. In particular, supposing that $\bm{\beta}$ and
$\bm{\nu}$ are already given and satisfy (13c) (13d) (13g) and (13j), then we
have the following result for the rest of $\mathbb{P}_{2}$’s KKT conditions:

(18) $\textstyle\begin{array}[]{l}\text{(\ref{P2KKTeqStationaritypn})
(\ref{P2KKTeqStationarityBn}) (\ref{P2KKTeqComplementarytaun})
(\ref{P2KKTeqComplementarylambda}) (\ref{P2KKTeqPrimalfeasibility2})
(\ref{P2KKTeqPrimalfeasibility3}) (\ref{P2KKTeqDualfeasibilitytau}) and
(\ref{P2KKTeqDualfeasibility}), denoted by}\\\ \text{set $\mathcal{K}$, form
the KKT conditions of Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$
below.}\end{array}$

where we have

(19) Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$:
$\displaystyle\max_{\bm{p},\bm{B}}\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})$
$\displaystyle\textnormal{subject to:
}~{}\textnormal{(\ref{constra:bandwidth}), (\ref{constra:rate})}.$

with $\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})$ defined as follows:

(20) $\displaystyle\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n}):$
$\displaystyle=\nu_{n}\cdot\big{(}F_{n}(p_{n},B_{n})-\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})\big{)}.$

Lemma 5.1 below states the relationship between $\mathbb{P}_{2}$ and
$\mathbb{P}_{3}$.

###### Lemma 5.1.

If we have ➀: $[\bm{p}^{*},\bm{B}^{*},\bm{\beta}^{*}]$ is a globally optimal
solution to Problem $\mathbb{P}_{2}$, then we get ➁: $\bm{\beta}^{*}$ denoting
$[\beta_{n}^{*}|_{n\in\mathcal{N}}]$ satisfies

(21)
$\displaystyle\beta_{n}^{*}=\frac{F_{n}(p_{n}^{*},B_{n}^{*})}{p_{n}^{*}+p_{n}^{cir}},\text{
for all }n\in\mathcal{N},$

and ➂: $[\bm{p}^{*},\bm{B}^{*}]$ is a globally optimal solution to Problem
$\mathbb{P}_{3}(\bm{\beta}^{*},\bm{\nu}^{*})$, where we have ➃: $\bm{\nu}^{*}$
denoting $[\nu_{n}^{*}|_{n\in\mathcal{N}}]$ is given by

(22) $\displaystyle\nu_{n}^{*}=\frac{1}{p_{n}^{*}+p_{n}^{cir}},\text{ for all
}n\in\mathcal{N}.$

###### Proof.

Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ belongs to convex optimization.
In particular, according to Lemma 4.6 and (8) (20), the objective function to
be maximized is concave, while the constraints are clearly convex with Lemma
4.5 (note that “concave $\geq$ constant” is a convex constraint as noted in
Footnote 2). Also, Slater’s condition holds for Problem
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$. In other words, there exists at least
one point $[\bm{p},\bm{B}]$ such that constraints (4a) and (4b) are satisfied
with strict inequalities. An example is as follows: with $B_{n}$ being
$\frac{B_{\text{total}}}{2N}$ for all $n\in\mathcal{N}$, set $p_{n}$ such that
$r_{n}(p_{n},B_{n})=2r_{n}^{\min}$ for all $n\in\mathcal{N}$. The above along
with Lemma 4.3 shows the first “$\Leftrightarrow$” result below:

$\textstyle\begin{rcases*}\text{\char
194}\Leftrightarrow\text{$[\bm{p}^{*},\bm{B}^{*},\bm{\beta}^{*},\bm{\nu}^{*}]$
satisfies the set $\mathcal{K}$ of conditions in (\ref{showP3KKT}).}\\\ \hskip
10.0pt\text{Results \char 193 and \char 195 hold; i.e.,
$[\bm{\beta}^{*},\bm{\nu}^{*}]$ satisfies (\ref{decidebetanstar}) and
(\ref{decidenunstar}).}\end{rcases*}$ (23)
$\textstyle\Leftrightarrow\text{$[\bm{p}^{*},\bm{B}^{*},\bm{\beta}^{*},\bm{\nu}^{*}]$
satisfies KKT conditions
(\ref{P2KKTeqStationaritypn})--(\ref{P2KKTeqDualfeasibility})}\Leftarrow\text{\char
192}$

where the second “$\Leftrightarrow$” above holds from (14) and (15), and the
last “$\Leftarrow$” above follows from (11). ∎

For additional understanding of Lemma 5.1, interested readers can refer to
Lemma 2.1 and Remark 2.1 of (jong2012efficient, ), where Lemma 2.1 of
(jong2012efficient, ) handles minimizing the sum of convex-concave ratios and
Remark 2.1 of (jong2012efficient, ) maximizes the sum of concave-convex
ratios.

With Lemma 5.1 presented above, we now describe how to solve Problem
$\mathbb{P}_{2}$ using $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$. Let
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$ denote
a globally optimal solution to $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$, where
$\bm{p}^{\\#}(\bm{\beta},\bm{\nu})=[p_{n}^{\\#}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}]$
and
$\bm{B}^{\\#}(\bm{\beta},\bm{\nu})=[B_{n}^{\\#}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}]$.
We further define

(24) $\textstyle\phi_{1,n}(\bm{\beta},\bm{\nu})$
$\textstyle:=-F_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu}))+\beta_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}}),$
(25) $\textstyle\phi_{2,n}(\bm{\beta},\bm{\nu})$
$\textstyle\\!:=\\!-1+\nu_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}}),$
$\textstyle\bm{\phi}_{1}(\bm{\beta},\bm{\nu})$
$\textstyle\\!:=\\![\phi_{1,n}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}],~{}\bm{\phi}_{2}(\bm{\beta},\bm{\nu})\\!:=\\![\phi_{2,n}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}],$
(26) $\textstyle\bm{\phi}(\bm{\beta},\bm{\nu})$
$\textstyle\\!:=\\![\bm{\phi}_{1}(\bm{\beta},\bm{\nu}),\bm{\phi}_{2}(\bm{\beta},\bm{\nu})].$

With $(\bm{p}^{*},\bm{B}^{*},\bm{\beta}^{*})$ denoting a globally optimal
solution to Problem $\mathbb{P}_{2}$ (and hence $(\bm{p}^{*},\bm{B}^{*})$
denoting a globally optimal solution to Problem $\mathbb{P}_{1}$), clearly
setting $(\bm{\beta},\bm{\nu})$ as $(\bm{\beta}^{*},\bm{\nu}^{*})$ of (21) and
(22) satisfies

(27) $\textstyle\bm{\phi}(\bm{\beta},\bm{\nu})=\bm{0}.$

Based on the above, solving Problem $\mathbb{P}_{2}$ and hence
$\mathbb{P}_{1}$ can be transformed into solving (27) to obtain
$\mathbb{P}_{3}(\bm{\beta}^{*},\bm{\nu}^{*})$, and then setting
$[\bm{p}^{*},\bm{B}^{*}]$ as
$[\bm{p}^{\\#}(\bm{\beta}^{*},\bm{\nu}^{*}),$$\bm{B}^{\\#}(\bm{\beta}^{*},\bm{\nu}^{*})]$,
a globally optimal solution to $\mathbb{P}_{3}(\bm{\beta}^{*},\bm{\nu}^{*})$,
according to Lemma 5.1. Based on the above idea, we present Algorithm 1 next,
where it will become clear that

solving $\mathbb{P}_{1}$ becomes solving a series of parametric convex optimi-
(28) zation $\mathbb{P}_{3}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$, with $i$
denoting the iteration index.

Readers may notice that our Lemma 5.1 provides just a necessary condition for
a global optimum of Problem $\mathbb{P}_{2}$. Lemma 5.2 below shows
“necessary” and “sufficient” for strictly concave utility, which holds for all
types of functions in simulations of Section 7.

###### Lemma 5.2.

If the utility function $f_{n}(\cdot)$ for any $n\in\mathcal{N}$ is strictly
concave (i.e., $f_{n}^{\prime}(\cdot)$ is decreasing) for $x>0$, the
“$\Leftarrow$” in (23) can be replaced by “$\Leftrightarrow$”, so that “➁ ➂ ➃”
$\Leftrightarrow$ “➀” actually holds in Lemma 5.1.

###### Proof.

From Theorem 2 on Page 2, for decreasing $f_{n}^{\prime}(\cdot)$, we can prove
that $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ has a unique global optimum
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$. We
further obtain that $(\bm{\beta}^{*},\bm{\nu}^{*})$ satisfying (27) (i.e., “➁
➂ ➃”) is unique. Since we have explained that $\mathbb{P}_{2}$ has at least
one global maximum, we know from the above this maximum is unique. Thus, for
strictly concave utility, “➁ ➂ ➃” $\Leftrightarrow$ “➀” holds in Lemma 5.1. ∎

1 Initialize feasible $[\bm{p}^{(0)}$, $\bm{B}^{(0)}]$, $i=0$, $\xi\in(0,1)$,
$\epsilon\in(0,1)$.

2 Calculate $\bm{\beta}^{(0)}=[\beta_{n}^{(0)}|_{n\in\mathcal{N}}]$ and
$\bm{\nu}^{(0)}=[\nu_{n}^{(0)}|_{n\in\mathcal{N}}]$ via
$\beta_{n}^{(0)}=\frac{c_{n}f_{n}(r_{n,\textnormal{s}}(p_{n}^{(0)},B_{n}^{(0)}))}{{p_{n}^{(0)}+p_{n}^{\textnormal{cir}}}}$
and $\nu_{n}^{(0)}=\frac{1}{p_{n}^{(0)}+p_{n}^{\textnormal{cir}}}.$
//Comment: Since we aim to find $(\bm{\beta}^{*},\bm{\nu}^{*})$ satisfying
(27) (i.e., “➁ ➂ ➃” in Lemma 5.1), the above initialization is intuitively
good since it mimics (21) in “➁” and (22) in “➃” of Lemma 5.1.

3repeat

4 Use Eq. (34) in Theorem 2 on Page 34 to solve
$\mathbb{P}_{3}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$, and obtain a solution
$[\bm{p}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}),\bm{B}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$.
//Comment: This line can use the bisection method in a straightforward manner,
so we put the details in Appendix B.2.

5 Use
$[\bm{p}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}),\bm{B}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$
obtained above to compute $\bm{\phi}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$
according to Eq. (26) on Page 26.

6 If $\bm{\phi}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$ is the zero vector, then
$[\bm{p}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}),\bm{B}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$
is the global optimal solution to Problem $\mathbb{P}_{1}$ and we finish the
algorithm.

7 Otherwise, let $J_{i}$ be the smallest integer that satisfies
$\textstyle\|\bm{\phi}(\bm{\beta}^{(i)}+\xi^{J_{i}}\bm{\sigma_{1}}^{(i)},\bm{\nu}^{(i)}+\xi^{J_{i}}\bm{\sigma_{2}}^{(i)})\|_{2}$
(29)
$\textstyle\leq(1-\xi^{J_{i}}\epsilon)\cdot\|\bm{\phi}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})\|_{2},$
where “$\|\cdot\|_{2}$” denotes the Euclidean norm, and the $n$th-dimension of
$\bm{\sigma_{1}}^{(i)}$ (resp. $\bm{\sigma_{2}}^{(i)}$) for $n\in\mathcal{N}$,
denoted by $\bm{\sigma_{1}}^{(i)}[n]$ (resp. $\bm{\sigma_{2}}^{(i)}[n]$), is
given by  (30)
$\textstyle\textstyle{\bm{\sigma_{1}}^{(i)}[n]:=-\frac{(\partial\phi_{1,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}))/(\partial\beta_{n})}{\phi_{1,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})}}~{}$
$\textstyle=-\frac{\text{(\ref{bothequal})}}{\text{RHS
of~{}(\ref{eqn:phi_1v2}) with $(\bm{\beta},\bm{\nu})$ being
$(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$}},$ (31)
$\textstyle\bm{\sigma_{2}}^{(i)}[n]:=-\frac{(\partial\phi_{2,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}))/(\partial\nu_{n})}{\phi_{2,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})}~{}$
$\textstyle=-\frac{\text{(\ref{bothequal})}}{\text{RHS
of~{}(\ref{eqn:phi_2v2}) with $(\bm{\beta},\bm{\nu})$ being
$(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$}},$ where RHS is short for the right-hand
side.
//Comment: Obtaining ${J_{i}}$ above involves evaluating $({J_{i}}+1)$ number
of $\bm{\phi}(\bm{\beta},\bm{\nu})$ for (7). To compute each of them, we need
to solve Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ via (34) on Page 34 to
obtain
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$, and
then use (26).

8 Update  (32)
$\textstyle[\bm{\beta}^{(i+1)},\bm{\nu}^{(i+1)}]\leftarrow[\bm{\beta}^{(i)}+\xi^{J_{i}}\bm{\sigma_{1}}^{(i)},\bm{\nu}^{(i)}+\xi^{{J_{i}}}\bm{\sigma_{2}}^{(i)}],$
where ${J_{i}}$ is obtained from (7).
//Comment: If ${J_{i}}$ happens to be $0$, then (32) becomes the standard
Newton method, as explained in the last paragraph on Page 13 of
(jong2012efficient, ). As shown by Problem 2 on Page 14 of (jong2012efficient,
), the standard Newton method may fail for some initial points, so we follow
(jong2012efficient, ) to find ${J_{i}}$ according to (7) instead of always
setting ${J_{i}}$ as $0$.

9 Let $i\leftarrow i+1$.

10until  _$\bm{\phi}(\bm{\beta}^{(i)},\bm{\nu}^{(i}))$ is close to $\bm{0}$_;

11Use the current $[\bm{\beta},\bm{\nu}]$ in (34) on Page 34 and return the
obtained
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$ as the
solution to Problem $\mathbb{P}_{1}$.

Algorithm 1 Our approach of computing a globally optimal solution
$[\bm{p},\bm{B}]$ (up to arbitrary accuracy) to Problem $\mathbb{P}_{1}$ of
Section 3.2 on weighted sum-UEE optimization.

###  5.2. Our Algorithm 1 to solve Problem $\mathbb{P}_{1}$

As explained in the previous subsection, we solve (27) first in order to
obtain a globally optimal solution to Problem $\mathbb{P}_{1}$. Root-finding
algorithms such as Newton’s method can be used to solve (27). Our Algorithm 1
actually uses a modified Newton method of (jong2012efficient, ), which always
converges to the desired solution. In contrast, the original Newton’s method
is sensitive to initialization (e.g., no convergence if starting at bad
initialization, as shown in Section 4 of (jong2012efficient, )).

Algorithm 1 starts with computing the initial
$[\bm{\beta}^{(0)},\bm{\nu}^{(0)}]$ from $[\bm{p}^{(0)}$, $\bm{B}^{(0)}]$, as
shown in the pseudocode. In the $i$-th iteration of Algorithm 1 ($i$ starts
from $0$), we update $[\bm{\beta}^{(i)},\bm{\nu}^{(i)}]$ to
$[\bm{\beta}^{(i+1)},\bm{\nu}^{(i+1)}]$ based on (7) (30) (31) (32), which
essentially present the modified Newton method to solve (27). The numerators
in (30) (31) use
$(\partial\phi_{1,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}))/(\partial\beta_{n})$
and $(\partial\phi_{2,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}))/(\partial\nu_{n})$,
which are shown in Appendix A to be equal to

(33) $\textstyle
p_{n}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})+p_{n}^{\textnormal{cir}}.$

The denominators in (30) (31) use
$\phi_{1,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$ and
$\phi_{2,n}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$, whose computations based on
(24) (25) require obtaining
$[\bm{p}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}),\bm{B}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$
by solving Problem $\mathbb{P}_{3}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})$. This is
the reason why we have (28).

We remark that in Algorithm 1, the iterative process of computing
$[\bm{p}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)}),\bm{B}^{\\#}(\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$
and then using it for updating $[\bm{\beta}^{(i)},\bm{\nu}^{(i)})]$ to
$[\bm{\beta}^{(i+1)},\bm{\nu}^{(i+1)})]$ is not the classical dual gradient
descent (DGD) (boyd2004convex, ) despite the resemblance, since $\bm{\beta}$
is not a Lagrange multiplier. Algorithm 1 solves (27) using the modified
Newton method, while DGD involves maximizing the dual function.

We formally state the solution quality of Algorithm 1 as follows.

###### Theorem 1.

Under Conditions 1 and 2 of Section 4, our proposed Algorithm 1 finds a
globally optimal solution to Problem $\mathbb{P}_{1}$ (up to arbitrary
accuracy).

###### Proof.

The analyses above in Sections 5.1 and 5.2, stated before Theorem 1, have
already provided the proof of Theorem 1. ∎

Next, we discuss the fast convergence and order-optimal time complexity of
Algorithm 1. As shown in Theorem 3.2 of (jong2012efficient, ), the modified
Newton method used in Algorithm 1 has global linear and local quadratic rates
of convergence.

To analyze the time complexity, we use floating point operations (flops). One
addition/subtraction/multiplication/division is one flop. We now analyze Lines
3–8, the main part of Algorithm 1. Suppose that in Line 1, we use the
bisection method to obtain $\lambda^{\\#}$ from (36), for which there are $K$
iterations and each iteration has $\mathcal{O}(N)$, where $K$ depends on the
error tolerance, as detailed in Appendix B.2. Hence, Line 1 consumes
$\mathcal{O}(KN)$. Lines 5, 6, and 8 cost $\mathcal{O}(N)$ flops. Line 7 takes
$\mathcal{O}((J_{i}+1)N)$ flops. Suppose the loop in Line 3 needs
$\mathcal{I}$ iterations before convergence ($\mathcal{I}$ is less than $10$
in our experiments to find a $0.01$-global optimum, which means the relative
difference between the objective-function values under the found solution and
the true global optimum is at most $0.01$). Then the time complexity of
Algorithm 1 is
$\mathcal{O}(\mathcal{I}KN+\sum_{i=0}^{\mathcal{I}-1}(J_{i}+1)N)$, which is
linear in $N$. This linear complexity is the best that any algorithm can do,
since we need to decide $N$ number of $[B_{n},p_{n}]$ for all $N$ users.
Hence, Algorithm 1 achieves the optimal time complexity in the order sense.

###  5.3. Solving Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$

From (28), solving Problem $\mathbb{P}_{1}$ requires solving a series of
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$. One approach is to use the Stanford CVX
tool (boyd2004convex, ). However, the worst-case complexity of global convex
optimization grows exponentially with the problem size $N$ from Section 1.4.2
of (boyd2004convex, ). Based on Theorem 2 below, we can solve
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ and hence $\mathbb{P}_{1}$ in linear
time with respect to $N$, as discussed in the previous subsection.

###### Theorem 2.

Under Conditions 1 and 2 of Section 4, any globally optimal solution
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$ to
Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ defined in (19) can be given as
follows:

(34)
$\textstyle\begin{cases}B_{n}^{\\#}(\bm{\beta},\bm{\nu})=\mathcal{B}_{n}(\lambda^{\\#})\text{
for all $n\in\mathcal{N}$,}\\\
p_{n}^{\\#}(\bm{\beta},\bm{\nu})=\frac{\sigma_{n}^{2}B_{n}^{\\#}(\bm{\beta},\bm{\nu})\cdot\psi_{n}(\lambda^{\\#})}{g_{n}}~{}\text{for
all $n\in\mathcal{N}$},\end{cases}$

with function $\mathcal{B}_{n}(\lambda)$ defined by

(35)
$\textstyle\mathcal{B}_{n}(\lambda):=\frac{\max\\{\gamma_{n}(\lambda),r_{n}^{\min}\\}}{\log_{2}\big{(}1+\psi_{n}(\lambda))},$

and $\lambda^{\\#}$ denoting the solution to

(36)
$\textstyle\sum\limits_{n\in\mathcal{N}}\mathcal{B}_{n}(\lambda)=B_{\text{total}},$

where $\psi_{n}(\lambda)$ and $\gamma_{n}(\lambda)$ are defined by

(37)
$\textstyle\psi_{n}(\lambda):=\exp\big{\\{}1+W\big{(}\frac{1}{e}(\frac{g_{n}\lambda}{\nu_{n}\beta_{n}\sigma_{n}^{2}}-1)\big{)}\big{\\}}-1,$
for $W(\cdot)$ being the principal branch of the Lambert W function
$\textstyle\text{($W(z)$ for $z\geq-e^{-1}$ is the solution of $x\geq-1$ to
the equation $xe^{x}=z$)},$ (38) $\textstyle\text{and
}\gamma_{n}(\lambda)-r_{n,\textnormal{e}}:=\begin{cases}\xi:=(f_{n}^{\prime})^{-1}\Big{(}\frac{\beta_{n}\sigma_{n}^{2}\cdot(1+\psi_{n}(\lambda))\ln{2}}{c_{n}g_{n}}\Big{)}\text{
}\\\ ~{}~{}~{}~{}~{}\text{when such result $\xi\geq 0$ exists},\\\
0,~{}\text{otherwise},\end{cases}$

with $(f_{n}^{\prime})^{-1}(\cdot)$ denoting the inverse function of the
derivative $f_{n}^{\prime}(\cdot)$.

Theorem 2 is proved in Appendix B, where we also elaborate on the bisection
method to obtain $\lambda^{\\#}$ from (36).

##  6\. Broad Usage of Our Technique

In this section, we review the optimization used in Algorithm 1 to obtain an
insightful technique, which can be used to solve many other problems in
wireless networks and mobile computing.

In Section 5.1, Problem $\mathbb{P}_{2}$ is not convex optimization since the
non-convex product $\beta_{n}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$ exists in
(7b), as shown in the sentences following (8). The solving process of
$\mathbb{P}_{2}$ is transformed into solving a series of parametric convex
optimization $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ where
$[\bm{\beta},\bm{\nu}]$ is given so that there is no non-convex product term
and we have convex optimization. The solving of each $\mathbb{P}_{3}$ is used
to update $[\bm{\beta},\bm{\nu}]$ under which $\mathbb{P}_{3}$ is solved again
with the new $[\bm{\beta},\bm{\nu}]$, where the update of
$[\bm{\beta},\bm{\nu}]$ is based on the KKT conditions of $\mathbb{P}_{2}$.

From the above discussion, we can identify the following:

Our technique to handle functions of product or quotient terms in
optimization: With “$\divideontimes$” denoting multiplication or division, if
there are terms $f_{n}(A_{n}(\bm{x})\divideontimes y_{n})|_{n\in\mathcal{N}}$
in an optimization problem $\mathbb{P}$, for functions
$f_{n},A_{n}|_{n\in\mathcal{N}}$ and variables $\bm{x}$ and
$\bm{y}=[y_{n}|_{n\in\mathcal{N}}]$, we can convert $\mathbb{P}$ into a series
of parametric convex optimization $\mathbb{Q}(\bm{y},\bm{z})$, where $\bm{z}$
comprises additional variables in the parameterization (e.g., $\bm{\nu}$ in
our “$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$”). In $\mathbb{Q}(\bm{y},\bm{z})$,
given $[\bm{y},\bm{z}]$, variables in $A_{n}(\bm{x})\divideontimes y_{n}$ just
have $\bm{x}$, so that $\mathbb{Q}$ can be easier to solve than $\mathbb{P}$,
or $\mathbb{Q}$ may even happen to be convex in $\bm{x}$. The solving of each
$\mathbb{Q}$ will be used to update $[\bm{y},\bm{z}]$ under which $\mathbb{Q}$
is solved again with the new $[\bm{y},\bm{z}]$, where the update of
$[\bm{y},\bm{z}]$ is based on the KKT conditions of $\mathbb{P}$.

With the above technique, we can address $f_{n}(A_{n}(\bm{x})\divideontimes
B_{n}(\bm{x}))|_{n\in\mathcal{N}}$ in optimization as well, for functions
$f_{n},A_{n},B_{n}|_{n\in\mathcal{N}}$ and variables $\bm{x}$. We replace
$A_{n}(\bm{x})\divideontimes B_{n}(\bm{x})$ by an auxiliary variable $z_{n}$
and enforce the constraint of $z_{n}$ being either no greater or no less than
$A_{n}(\bm{x})\divideontimes B_{n}(\bm{x})$ (depending on the specific
problem), where the constraint can be further converted into a relationship
between $A_{n}(\bm{x})$ and $z_{n}\divideontimes B_{n}(\bm{x})$, like how we
transform $\mathbb{P}_{1}$ of (4) to $\mathbb{P}_{2}$ of (7).

To summarize, our technique can be useful for various optimization problems
involving product or quotient terms. In addition, the technique often obtains
a global optimum, as in Theorem 1. The above finding goes beyond the sum-of-
ratios optimization of (jong2012efficient, ), although our original motivation
comes from (jong2012efficient, ). The following discussion shows that our
above finding is very likely to be new.

Two recent papers (shen2018fractional, ; shen2018fractional2, ) by Shen and Yu
have been considered breakthroughs in fractional programming, as seen from
their high citations (692 and 190, respectively, as of 10 March 2023 in Google
Scholar). However, they find neither local nor global optimum. In contrast,
our technique above will find a global optimum. Interested readers can refer
to Appendix F.

Our above technique can be applied to many optimization problems in wireless
networks and mobile computing, as illustrated by two examples below. In
interference-constrained wireless networks, globally solving the weighted sum-
rate maximization (WSRM) efficiently was an open problem for years before it
was addressed by (qian2009mapel, ), since a user’s rate (per unit bandwidth)
given by
$\log_{2}(1+\frac{\text{TransmitPower}}{\text{Interference}+\text{Noise}})$
involves a fraction inside a logarithm, which is difficult to deal with. Our
technique above will find a global optimum for WSRM and other problems
involving the above rate expression, while the polyblock-based approach of
(qian2009mapel, ) relies on the structure of WSRM and may not be applicable to
other problems. In mobile edge computing, with $\gamma$ denoting the
offloading ratio of computation tasks, (zhao2021energy, ) minimizes the system
cost, given by
$\gamma\cdot\text{EdgeComputingCost}+(1-\gamma)\cdot\text{LocalComputingCost}$.
The multiplication above means no joint convexity in $\gamma$ and other
variables. Then (zhao2021energy, ) uses alternating optimization which is
neither locally nor globally optimal, while our technique will find a global
optimum.

##  7\. Simulation

The utility functions for simulations are presented in Section 7.1 and
validated by real data in Section 7.2. Then we describe simulation settings in
Section 7.3, before reporting results in other subsections.

###  7.1. Utility functions for simulation

We provide three types of utility functions below since the Metaverse offers
diverse applications. In Section 7.2, we validate these functions using real
data.

Type 1 utility function:  We have

(39) $\displaystyle f_{n}(x)=\kappa_{n}\ln(b_{n}+a_{n}x),$

where $a_{n},\kappa_{n}>0,b_{n}\geq 0$. This type is used in
(yang2012crowdsourcing, ) for sensing tasks. In simulations starting from
Section 7.3, we let $b_{n}=1$.

Type 2 utility function:  We have

(40) $\displaystyle f_{n}(x)=\kappa_{n}\cdot(1-e^{-a_{n}x+c_{n}}),$

where $a_{n},\kappa_{n}>0$, and $e$ denotes Euler’s number. This type is
motivated by (liu2018edge, ) on augmented reality. We let $c_{n}=0$ in
simulations.

Type 3 utility function:  We have

(41) $\displaystyle f_{n}(x)=\kappa_{n}({x}+d_{n})^{a_{n}},$

where $\kappa_{n}>0$, $d_{n}\geq 0$, and $0<a_{n}<1$. This function form has
been used in prior work on congestion control (mo2000fair, ) and mobile data
subsidization (xiong2020reward, ). We let $d_{n}=0$ in simulations. In the
terminologies of economics, $\kappa_{n}{r_{n,\textnormal{s}}}^{a_{n}}$ can be
viewed as a Cobb–Douglas utility with respect to $r_{n,\textnormal{s}}$, while
$\kappa_{n}\cdot{(r_{n}-r_{n,\textnormal{e}})}^{a_{n}}$ can be regarded as a
Stone–Geary utility with respect to $r_{n}$; see Page 7 of (da2021welfare, ).

It is straightforward to show that the above three types for the utility
function all satisfy Condition 2 of Section 4.2. These three types are what we
will use in simulations from Section 7.3. We emphasize that our theoretical
results (e.g., Algorithm 1 as well as Theorems 1 and 2 in Section 5) of this
paper apply to any utility function satisfying Condition 2 of Section 4.2.

###  7.2. Real data validating utility functions above

We now validate Section 7.1’s utility functions with the SSV360 (ssv360, ) and
Netflix datasets (Netflix, ) from real-world experiments.

SSV360 dataset. This dataset of (ssv360, ) captures users’ assessment of 360°
videos when wearing HTC Vive Pro virtual reality headsets. Each data point
represents a user’s subjective quality assessment of a 360° scene, under
standing or seated viewing (SSV). In the dataset, having data points under
different video bitrates yet the same resolution is due to different
quantization parameters used in video compression. The wireless data rate
should be large enough to ensure a smooth watching experience at the given
video bitrate (liu2018edge, ). We let the bitrate be a constant fraction (say
$\theta$) of the wireless rate. Since changing the bitrate
$r_{\text{bitrate}}$ to the wireless rate $r_{\text{wireless}}$ just involves
replacing $r_{\text{bitrate}}$ with $r_{\text{wireless}}/\theta$, we perform
curve-fitting with the bitrate to validate the utility functions. The curves
in Fig. 2(a) are for the scenarios of “user 1 seated”, “user 2 seated”, and
“user 1 standing” respectively, to watch the same 360° scene “FormationPace”
(ssv360, ) with 2K resolution (i.e., $2048\times 1080$ pixels).

In the SSV360 dataset, the score follows the widely used Absolute Category
Rating (ACR) (gutierrez2022vqeg, ) and is an integer from $1$ to $5$. To
obtain better curve-fitting results, we further use the Netflix dataset, where
the score (i.e., the $y$-axis) ranges from $0$ to $100$.

Netflix dataset. In this dataset (Netflix, ), which is a part of Netflix’s
Emmy Award-winning Video Multimethod Assessment Fusion (VMAF) project, each
data point exhibits users’ mean opinion score in $[0,100]$ for a video at a
given resolution and a given bitrate. Because there are not enough data points
that have different bitrates yet the same resolution, we treat both resolution
and bitrate as variables for curve-fitting. The results are shown in Fig.
2(b).

The expressions for the curves in Fig. 2 are in the table below.

Dataset |  | Scenario in (ssv360, )
---
or video in (Netflix, )
| Utility function from curve-fitting for
---
normalized bitrate $x$ and normalized
resolution $y$ explained in the caption of Fig. 2
SSV360 in (ssv360, ) | user 1 seated | Type 1: $0.5424\ln(1+37.2965x)$
user 2 seated |  | Type 2: $2.9351(1-e^{-2.1224x})$
---
user 1 standing |  | Type 3: $3.2956{(x/15.94)}^{0.2733}$
---
Netflix in (Netflix, ) | ElFuente1 | Type 1: $33.4215\ln(1+0.784x+10.0826y)$
BigBuckBunny |  | Type 2: $103.3464(1-e^{-0.23166x-2.9792y})$
---
BirdsInCage |  | Type 3: $61.8622(x/15+y/1.1664)^{0.5301}$
---

The existence of $y$ in some expressions above can be understood that the
coefficients in (39) (40) (41) depend on $y$. In simulations below, we fix $y$
so that the utility function depends on only the rate.

###  7.3. Parameter setting

We first state settings that apply to all simulations. Based on
(zhou2022resource, ), we model the path loss between each legitimate user and
the Metaverse server as $128.1+37.6\log(distance)$ along with 8 decibels (dB)
for the standard deviation of shadow fading, and the unit of $distance$ is
kilometer. The power spectral density of Gaussian noise ${\sigma_{n}}^{2}$ is
$-174$ dBm/Hz (i.e., 4 zeptowatts/Hz, the value for thermal noise at 20 °C
room temperature (huang2013noise, )).

In addition, some default settings are as follows, unless otherwise specified.
$N$ denoting the number of legitimate users is 30. The weight parameter
$c_{n}$ is set to $1$ for all users (unless configured otherwise), which means
the weighted sum-UEE just becomes sum-UEE by default. The default total
bandwidth $B_{\textnormal{total}}$ is 20 MHz. The circuit power $p_{n}^{cir}$
is 2 dBm (i.e., 1.6 milliwatts) for each $n$. Both the eavesdropping rate
$r_{n,e}$ and the minimum transmission rate $r_{n}^{\min}$ are $20$ kilobits
per second (Kbps) by default. For the utility functions, we set
$\kappa_{n}=1$, $a_{n}=0.5$, $b_{n}=1$, $c_{n}=0$, and $d_{n}=0$ by default.
In all simulations, we stop the algorithm after obtaining a $0.01$-global
optimum, whose meaning is discussed at the end of Section 5.2.

###  7.4. Comparison of different algorithms

We compare our Algorithm 1 with the following baselines:

  * (i)

Optimize $\bm{B}$ only: Here we let $p_{n}$ for each $n$ be $1$ milliwatt
(i.e., $10^{-3}$ W), which will be substituted into Problem $\mathbb{P}_{1}$.
Then “optimizing $\bm{B}$ only” becomes convex optimization, for which the KKT
conditions are analyzed to obtain the solution.

  * (ii)

Optimize $\bm{p}$ only: In this case, we let $B_{n}$ for each $n$ be
$B_{\text{total}}/N$, which will be substituted into Problem $\mathbb{P}_{1}$.
Then “optimizing $\bm{p}$ only” belongs to convex optimization, for which the
KKT conditions are inspected to acquire the solution.

  * (iii)

Alternating optimization: Starting with a feasible initialization, we perform
“(i)” and “(ii)” above in an alternating manner, until convergence (when the
relative improvement between two consecutive iterations is negligible).

For the detailed analyses of the baseline algorithms, interested readers can
refer to Appendix C.

Figure 2. The curve-fitting results using Types 1, 2, and 3 utility functions.
In both subfigures, the normalized video bitrate is obtained from dividing the
bitrate by the maximum value of $15.94$ Mbps (resp., $15$ Mbps) in the SSV360
(resp., Netflix) dataset. In Fig. 2(b), the normalized resolution is the
result of dividing the resolution by the maximum resolution of $1920\times
1080$ pixels (i.e., $1080$p) in the Netflix dataset.  Figure 3. (a).
Algorithms with respect to the total bandwidth. (b). Running time and
objective-function value under each algorithm, where AO is short for
alternating optimization.  Figure 4. Sum-UEE for high-, medium-, and low-
priority users, under different $c_{n}$ values, for Type 1, 2, or 3 utility.
Figure 5. Setting different $r_{n,\textnormal{e}}$ for two user groups, where
the legend “$(s,t)$” means that $r_{n,\textnormal{e}}$ for user $U_{n}$ in
Group 1 is set as $s\cdot r_{n}^{\min}$ while $r_{n,\textnormal{e}}$ for user
$U_{n}$ in Group 2 equals $t\cdot r_{n}^{\min}$.

We compare Algorithm 1 with the above baselines in Fig. 3, where Type 3
utility function is used. Fig. 3(a) plots the sum-UEE with respect to the
total bandwidth $B_{\textnormal{total}}$, and shows that our proposed
algorithm achieves a larger sum-UEE than all the baselines. For each curve, as
$B_{\textnormal{total}}$ increases, the sum-UEE grows but at a slower rate.
Formally proving this is not in this paper’s scope, but it seems intuitive
since each user’s UEE is an increasing and concave function of the bandwidth,
given the transmission power.

Fig. 3(b) displays the convergence performance of our proposed algorithm and
alternating optimization (AO). Our algorithm always obtains a better sum-UEE
and converges much faster. In particular, on a laptop with 8GB of RAM and
256GB of storage, the proposed algorithm converges within 3 seconds, but the
AO approach takes around $20$, $24$, $36$ seconds under $N=10,20,30$,
respectively.

###  7.5. The priority levels of users

Here we explore the influence of the priority of users under different utility
functions. We consider that $30$ users are evenly classified into three
priority levels, corresponding to different weights $c_{n}$. Larger $c_{n}$
means more weight in our studied optimization. For example, the legend
“$c_{n}=100,10,1$” in Fig. 4 means that $10$ users with $c_{n}$ being $100$
(resp., $10$ and $1$) have high (resp., medium and low) priority.

Fig. 4(a) (b) and (c) use utility functions of Types 1, 2, and 3,
respectively. The sum-UEE of users in each priority group is plotted. In each
subfigure, the bar charts show that the sum-UEE of the high-priority group is
the largest, while that of the low-priority group is the lowest, matching the
intuition, since higher priority means larger $c_{n}$ and “more say” in the
weighted sum-UEE optimization. In addition, the numbers in Fig. 4(c) for Type
3 utility $u_{3}:=r_{n,\textnormal{s}}^{0.5}$ are greater than the
corresponding ones in Fig. 4(a) for Type 1 utility
$u_{1}:=\ln(1+0.5r_{n,\textnormal{s}})$, which are further larger than those
in Fig. 4(b) for Type 2 utility $u_{2}:=1-e^{-0.5r_{n,\textnormal{s}}}$. The
above is consistent with $u_{3}>u_{1}>u_{2}$ for large $r_{n,\textnormal{s}}$
(in the unit of bps).

In Fig. 4’s subfigures, from Case 1 of “$c_{n}=100,10,1$” to Case 2 of
“$c_{n}=100,10,5$”, and further to Case 3 of “$c_{n}=20,10,5$”, the relative
dominance of high-priority group decreases while the relative weight of low-
priority group increases, which accords with declining (resp., rising) sum-UEE
of high-priority (resp., low-priority) group from Case 1 to Case 2, and
further to Case 3. For the medium-priority group, as expected, the sum-UEE
decreases from Case 1 to Case 2 (though not clear in the plots without zooming
in), and increases from Case 2 to Case 3. The above demonstrates the impact of
the weight $c_{n}$ as the priority level.

###  7.6. Impact of individual rate constraints

Now we report the effect of varying $r_{n,\textnormal{e}}$. The number of
users $N$ is set as 20, and we divide them equally into two groups with
$r_{n,\textnormal{e}}$ as $s\cdot r_{n}^{\min}$ and $t\cdot r_{n}^{\min}$,
respectively, where $r_{n}^{\min}$ is the default $20$ Kbps. Each group’s
average transmission power, average allocated bandwidth, and average UEE are
plotted in Fig. 5(a), (b), and (c), respectively. Also, Fig. 5 uses Type 3
utility function and evaluates $(s,t)$ as $(0,0)$, $(0,0.5)$, and $(0.5,0.5)$,
respectively. From $(0,0)$ to $(0,0.5)$ (resp., $(0,0.5)$ to $(0.5,0.5)$), the
second (resp., first) group’s average transmission power increases. This is
intuitive since raising a group’s $r_{n,\textnormal{e}}$ with everything else
unchanged requires the group to enlarge the transmission power and hence the
data rate. The bandwidth allocation does not vary much under the cases of
$(0,0)$, $(0,0.5)$, and $(0.5,0.5)$. For each group, the average UEE slightly
drops as $r_{n,\textnormal{e}}$ grows, which seems intuitive since each user’s
utility $f_{n}(r_{n}(p_{n},B_{n})-r_{n,\textnormal{e}})$ is negatively
correlated with $r_{n,\textnormal{e}}$.

For more simulation results (e.g., how our proposed algorithm performs when
the number of users changes or when there are heterogeneous types of utility
functions among the users), interested readers can refer to Appendices E.1 and
E.2.

##  8\. Conclusion

In this paper, in a wireless network for the Metaverse, we have studied the
weighted optimization of all users’ utility-energy efficiency (UEE) under
physical-layer security. The formulated problem belongs to non-convex
optimization, and we solve it via a transform to parametric convex
optimization. The resulting algorithm is optimal in terms of both the solution
quality and the order of time complexity. Simulation results are provided with
utility functions validated by real data. We envision more research to adopt
our transform technique due to its broad applicability to other problems in
wireless networks and mobile computing.

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Appendices:

We introduce some notation to be used in the appendices. For a scalar function
$f(x_{1},x_{2},\ldots,x_{M})$ of $M$ variables $x_{1},x_{2},\ldots,x_{M}$, we
use $\nabla_{x_{m}}f(x_{1},x_{2},\ldots,x_{M})$ where $m\in\\{1,2,\ldots,M\\}$
to denote the partial derivative of $f(x_{1},x_{2},\ldots,x_{M})$ with respect
to $x_{m}$, and use
$\nabla_{x_{m}}f(x_{1},x_{2},\ldots,x_{M})|_{x_{m}=x_{m}^{*}}$ to denote the
corresponding result when $x_{m}$ equals a given value $x_{m}^{*}$. For a
$K$-element set $\\{x_{i_{1}},x_{i_{2}},\ldots,x_{i_{K}}\\}$ of variables,
which is a subset of $\\{x_{1},x_{2},\ldots,x_{M}\\}$, we define
$\nabla_{x_{i_{1}},x_{i_{2}},\ldots,x_{i_{K}}}f(x_{1},x_{2},\ldots,x_{M})$ as
the vector
$[\nabla_{x_{i_{k}}}f(x_{1},x_{2},\ldots,x_{M})|_{k=1,2,\ldots,K}]$.

##  Appendix A Explaining (33)

To establish (33), we will prove

(A.1)
$\textstyle\text{$(\partial\phi_{1,n}(\bm{\beta},\bm{\nu}))/(\partial\beta_{n})$
and $(\partial\phi_{2,n}(\bm{\beta},\bm{\nu}))/(\partial\nu_{n})$ equal
}p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}}.$

The definitions of $\phi_{1,n}(\bm{\beta},\bm{\nu})$ and
$\phi_{2,n}(\bm{\beta},\bm{\nu})$ in (24) and (25) use
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$, which
denotes a globally optimal solution to $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$.
Hence, below we analyze $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$.

Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ belongs to convex optimization
and Slater’s condition holds, as shown in the proof of Lemma 5.1. Then the
Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient to obtain the
globally optimal solution, as stated in Lemma 4.3. To this end, we define the
Lagrange function:

$\displaystyle\textstyle{L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})=-\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n},B_{n}|\beta_{n},\nu_{n})}$
(A.2)
$\displaystyle\textstyle{\quad+\sum_{n\in\mathcal{N}}\tau_{n}\cdot(r_{n}^{\min}-r_{n})+\lambda\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\text{total}})},$

where $\bm{\tau}$ and $\lambda$ are called Lagrange multipliers.

The KKT conditions of Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ are as
follows, with $L_{\mathbb{P}_{3}}$ short for
$L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})$:

Stationarity: (A.3a) $\textstyle\frac{\partial L_{\mathbb{P}_{3}}}{\partial
p_{n}}=0,\text{ for all }n\in\mathcal{N},$ (A.3b) $\textstyle\frac{\partial
L_{\mathbb{P}_{3}}}{\partial B_{n}}=0,\text{ for all }n\in\mathcal{N},$
Complementary slackness: (A.3c)
$\textstyle\tau_{n}\cdot(r_{n}^{\min}-r_{n}(p_{n},B_{n}))=0,\text{ for all
}n\in\mathcal{N},$ (A.3d)
$\textstyle\lambda\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\text{total}})=0;$
Primal feasibility: (A.3e) $\textstyle r_{n}(p_{n},B_{n})\geq
r_{n}^{\min},\text{for all }n\in\mathcal{N},$ (A.3f)
$\textstyle\sum_{n\in\mathcal{N}}B_{n}\leq B_{\text{total}};$ Dual
feasibility: (A.3g) $\textstyle\tau_{n}\geq 0,\text{for all }n\in\mathcal{N}.$
(A.3h) $\textstyle\lambda\geq 0.$

Recall that we use
$[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$ to
denote a globally optimal solution to $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$,
where
$\bm{p}^{\\#}(\bm{\beta},\bm{\nu})=[p_{n}^{\\#}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}]$
and
$\bm{B}^{\\#}(\bm{\beta},\bm{\nu})=[B_{n}^{\\#}(\bm{\beta},\bm{\nu})|_{n\in\mathcal{N}}]$.
Hence, $[\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu})]$
satisfies the KKT conditions above. Using (A.2) (A.3c) and (A.3d), we know

$\textstyle\textstyle{L_{\mathbb{P}_{3}}(\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu}),\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})}$
$\textstyle=-\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})$
$\textstyle\textstyle{\quad+\sum_{n\in\mathcal{N}}\tau_{n}\cdot(r_{n}^{\min}-r_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})))}$
$\textstyle\quad+\lambda\cdot(\sum_{n\in\mathcal{N}}B_{n}^{\\#}(\bm{\beta},\bm{\nu})-B_{\text{total}})$
(A.4)
$\textstyle=-\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n}).$

For notation simplicity, we group $\bm{\beta}$ and $\bm{\nu}$ together to
define $\bm{\alpha}$; i.e., $\bm{\alpha}:=[\bm{\beta},\bm{\nu}]$. Note that
(A.4) above holds for any $\bm{\alpha}$. Hence, it holds that

$\textstyle\nabla_{\bm{\alpha}}\left(\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})\right)$
(A.5)
$\textstyle=-\nabla_{\bm{\alpha}}\left(L_{\mathbb{P}_{3}}(\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu}),\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})\right).$

Based on (A.5), to get
$\nabla_{\bm{\alpha}}\left(\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})\right)$,
we compute
$\nabla_{\bm{\alpha}}\left(L_{\mathbb{P}_{3}}(\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu}),\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})\right)$
and take its negative. We have

$\textstyle\nabla_{\bm{\alpha}}\left(L_{\mathbb{P}_{3}}(\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu}),\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})\right)$
$\textstyle=\sum_{n\in\mathcal{N}}\left\\{\left[(\nabla_{p_{n}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right]\cdot\nabla_{\bm{\alpha}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})\right\\}$
$\textstyle\quad+\sum_{n\in\mathcal{N}}\left\\{\left[(\nabla_{b_{n}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right]\cdot\nabla_{\bm{\alpha}}b_{n}^{\\#}(\bm{\beta},\bm{\nu})\right\\}$
(A.6)
$\textstyle\quad+\left[(\nabla_{\bm{\alpha}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right].$

From (A.3a) and (A.3b), we have

$\textstyle(\nabla_{p_{n}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}$ $\textstyle=0,\text{
and}$
$\textstyle\nabla_{b_{n}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}$ $\textstyle=0,$

which are used in (A.6) to obtain

$\textstyle\nabla_{\bm{\alpha}}\left(L_{\mathbb{P}_{3}}(\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\bm{B}^{\\#}(\bm{\beta},\bm{\nu}),\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu})\right)$
(A.7)
$\textstyle=\left[(\nabla_{\bm{\alpha}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right].$

From (A.2) (A.5) and (A.7), it holds that

$\textstyle\nabla_{\bm{\alpha}}\left(\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})\right)$
$\textstyle=-\left[(\nabla_{\bm{\alpha}}L_{\mathbb{P}_{3}}(\bm{p},\bm{B},\bm{\tau},\lambda\mid\bm{\beta},\bm{\nu}))|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right]$ (A.8)
$\textstyle=\sum_{n\in\mathcal{N}}\left[\left(\nabla_{\bm{\alpha}}\mathcal{F}_{n}(p_{n},B_{n}|\beta_{n},\nu_{n})\right)|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right].$

Using the above and (20), we further acquire

$\textstyle\nabla_{\beta_{n}}\left(\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})\right)$
$\textstyle=\sum_{n\in\mathcal{N}}\left[\left(\nabla_{\beta_{n}}\mathcal{F}_{n}(p_{n},B_{n}|\beta_{n},\nu_{n})\right)|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right]$ (A.9)
$\textstyle=-\sum_{n\in\mathcal{N}}\left[\nu_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\right],$

and

$\textstyle\nabla_{\nu_{n}}\left(\sum_{n\in\mathcal{N}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})\right)$
$\textstyle=\sum_{n\in\mathcal{N}}\left[\left(\nabla_{\nu_{n}}\mathcal{F}_{n}(p_{n},B_{n}|\beta_{n},\nu_{n})\right)|_{\begin{subarray}{l}\bm{p}=\bm{p}^{\\#}(\bm{\beta},\bm{\nu}),\\\
\bm{b}=\bm{b}^{\\#}(\bm{\beta},\bm{\nu})\end{subarray}}\right]$ (A.10)
$\textstyle=\sum_{n\in\mathcal{N}}\left[F_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu}))-\beta_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\right].$

Since (A.9) and (A.10) hold for any $\bm{\beta}$ and $\bm{\nu}$, we obtain

$\textstyle\nabla_{\beta_{n}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})$
(A.11)
$\textstyle=-\left[\nu_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\right],$

and

$\textstyle\nabla_{\nu_{n}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})$
(A.12)
$\textstyle=\left[F_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu}))-\beta_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\right].$

From the definition of $\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})$ in
(20), we also have

$\textstyle\nabla_{\beta_{n}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})$
$\textstyle=\nu_{n}\cdot\big{[}\nabla_{p_{n}}F_{n}(p_{n},B_{n})|_{p_{n}=p_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\beta_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})$
(A.13)
$\textstyle\quad+\nabla_{B_{n}}F_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\beta_{n}}B_{n}^{\\#}(\bm{\beta},\bm{\nu})-(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\big{]},$

and

$\textstyle\nabla_{\nu_{n}}\mathcal{F}_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu})|\beta_{n},\nu_{n})$
$\textstyle=\left[F_{n}(p_{n}^{\\#}(\bm{\beta},\bm{\nu}),B_{n}^{\\#}(\bm{\beta},\bm{\nu}))-\beta_{n}\cdot(p_{n}^{\\#}(\bm{\beta},\bm{\nu})+p_{n}^{\textnormal{cir}})\right]$
$\textstyle\quad+\nu_{n}\cdot\big{[}\nabla_{p_{n}}F_{n}(p_{n},B_{n})|_{p_{n}=p_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\nu_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})$
$\textstyle\quad+\nabla_{B_{n}}F_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\nu_{n}}B_{n}^{\\#}(\bm{\beta},\bm{\nu})$
(A.14)
$\textstyle\quad-\beta_{n}\cdot\nabla_{\nu_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})\big{]}.$

Comparing (A.11) and (A.13), and comparing (A.12) and (A.14), since we always
enforce $\nu_{n}>0$, we have proved

$\textstyle\big{[}\nabla_{p_{n}}F_{n}(p_{n},B_{n})|_{p_{n}=p_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\beta_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})$
(A.15)
$\textstyle+\nabla_{B_{n}}F_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\beta_{n}}B_{n}^{\\#}(\bm{\beta},\bm{\nu})\big{]}=0,$

and

$\textstyle\big{[}\nabla_{p_{n}}F_{n}(p_{n},B_{n})|_{p_{n}=p_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\nu_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})$
(A.16)
$\textstyle+\nabla_{B_{n}}F_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}(\bm{\beta},\bm{\nu})}\cdot\nabla_{\nu_{n}}B_{n}^{\\#}(\bm{\beta},\bm{\nu})-\beta_{n}\cdot\nabla_{\nu_{n}}p_{n}^{\\#}(\bm{\beta},\bm{\nu})\big{]}=0.$

Based on the above, and the definitions of $\phi_{1,n}(\bm{\beta},\bm{\nu})$
and $\phi_{2,n}(\bm{\beta},\bm{\nu})$ in (24) and (25), the desired result
(A.1) is proved.

##  Appendix B Solving Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$

We will prove Theorem 2 and use it to solve Problem
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$.

###  B.1. Proof of Theorem 2 which characterizes the solution to Problem
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$

Some texts below are repeated from Appendix A. Problem
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ belongs to convex optimization and
Slater’s condition holds, as shown in the proof of Lemma 5.1. Then the
Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient to obtain the
globally optimal solution, as stated in Lemma 4.3.

Let $[\bm{p}^{\\#},\bm{B}^{\\#},\bm{\tau}^{\\#},\lambda^{\\#}]$ satisfy the
KKT conditions of Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$. Then, after
defining

(A.17)
$\displaystyle\textstyle{\vartheta_{n}^{\\#}:=\frac{g_{n}p_{n}^{\\#}}{\sigma_{n}^{2}B_{n}^{\\#}}},$

we obtain the following KKT conditions:

$\textstyle\frac{\partial L_{\mathbb{P}_{3}}}{\partial p_{n}}$
$\textstyle=-(\nu_{n}c_{n}f_{n}^{\prime}(r_{n,\textnormal{s}}(p_{n}^{\\#},B_{n}^{\\#}))+\tau_{n}^{\\#})\frac{g_{n}}{\sigma_{n}^{2}(1+\vartheta_{n}^{\\#})\ln{2}}$
(A.18) $\textstyle\quad+\nu_{n}\beta_{n}=0,~{}\forall n\in\mathcal{N},$
$\textstyle\frac{\partial L_{\mathbb{P}_{3}}}{\partial B_{n}}$
$\textstyle=-\nabla_{B_{n}}\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})|_{B_{n}=B_{n}^{\\#},p_{n}=p_{n}^{\\#}}$
(A.19)
$\textstyle\quad~{}-\tau_{n}^{\\#}\nabla_{B_{n}}r_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#},p_{n}=p_{n}^{\\#}}\\!+\\!\lambda^{\\#}$
$\textstyle=-\Big{(}\nu_{n}c_{n}f_{n}^{\prime}(r_{n,\textnormal{s}}(p_{n}^{\\#},B_{n}^{\\#}))+\tau_{n}^{\\#}\Big{)}\Big{(}\log_{2}(1\\!+\\!\vartheta_{n}^{\\#})$
(A.20)
$\textstyle\quad~{}-\frac{\vartheta_{n}^{\\#}}{(1+\vartheta_{n}^{\\#})\ln{2}}\Big{)}\\!+\\!\lambda^{\\#}\\!=\\!0,~{}\forall
n\in\mathcal{N},$ (A.21)
$\textstyle\lambda^{\\#}\cdot(\sum_{n\in\mathcal{N}}B_{n}^{\\#}-B_{\text{total}})=0,$
(A.22)
$\textstyle\tau_{n}^{\\#}\cdot(r_{n}^{\min}-r_{n}(p_{n}^{\\#},B_{n}^{\\#}))=0,~{}\forall
n\in\mathcal{N},$

where (A.18) and (A.20) refer to the stationarity conditions, while (A.21) and
(A.22) are called complementary slackness. We show the intermediate step
(A.19) since it will be useful later. For the conditions of primal feasibility
(i.e., (4a) and (4b) for $[\bm{p}^{\\#},\bm{B}^{\\#}]$) and dual feasibility
(i.e., $\tau_{n}^{\\#}\geq 0$ for all $n\in\mathcal{N}$ and $\lambda^{\\#}\geq
0$), we will write them out at the places where we need them.

Next, we aim to simplify (A.18)-(A.22) step-by-step to obtain
$[\bm{p}^{\\#},\bm{B}^{\\#},\bm{\tau}^{\\#},\lambda^{\\#}]$. To begin with,
Condition 2 on Page 2 ensures
$f^{\prime}(r_{n,\textnormal{s}}(p_{n}^{\\#},B_{n}^{\\#}))>0$. Using this
along with $\tau_{n}^{\\#}\geq 0$ and555Using
$r_{n}(p_{n}^{\\#},B_{n}^{\\#})\geq r_{n}^{\min}$ from (4b) and
$r_{n}^{\min}>0$ from Condition 1 on Page 1, we have
$r_{n}(p_{n}^{\\#},B_{n}^{\\#})>0$ which implies $p_{n}^{\\#}>0$ and
$B_{n}^{\\#}>0$, inducing $\vartheta_{n}^{\\#}>0$. For any $x>0$, we can prove
$\log_{2}(1+x)-\frac{x}{(1+x)\ln{2}}>0$, so that
$\log_{2}(1+\vartheta_{n}^{\\#})-\frac{\vartheta_{n}^{\\#}}{(1+\vartheta_{n}^{\\#})\ln{2}}>0$.
$\log_{2}(1+\vartheta_{n}^{\\#})-\frac{\vartheta_{n}^{\\#}}{(1+\vartheta_{n}^{\\#})\ln{2}}>0$
in (A.20), we know $\lambda^{\\#}>0$ so that (A.21) becomes

(A.23) $\textstyle\sum_{n\in\mathcal{N}}B_{n}^{\\#}=B_{\text{total}}.$

We note that both (A.18) and (A.20) have the term
$\nu_{n}c_{n}f^{\prime}(r_{n,\textnormal{s}}(p_{n}^{\\#},B_{n}^{\\#}))+\tau_{n}^{\\#}$,
which is strictly positive due to $\tau_{n}^{\\#}\geq 0$ and
$f^{\prime}(r_{n,\textnormal{s}}(p_{n}^{\\#},B_{n}^{\\#}))>0$ explained above.
Thus, from (A.18) and (A.20), we get

(A.24)
$\textstyle\frac{\Big{(}\log_{2}\big{(}1+\vartheta_{n}^{\\#}\big{)}-\frac{\vartheta_{n}^{\\#}}{(1+\vartheta_{n}^{\\#})\ln
2}\Big{)}\nu_{n}}{\frac{g_{n}}{\sigma_{n}^{2}(1+\vartheta_{n}^{\\#})\ln
2}}=\frac{\lambda^{\\#}}{\beta_{n}}.$

From the above equation (A.24), we solve $\vartheta_{n}^{\\#}$ given
$\lambda^{\\#}$. Denoting the solution as $\psi_{n}(\lambda)$ to highlight its
dependence on $\lambda$, we have:

(A.25) $\textstyle\vartheta_{n}^{\\#}=\psi_{n}(\lambda^{\\#}),\textnormal{ for
}\textstyle{\psi_{n}(\lambda):=\exp\big{\\{}1+W\big{(}\frac{1}{e}(\frac{g_{n}\lambda}{\nu_{n}\beta_{n}\sigma_{n}^{2}}-1)\big{)}\big{\\}}-1}.$

Once we have $\lambda^{\\#}$, (A.17) and (A.25) mean that
$\vartheta_{n}^{\\#}$ denoting
$\frac{g_{n}p_{n}^{\\#}}{\sigma_{n}^{2}B_{n}^{\\#}}$ is decided. To derive
$p_{n}^{\\#}$ and $B_{n}^{\\#}$ given $\lambda^{\\#}$, we need another
condition of $p_{n}^{\\#}$ and $B_{n}^{\\#}$. To this end, we notice (A.20),
but (A.20) invovles $\tau_{n}^{\\#}\geq 0$. If $\tau_{n}^{\\#}=0$, then (A.20)
together with (A.25) will decide $p_{n}^{\\#}$ and $B_{n}^{\\#}$ given
$\lambda^{\\#}$. Therefore, we will discuss Case 1: $\tau_{n}^{\\#}=0$ and
Case 2: $\tau_{n}^{\\#}>0$ respectively for each $n\in\mathcal{N}$. From the
above explanation, we first try to express $p_{n}^{\\#}$ and $B_{n}^{\\#}$ as
expressions of $\lambda^{\\#}$, and then substitute these expressions into our
conditions to obtain $\lambda^{\\#}$.

Before elaborating on the two cases, we note (A.19) and define a function
$\gamma_{n}(\lambda)$ which will facilitate discussing the two cases.
Specifically, given $\lambda$, then under the constraint of

(A.26)
$\displaystyle\textstyle{\frac{g_{n}p_{n}}{\sigma_{n}^{2}B_{n}}=\psi_{n}(\lambda),}~{}\textnormal{
for $\psi_{n}(\lambda)$ defined in~{}(\ref{solvepsin2})},$

we define $\gamma_{n}(\lambda)$ as the result of $r_{n}(p_{n},B_{n})\geq
r_{n,\textnormal{e}}$ to ensure (we will discuss soon when such
$r_{n}(p_{n},B_{n})$ does not exist)

(A.27)
$\textstyle\nabla_{B_{n}}\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})=\lambda.$

From
$r_{n}(p_{n},B_{n})=B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})$,
$B_{n}$ ensuring (A.26) and (A.27) is given by

(A.28)
$\textstyle\textstyle{\frac{\gamma_{n}(\lambda)}{\log_{2}(1+\psi_{n}(\lambda))}}.$

We aim to obtain the expression of $\gamma_{n}(\lambda)$ for
$r_{n}(p_{n},B_{n})$ from (A.26) and (A.27). From (20) and (A.27),

$\textstyle f_{n}^{\prime}(r_{n}(p_{n},B_{n})-r_{n,\textnormal{e}})$ (A.29)
$\textstyle=\textstyle{\frac{\lambda}{\nu_{n}c_{n}\cdot(\log_{2}\big{(}1+\psi_{n}(\lambda)\big{)}-\frac{\psi_{n}(\lambda)}{(1+\psi_{n}(\lambda))\ln
2})}}=\textstyle{\frac{\beta_{n}\sigma_{n}^{2}(1+\psi_{n}(\lambda))\ln{2}}{c_{n}g_{n}}},$

where the last step uses
$\frac{(\log_{2}(1+\psi_{n}(\lambda))-\frac{\psi_{n}(\lambda)}{(1+\psi_{n}(\lambda))\ln
2})\nu_{n}}{\frac{g_{n}}{\sigma_{n}^{2}(1+\psi_{n}(\lambda))\ln
2}}=\frac{\lambda}{\beta_{n}}$ from (A.24) and (A.25).

From (A.29), we know $r_{n}(p_{n},B_{n})\geq r_{n,\textnormal{e}}$ ensuring
(A.27) may not exist for all $\lambda$, since we do not know the range of
$f_{n}^{\prime}$. Whenever such $r_{n}(p_{n},B_{n})\geq r_{n,\textnormal{e}}$
does not exist, we just define $\gamma_{n}(\lambda)$ as
$r_{n,\textnormal{e}}$. The above leads to the desired expression of
$\gamma_{n}(\lambda)$ in Theorem 2 on Page 2.

We now discuss the two cases for each $n\in\mathcal{N}$:

  * •

Case 1: $\tau_{n}^{\\#}=0$. In this case, (A.17) (A.19) and (A.25) mean that
setting $p_{n}$, $B_{n}$, and $\lambda$ as $p_{n}^{\\#}$, $B_{n}^{\\#}$, and
$\lambda^{\\#}$ respectively ensures (A.26) and (A.27), where we note the
primal feasibility condition (4b) along with Condition 1 on Page 1 means
$r_{n}(p_{n}^{\\#},B_{n}^{\\#})\geq r_{n}^{\min}\geq r_{n,\textnormal{e}}$.
Noting the above and (A.28), we obtain
$B_{n}^{\\#}=\frac{\gamma_{n}(\lambda^{\\#})}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}\geq\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$.

  * •

Case 2: $\tau_{n}^{\\#}>0$. In this case, (A.22) means
$r_{n}(p_{n}^{\\#},B_{n}^{\\#})=r_{n}^{\min}$, which along with (A.17) and
(A.25) induces
$B_{n}^{\\#}=\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$. Also,
(A.19) means

$\textstyle\nabla_{B_{n}}\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})|_{B_{n}=B_{n}^{\\#},p_{n}=p_{n}^{\\#}}$
(A.30)
$\textstyle=\lambda^{\\#}-\tau_{n}^{\\#}\nabla_{B_{n}}r_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}}<\lambda^{\\#},$

where the last step uses $\tau_{n}^{\\#}>0$ and
$\nabla_{B_{n}}r_{n}(p_{n},B_{n})|_{B_{n}=B_{n}^{\\#}}>0$ (note
$B_{n}^{\\#}>0$ as explained in Footnote 5). The above means setting $p_{n}$,
$B_{n}$, and $\lambda$ as $p_{n}^{\\#}$,
$\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$, and
$\lambda^{\\#}$ respectively ensures (A.26) and (A.30). Moreover, when
$\gamma_{n}(\lambda^{\\#})>0$ exists, (A.17) (A.25) and (A.28) means setting
$p_{n}$, $B_{n}$, and $\lambda$ as $p_{n}^{\\#}$,
$\frac{\gamma_{n}(\lambda^{\\#})}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$, and
$\lambda^{\\#}$ respectively ensures (A.26) and (A.27). Comparing(A.27) and
(A.30), and noting that
$\nabla_{B_{n}}\mathcal{F}_{n}(p_{n},B_{n}\,|\,\beta_{n},\nu_{n})=\nu_{n}c_{n}f_{n}^{\prime}(r_{n,\textnormal{s}}(p_{n},B_{n}))\nabla_{B_{n}}r_{n,\textnormal{s}}(p_{n},B_{n})$
is strictly decreasing function with respect to666This holds since
$\nabla_{B_{n}}r_{n,\textnormal{s}}(p_{n},B_{n})$ is strictly decreasing with
respect to $B_{n}$ and positive, and
$f_{n}^{\prime}(r_{n,\textnormal{s}}(p_{n},B_{n}))$ is non-increasing with
respect to $B_{n}$ and positive given Condition 2 on Page 2. $B_{n}$, we
obtain
$B_{n}^{\\#}=\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}>\frac{\gamma_{n}(\lambda^{\\#})}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$.
When we cannot find $\gamma_{n}(\lambda^{\\#})\geq r_{n,\textnormal{e}}$ for
(A.27), as already explained, we just set $\gamma_{n}(\lambda^{\\#})$ as
$r_{n,\textnormal{e}}$ and still have
$B_{n}^{\\#}=\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}\geq\frac{\gamma_{n}(\lambda^{\\#})}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}$
due to $r_{n}^{\min}\geq r_{n,\textnormal{e}}$.

Summarizing the two cases, we conclude for any $n\in\mathcal{N}$ that

$\textstyle B_{n}^{\\#}$
$\textstyle=\textstyle{\max\\{\frac{\gamma_{n}(\lambda^{\\#})}{\log_{2}\big{(}1+\psi_{n}(\lambda^{\\#}))},~{}\frac{r_{n}^{\min}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}\\}}$
(A.31)
$\textstyle=\textstyle{\frac{\max\\{\gamma_{n}(\lambda^{\\#}),~{}r_{n}^{\min}\\}}{\log_{2}(1+\psi_{n}(\lambda^{\\#}))}}.$

Then (A.29) and
$r_{n,\textnormal{s}}(p_{n},B_{n})=r_{n}(p_{n},B_{n})-r_{n,\textnormal{e}}$
Now we know how to compute $B_{n}^{\\#}$ in (A.31) given $\lambda^{\\#}$. Then
$\lambda^{\\#}$ is decided such that $B_{n}^{\\#}|_{n\in\mathcal{N}}$ from
(A.31) together satisfy (A.23). Finally, after $\lambda^{\\#}$ and
$B_{n}^{\\#}$ are obtained, $p_{n}^{\\#}$ is computed as
$\frac{\sigma_{n}^{2}B_{n}^{\\#}\cdot\psi_{n}(\lambda^{\\#})}{g_{n}}$ based on
(A.17) and (A.25). To summarize, we have proved Eq. (34) of Theorem 2 on Page
2; i.e., Theorem 2 is proved. ∎

For strictly concave utility, Lemma B.1 below shows that
$\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$ has a unique globally optimal solution
given by Theorem 2.

###### Lemma B.1.

In Theorem 2, with an additional condition that the function $f_{n}(x)$ for
any $n\in\mathcal{N}$ is strictly concave, then Theorem 2 gives the unique
globally optimal solution of $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$.

###### Proof.

We will show the following three results, where $\Psi(\lambda)$ denotes
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(\lambda)$.

  * ❶

$\Psi(\lambda)$ is strictly decreasing in $\lambda\geq 0$,

  * ❷

$\lim_{\lambda\to 0^{+}}\Psi(\lambda)=\infty$, and

  * ❸

$\lim_{\lambda\to\infty}\Psi(\lambda)=0$.

Proving Result ❶: In Eq. (34), we also define function
$\mathcal{B}_{n}(\lambda)$. It is used to better explain the proof here for
Result (ii). Note we always enforce $\lambda\geq 0$ below. Here we consider
the function $f_{n}(\cdot)$ for any $n\in\mathcal{N}$ is strictly concave
(i.e., $f_{n}^{\prime}(\cdot)$ is strictly decreasing) for $x\geq 0$.
Combining this with the fact that $\psi_{n}(\lambda)$ in (A.25) is increasing
in $\lambda$, we know that $\gamma_{n}(\lambda)$ in (34) is non-increasing in
$\lambda$. Since $\psi_{n}(\lambda)$ is increasing and $\gamma_{n}(\lambda)$
is decreasing, $\mathcal{B}_{n}(\lambda)$ is strictly decreasing in $\lambda$,
so that Result ❶ is proved.

Proving Results ❷ and ❸: Since $f_{n}^{\prime}(x)$ here is strictly
decreasing, $\gamma_{n}(\lambda)$ defined in (34) is at most
$r_{n,\textnormal{e}}+\zeta$ for
$\zeta:=(f_{n}^{\prime})^{-1}\big{(}\frac{\beta_{n}\sigma_{n}^{2}\ln{2}}{c_{n}g_{n}}\big{)}$
when $\zeta\geq 0$ exists, and equals $r_{n,\textnormal{e}}$ otherwise.
Anyways, $\gamma_{n}(\lambda)$ is upper bounded by a constant. From (A.25), we
know $\lim_{\lambda\to 0^{+}}\psi_{n}(\lambda)=0$ and
$\lim_{\lambda\to\infty}\psi_{n}(\lambda)=\infty$. Then from (34), we have
$\lim_{\lambda\to 0^{+}}\mathcal{B}_{n}(\lambda)=0$ and
$\lim_{\lambda\to\infty}\mathcal{B}_{n}(\lambda)=\infty$, so that Results ❷
and ❸ are proved.

As noted in (A.25), $\lambda^{\\#}$ is the solution of $\lambda$ to
$\sum\limits_{n\in\mathcal{N}}\mathcal{B}_{n}(\lambda)=B_{\text{total}}$.
Clearly, $\lambda^{\\#}$ is unique given Results ❶ ❷ and ❸ above. Thus, the
desired result is proved. ∎

###  B.2. Algorithm to solve Problem $\mathbb{P}_{3}(\bm{\beta},\bm{\nu})$
based on Theorem 2

We still let $\Psi(\lambda)$ denote
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(\lambda)$. Similar to Lemma B.1 for
strictly concave utility, we can prove for concave utility, $\Psi(\lambda)$ is
non-increasing as $\lambda$ increases. This motivates us to use the bisection
method to find $\lambda^{\\#}$ from (36).

For the bisection method, we use $0$ as the initial lower bound. We can find
the initial upper bound as follows. Starting with a random positive number
$\Lambda$. If $\Psi(\Lambda)$ denoting
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(\Lambda)$ is less than
$B_{\textnormal{total}}$, we use $\Lambda$ as the initial upper bound. If
$\Psi(\Lambda)$ equals $B_{\textnormal{total}}$, then $\Lambda$ is just our
desired $\lambda^{\\#}$. If $\Psi(\Lambda)$ is greater than
$B_{\textnormal{total}}$, we check $\Psi(2\Lambda)$ denoting
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(2\Lambda)$. Similarly, if
$\Psi(2\Lambda)$ is less than $B_{\textnormal{total}}$, we use $2\Lambda$ as
the initial upper bound. If $\Psi(2\Lambda)$ equals $B_{\textnormal{total}}$,
then $2\Lambda$ is just our desired $\lambda^{\\#}$. If $\Psi(2\Lambda)$ is
greater than $B_{\textnormal{total}}$, we check $\Psi(2^{2}\Lambda)$ denoting
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(2^{2}\Lambda)$. The process continues.
Basically, we find $i$ such that $\Psi(2^{i-1}\Lambda)$ is greater than
$B_{\textnormal{total}}$, and $\Psi(2^{i}\Lambda)$ is less than
$B_{\textnormal{total}}$, then we use $2^{i}\Lambda$ as the initial upper
bound. If there exists $i$ which makes $\Psi(2^{i}\Lambda)$ equal
$B_{\textnormal{total}}$, then $2^{i}\Lambda$ is just our desired
$\lambda^{\\#}$.

With the initial lower bound and the initial upper bound explained above, the
remaining process to find $\lambda^{\\#}$ follows from the standard bisection
method. In each iteration, the bisection method divides the interval $[a,b]$
in two parts by computing the midpoint $c=(a+b)/2$ of the interval and the
value of $\Psi(c)$. If $\Psi(c)$ equals $B_{\textnormal{total}}$, then the
process has succeeded and $c$ is just our desired $\lambda^{\\#}$. Otherwise,
if $\Psi(c)$ is greater than $B_{\textnormal{total}}$, we update $a$ to $c$ so
that the next iteration starts with the interval $[c,b]$; if $\Psi(c)$ is less
than $B_{\textnormal{total}}$, we update $b$ to $c$ so that the next iteration
starts with the interval $[a,c]$. The bisection method converges when
$\Psi(c)$ is close to (but should be no greater than)
$B_{\textnormal{total}}$.

Note that for each $\lambda$, computing $\Psi(\lambda)$ denoting
$\sum_{n\in\mathcal{N}}\mathcal{B}_{n}(\lambda)$ costs $\mathcal{O}(N)$ time.
The number of iterations to find the initial upper bound depends on the
initialization, while the number of iterations for the bisection method
depends on the error tolerance. With the initial lower bound $0$, the initial
upper bound $H$, and the error tolerance $\epsilon$, the number of iterations
for the bisection method is $\mathcal{O}(\log_{2}{\frac{H}{\epsilon}})$.

##  Appendix C Baseline Algorithms for Comparison

As shown in Section 7.4, we compare our Algorithm 1 with the following
baselines: “Optimizing $\bm{B}$ only”, “Optimizing $\bm{p}$ only”, and
“Alternating optimization”. We detail them below.

###  C.1. Optimizing $\bm{p}$ only

Given $B_{n}$, we define $p_{n}^{\min}$ as the value of $p_{n}$ which causes
$r_{n}(p_{n},B_{n})$ to be $r_{n}^{\min}$. Formally,

(A.32)
$p_{n}^{\min}:=\frac{(2^{\frac{r_{n}^{\min}}{B_{n}}}-1){\sigma_{n}}^{2}B_{n}}{g_{n}}$
so that
$B_{n}\log_{2}(1+\frac{g_{n}p_{n}^{\min}}{{\sigma_{n}}^{2}B_{n}})=r_{n}^{\min}$.

Then $r_{n}\geq r_{n}^{\min}$ in (4b) of Problem $\mathbb{P}_{1}$ means
$p_{n}\geq p_{n}^{\min}$. Then “Optimizing $\bm{p}$ only” just means for each
$n\in\mathcal{N}$, maximizing $\varphi_{n}(p_{n},B_{n})$ subject to $p_{n}\geq
p_{n}^{\min}$.

###### Lemma C.1.

For each $n\in\mathcal{N}$, the following results hold.

  * (i)

$\varphi_{n}(p_{n},B_{n})$ is pseudoconvex and semistrictly quasiconvex for
$p_{n}\in[p_{n}^{\min},\infty)$ and $B_{n}\in(0,\infty)$.

  * (ii)

To optimize the weighted sum-UEE (which means maximizing
$\varphi_{n}(p_{n},B_{n})$ given $B_{n}$ for $p_{n}\in[p_{n}^{\min},\infty)$),
we just need to compute a stationary point of $\varphi_{n}(p_{n},B_{n})$ with
respect to $p_{n}$ given $B_{n}$ and have a comparison with $p_{n}^{\min}$.
The maximum of them will be a point at which $\varphi_{n}(p_{n},B_{n})$
achieves the maximum.

Given Section 7.1, below we consider $b_{n}=1$, $c_{n}=0$, and $d_{n}=0$ so
that the three types of utility functions are as follows:

  * •

Type 1 utility function: $f_{n}(x)=\kappa_{n}\ln(1+a_{n}x)$ with
$\kappa_{n}>0$ and $0<a_{n}<1$,

  * •

Type 2 utility function: $f_{n}(x)=\kappa_{n}\cdot(1-e^{-a_{n}x})$ with
$a_{n},\kappa_{n}>0$,

  * •

Type 3 utility function: $f_{n}(x)=\kappa_{n}{x}^{a_{n}}$ with $\kappa_{n}>0$
and $0<a_{n}<1$.

###### Lemma C.2.

For Type 1 utility function: $f_{n}(x)=\kappa_{n}\ln(1+a_{n}x)$ with
$\kappa_{n}>0$ and $0<a_{n}<1$, $\varphi_{n}(p_{n},B_{n})$ given $B_{n}$
achieves its maximum at $p_{n}$ given by the maximum of the following two
numbers: $p_{n}^{\min}$ of Eq. (A.32), and the solution $p_{n}$ to

$\textstyle\textstyle{\ln(1+a_{n}\cdot(B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})-r_{n,\textnormal{e}}))}$
(A.33)
$\textstyle=\textstyle{W(\frac{a_{n}g_{n}B_{n}}{({\sigma_{n}}^{2}B_{n}+g_{n}p_{n})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}}))},$

where $W(\cdot)$ is the principal branch of the Lambert W function.

###### Lemma C.3.

For Type 2 utility function: $f_{n}(x)=\kappa_{n}\cdot(1-e^{-a_{n}x})$ with
$a_{n},\kappa_{n}>0$, $\varphi_{n}(p_{n},B_{n})$ given $B_{n}$ achieves its
maximum at $p_{n}$ given by

$\textstyle{\max\\{p_{n}^{\min}\textnormal{ of
Eq.~{}(\ref{eqdefpnmin})},~{}(\chi_{n}-1)\frac{{\sigma_{n}}^{2}B_{n}}{g_{n}}\\}},$

where $\chi_{n}$ satisfies

(A.34) $\textstyle\textstyle{{\chi_{n}}^{a_{n}B_{n}/(\ln
2)}e^{-a_{n}r_{n,\textnormal{e}}}+\frac{{\sigma_{n}}^{2}B_{n}-g_{n}p_{n}^{\textnormal{cir}}}{\chi_{n}{\sigma_{n}}^{2}\ln
2}=\frac{a_{n}B_{n}}{\ln 2}+1}.$

###### Lemma C.4.

For Type 3 utility function: $f_{n}(x)=\kappa_{n}{x}^{a_{n}}$ with
$\kappa_{n}>0$ and $0<a_{n}<1$, $\varphi_{n}(p_{n},B_{n})$ given $B_{n}$
achieves its maximum at $p_{n}$ being

(A.39)
$\textstyle\textstyle{\max\bigg{\\{}\begin{array}[]{l}p_{n}^{\min}\textnormal{}\\\
\textnormal{of
Eq.~{}(\ref{eqdefpnmin})},\end{array}\bigg{[}\textstyle{\begin{array}[]{l}\exp\big{(}a_{n}+\frac{r_{n,\textnormal{e}}\ln
2}{B_{n}}+\\\
W((\frac{a_{n}g_{n}p_{n}^{\textnormal{cir}}}{{\sigma_{n}}^{2}B_{n}}-a_{n})e^{-a_{n}-\frac{r_{n,\textnormal{e}}\ln
2}{B_{n}}})\big{)}-1\end{array}}\bigg{]}\frac{{\sigma_{n}}^{2}B_{n}}{g_{n}}}\bigg{\\}}.$

The proofs of Lemmas C.1, C.2, C.3, C.4 are provided in Section D below.

###  C.2. Optimizing $\bm{B}$ only

Given $p_{n}$, we define $B_{n}^{\min}$ as the value of $B_{n}$ which causes
$r_{n}(p_{n},B_{n})$ to be $r_{n}^{\min}$; i.e.,
$B_{n}^{\min}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}^{\min}})=r_{n}^{\min}$.

Then we have the following optimization.

(A.40a)
$\textstyle\max_{B_{n}}~{}\sum_{n\in\mathcal{N}}c_{n}\varphi_{n}(p_{n},B_{n})$
(A.40b) subject to: $\textstyle\sum_{n\in\mathcal{N}}B_{n}\leq
B_{\textnormal{total}},$ (A.40c) $\textstyle B_{n}\geq B_{n}^{\min},\text{for
any }n\in\mathcal{N}.$

From the first result of Lemma 4.6, $f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))$
is concave with respect to $B_{n}$ given $p_{n}$. Hence,
$\varphi_{n}(p_{n},B_{n})$ is concave with respect to $B_{n}$ given $p_{n}$.
Then the above problem belongs to convex optimization. The Lagrange function
of the problem is as follows:

$\textstyle L(p_{n},B_{n},\alpha_{n},\zeta)$
$\textstyle=-\sum_{n\in\mathcal{N}}c_{n}\varphi_{n}(p_{n},B_{n})+\sum_{n\in\mathcal{N}}\alpha_{n}\cdot(B_{n}^{\min}-B_{n})$
(A.41)
$\textstyle+\zeta\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\textnormal{total}})$

After applying KKT conditions to (C.2), we get:

(A.42) $\textstyle\frac{\partial L}{\partial
p_{n}}=-c_{n}\nabla_{p_{n}}\varphi_{n}(p_{n},B_{n})=0$ (A.43)
$\textstyle\frac{\partial L}{\partial
B_{n}}=-c_{n}\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})-\alpha_{n}+\zeta=0$ (A.44)
$\textstyle\alpha_{n}\cdot(B_{n}^{\min}-B_{n})=0$ (A.45)
$\textstyle\zeta\cdot(\sum_{n\in\mathcal{N}}B_{n}-B_{\textnormal{total}})=0$

Since $\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})>0$ and $\alpha_{n}\geq 0$, only
$\zeta>0$ could make (A.43) hold. Hence from (A.45), we could tell that
$\sum_{n\in\mathcal{N}}B_{n}=B_{\textnormal{total}}$. Let
$\widehat{B_{n}}(\zeta)$ be the solution of $B_{n}$ to the following equation:

(A.46) $\displaystyle c_{n}\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})=\zeta.$

It is straightforward to derive the expression of $\widehat{B_{n}}(\zeta)$
based on (A.46). For any $x>0$, we can prove
$\log_{2}(1+x)-\frac{x}{(1+x)\ln{2}}>0$. Then we can prove that
$\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})$ is decreasing as $B_{n}$ increases.
For each $n\in\mathcal{N}$, there are two possible cases:

  * •

If $\alpha_{n}=0$, then $B_{n}\geq B_{n}^{\min}$,
$c_{n}\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})=\zeta$ so that
$B_{n}=\widehat{B_{n}}(\zeta)\geq B_{n}^{\min}$.

  * •

If $\alpha_{n}>0$, then $B_{n}=B_{n}^{\min}$,
$c_{n}\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})=\zeta-\alpha_{n}<\zeta$. Thus
$B_{n}=B_{n}^{\min}>\widehat{B_{n}}(\zeta)$.

Summarizing both cases and we can derive $B_{n}$ as follows:

(A.47) $\textstyle B_{n}=\max\\{\widehat{B_{n}}(\zeta),~{}B_{n}^{\min}\\}$

and $\zeta$ could be derived from:

(A.48)
$\textstyle\sum_{n\in\mathcal{N}}\max\\{\widehat{B_{n}}(\zeta),B_{n}^{\min}\\}=B_{\text{total}}.$

As $\nabla_{B_{n}}\varphi_{n}(p_{n},B_{n})$ is decreasing as $B_{n}$
increases. Then $\widehat{B_{n}}(\zeta)$ decreases as $\zeta$ increases. Let
$\zeta^{\\#}$ be the solution of $\zeta$ to (A.48). We use the bisection
search to find $\zeta^{\\#}$. The following discussion is similar to that of
Appendix B.2. We let $F(\zeta)$ denote
$\sum_{n\in\mathcal{N}}\max\\{\widehat{B_{n}}(\zeta),B_{n}^{\min}\\}$. Then
$F(\zeta)$ is non-increasing as $\zeta$ increases.

For the bisection method, we use $0$ as the initial lower bound. We can find
the initial upper bound as follows. Starting with a random positive number
$\theta$. If $F(\theta)$ is less than $B_{\textnormal{total}}$, we use
$\theta$ as the initial upper bound. If $F(\theta)$ equals
$B_{\textnormal{total}}$, then $\theta$ is just our desired $\zeta^{\\#}$. If
$F(\theta)$ is greater than $B_{\textnormal{total}}$, we check $F(2\theta)$.
Similarly, if $F(2\theta)$ is less than $B_{\textnormal{total}}$, we use
$2\theta$ as the initial upper bound. If $F(2\theta)$ equals
$B_{\textnormal{total}}$, then $2\theta$ is just our desired $\zeta^{\\#}$. If
$F(2\theta)$ is greater than $B_{\textnormal{total}}$, we check
$F(2^{2}\theta)$. The process continues. Basically, we find $i$ such that
$F(2^{i-1}\theta)$ is greater than $B_{\textnormal{total}}$, and
$F(2^{i}\theta)$ is less than $B_{\textnormal{total}}$, then we use
$2^{i}\theta$ as the initial upper bound. If there exists $i$ which makes
$F(2^{i}\theta)$ equal $B_{\textnormal{total}}$, then $2^{i}\theta$ is just
our desired $\zeta^{\\#}$.

With the initial lower bound and the initial upper bound explained above, the
remaining process to find $\zeta^{\\#}$ follows from the standard bisection
method. In each iteration, the bisection method divides the interval $[a,b]$
in two parts by computing the midpoint $c=(a+b)/2$ of the interval and the
value of $F(c)$. If $F(c)$ equals $B_{\textnormal{total}}$, then the process
has succeeded and $c$ is just our desired $\zeta^{\\#}$. Otherwise, if $F(c)$
is greater than $B_{\textnormal{total}}$, we update $a$ to $c$ so that the
next iteration starts with the interval $[c,b]$; if $F(c)$ is less than
$B_{\textnormal{total}}$, we update $b$ to $c$ so that the next iteration
starts with the interval $[a,c]$. The bisection method converges when $F(c)$
is close to (but should be no greater than) $B_{\textnormal{total}}$.

After using the bisection method to find $\zeta^{\\#}$, we compute $B_{n}$ as
$\max\\{\widehat{B_{n}}(\zeta^{\\#}),~{}B_{n}^{\min}\\}$ for each
$n\in\mathcal{N}$.

###  C.3. Alternating optimization

The algorithm for alternating optimization is to combine the algorithms of
“optimizing $\bm{p}$ only” in C.1 and “optimizing $\bm{B}$ only” in C.2 to
perform optimization in an alternating manner.

Specifically, we treat “optimizing $p_{n}$ only” first and then “optimizing
$B_{n}$ only” as a round. After each round, we will compare the new solution
with that of the last round. If the relative difference between them is less
than our pre-determined threshold, we consider that alternating optimization
of $\bm{p}$ and $\bm{B}$ has converged.

##  Appendix D Proof of Lemmas for Appendix C

###  D.1. Proof of Lemma C.1

We first have the following properties of $\varphi_{n}(p_{n},B_{n})$.

  * •

From Lemma 4.6, the numerator $f_{n}(r_{n,\textnormal{s}}(p_{n},B_{n}))$ of
$\varphi_{n}(p_{n},B_{n})$ is jointly concave in $p_{n}$ and $B_{n}$. Then
from Result (iii) of Theorem 2.3.8 in777Theorem 2.3.8 of
(cambini2008generalized, ) and Lemma 2.2 of (ivanov2020characterization, ) are
about “semistrictly quasiconvex”, but can easily be extended to “semistrictly
quasiconcave” since a function $g(\cdot)$ is semistrictly quasiconvex if and
only if $-g(\cdot)$ is semistrictly quasiconcave. Specifically, a function
$g(\cdot)$ is semistrictly quasiconvex (resp., semistrictly quasiconcave) if
and only if for any $x,y$, the result $g(y)<g(x)$ (resp., $g(y)>g(x)$) implies
that $g(x+t(y-x))$ is smaller (resp., greater) than $g(x)$ for any
$t\in(0,1)$.  (cambini2008generalized, ), $\varphi_{n}(p_{n},B_{n})$ is
semistrictly quasiconcave which means that Result (i) of Lemma C.1 is proved,
where the definition of “semistrictly quasiconcave” is provided in Footnote 7.

  * •

From Lemma 2.2 of (ivanov2020characterization, ), a scalar function $g(\cdot)$
over a convex set $\mathcal{X}$ is semistrictly quasiconcave if and only if
any closed segment $\mathcal{S}\subset\mathcal{X}$ can be split into three
segments $\mathcal{S}_{1},\mathcal{S}_{2},\mathcal{S}_{3}$ such that
$g(\cdot)$ is increasing in $\mathcal{S}_{1}$, constant in $\mathcal{S}_{2}$,
and decreasing in $\mathcal{S}_{3}$. Note that
$\mathcal{S}_{1},\mathcal{S}_{2},\mathcal{S}_{3}$ can be $\emptyset$.

Then we define $p_{n}^{\operatorname{sec}}$ as the transmission power as:

(A.49)
$p_{n}^{\operatorname{sec}}:=\frac{(2^{\frac{r_{n,\textnormal{e}}}{B_{n}}}-1){\sigma_{n}}^{2}B_{n}}{g_{n}}$
so that
$B_{n}\log_{2}(1+\frac{g_{n}p_{n}^{\operatorname{sec}}}{{\sigma_{n}}^{2}B_{n}})=r_{n,\textnormal{e}}$.

Combining the above with Eq. (3), we could derive that:
$\varphi_{n}(p_{n}^{\operatorname{sec}},B_{n})=0$,
$\lim_{p_{n}\to\infty}\varphi_{n}(p_{n},B_{n})=0$, and
$\varphi_{n}(p_{n},B_{n})>0$ for
$p_{n}\in(p_{n}^{\operatorname{sec}},\infty)$, we know that888For the specific
types of utility functions $f_{n}(\cdot)$ used in Section 7.1 of this paper,
we can show $\widehat{p_{n}}=\widetilde{p_{n}}$ in principle, but it is not
the focus of our paper and does not impact the validness of our results. there
exists $\widehat{p_{n}}$ and $\widetilde{p_{n}}$ (possibly the same) such that
$\varphi_{n}(p_{n},B_{n})$ is increasing in
$[p_{n}^{\operatorname{sec}},\widehat{p_{n}}]$, constant in
$[\widehat{p_{n}},\widetilde{p_{n}}]$, and decreasing in
$[\widetilde{p_{n}},\infty)$.

The condition $r_{n}^{\min}\geq r_{n,\textnormal{e}}$ means $p_{n}^{\min}\geq
p_{n}^{\operatorname{sec}}$. Hence, the above analysis induces the following
cases.

  * •

If $p_{n}^{\min}<\widehat{p_{n}}$, then $\varphi_{n}(p_{n},B_{n})$ is
increasing in $[p_{n}^{\min},\widehat{p_{n}}]$, constant in
$[\widehat{p_{n}},\widetilde{p_{n}}]$, and decreasing in
$[\widetilde{p_{n}},\infty)$.

  * •

If $\widehat{p_{n}}\leq p_{n}^{\min}\leq\widetilde{p_{n}}$, then
$\varphi_{n}(p_{n},B_{n})$ is constant in
$[\widehat{p_{n}},\widetilde{p_{n}}]$, and decreasing in
$[\widetilde{p_{n}},\infty)$.

  * •

If $\widetilde{p_{n}}<p_{n}^{\min}$, then $\varphi_{n}(p_{n},B_{n})$ is
decreasing in $[p_{n}^{\min},\infty)$.

Based on the above, Result (ii) of Lemma C.1 is also proved. ∎

###  D.2. Proof of Lemma C.2

Given Section 7.1, we consider $b_{n}=1$ so that Type 1 utility function is
$f_{n}(x)=\kappa_{n}\ln(1+a_{n}x)$ with $\kappa_{n}>0$ and $0<a_{n}<1$.

We write $r_{n,\textnormal{s}}(p_{n},B_{n})$ as $r_{n,\textnormal{s}}$ for
simplicity below. Given $B_{n}$, by letting the derivative of
$\varphi_{n}(p_{n},B_{n})$ with respect to $p_{n}$ be zero, we obtain

(A.50)
$\displaystyle\frac{\kappa_{n}a_{n}}{1+a_{n}r_{n,\textnormal{s}}}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})=\kappa_{n}\ln(1+a_{n}r_{n,\textnormal{s}}),$

which induces

(A.51) $\displaystyle
a_{n}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})=(1+a_{n}r_{n,\textnormal{s}})\ln(1+a_{n}r_{n,\textnormal{s}}).$

This means

(A.52)
$\displaystyle\ln(1+a_{n}r_{n,\textnormal{s}})=W(a_{n}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})),$

for $W(\cdot)$ being the principal branch of the Lambert W function; i.e.,
$W(z)$ for $z\geq-e^{-1}$ is the solution of $x\geq-1$ to the equation
$xe^{x}=z$.

From
$r_{n,\textnormal{s}}(p_{n},B_{n}):=B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})-r_{n,\textnormal{e}}$,
we have

$\displaystyle(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})$
$\displaystyle=\frac{\frac{g_{n}}{{\sigma_{n}}^{2}}}{(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$ (A.53)
$\displaystyle=\frac{g_{n}B_{n}}{({\sigma_{n}}^{2}B_{n}+g_{n}p_{n})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}}).$

Then we can solve

$\displaystyle\ln(1+a_{n}\cdot(B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})-r_{n,\textnormal{e}}))$
(A.54)
$\displaystyle=W(\frac{a_{n}g_{n}B_{n}}{({\sigma_{n}}^{2}B_{n}+g_{n}p_{n})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}})).$

Using Result (ii) of Lemma C.1, the optimal $p_{n}$ given $B_{n}$ for the
weighted sum-UEE optimization is given by the maximum of the following two
numbers: $p_{n}^{\min}$ of Eq. (A.32), and the solution $p_{n}$ to the above
Eq. (A.54). ∎

###  D.3. Proof of Lemma C.3

Given Section 7.1, we consider $c_{n}=0$ so that Type 2 utility function is
$f_{n}(x)=\kappa_{n}\cdot(1-e^{-a_{n}x})$ with $a_{n},\kappa_{n}>0$.

We write $r_{n,\textnormal{s}}(p_{n},B_{n})$ as $r_{n,\textnormal{s}}$ for
simplicity below. Given $B_{n}$, by letting the derivative of
$\varphi_{n}(p_{n},B_{n})$ with respect to $p_{n}$ be zero, we obtain

(A.55)
$\displaystyle\kappa_{n}a_{n}e^{-a_{n}r_{n,\textnormal{s}}}(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})=\kappa_{n}\cdot(1-e^{-a_{n}r_{n,\textnormal{s}}}),$

which induces

(A.56) $\displaystyle
a_{n}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})=e^{a_{n}r_{n,\textnormal{s}}}-1.$

With $\chi_{n}:=1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}}$, we further obtain

$\displaystyle(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})$
$\displaystyle=\frac{\frac{g_{n}}{{\sigma_{n}}^{2}}}{(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}})$
$\displaystyle=\frac{(\chi_{n}-1){\sigma_{n}}^{2}B_{n}+g_{n}p_{n}^{\textnormal{cir}}}{\chi_{n}{\sigma_{n}}^{2}\ln
2}$ (A.57) $\displaystyle=\frac{B_{n}}{\ln
2}+\frac{g_{n}p_{n}^{\textnormal{cir}}-{\sigma_{n}}^{2}B_{n}}{\chi_{n}{\sigma_{n}}^{2}\ln
2}.$

Then

$\displaystyle a_{n}\cdot(\frac{B_{n}}{\ln
2}+\frac{g_{n}p_{n}^{\textnormal{cir}}-{\sigma_{n}}^{2}B_{n}}{\chi_{n}{\sigma_{n}}^{2}\ln
2})$
$\displaystyle=e^{a_{n}B_{n}\log_{2}\chi_{n}}e^{-a_{n}r_{n,\textnormal{e}}}-1$
(A.58) $\displaystyle={\chi_{n}}^{a_{n}B_{n}/(\ln
2)}e^{-a_{n}r_{n,\textnormal{e}}}-1.$

We further get

(A.59) $\displaystyle{\chi_{n}}^{a_{n}B_{n}/(\ln
2)}e^{-a_{n}r_{n,\textnormal{e}}}+\frac{{\sigma_{n}}^{2}B_{n}-g_{n}p_{n}^{\textnormal{cir}}}{\chi_{n}{\sigma_{n}}^{2}\ln
2}=a_{n}\frac{B_{n}}{\ln 2}+1.$

Using Result (ii) of Lemma C.1, the optimal $p_{n}$ given $B_{n}$ for the
weighted sum-UEE optimization is given by the maximum of the following two
numbers: $p_{n}^{\min}$ of Eq. (A.32), and
$(\chi_{n}-1)\frac{{\sigma_{n}}^{2}B_{n}}{g_{n}}$, where $\chi_{n}$ is the
solution to the above Eq. (A.59). ∎

###  D.4. Proof of Lemma C.4

Given Section 7.1, we consider $d_{n}=0$ so that Type 3 utility function is
$f_{n}(x)=\kappa_{n}{x}^{a_{n}}$ with $\kappa_{n}>0$ and $0<a_{n}<1$.

We write $r_{n,\textnormal{s}}(p_{n},B_{n})$ as $r_{n,\textnormal{s}}$ for
simplicity below. Given $B_{n}$, by letting the derivative of
$\varphi_{n}(p_{n},B_{n})$ with respect to $p_{n}$ be zero, we obtain

(A.60)
$\displaystyle\kappa_{n}a_{n}{r_{n,\textnormal{s}}}^{a_{n}-1}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})=\kappa_{n}{r_{n,\textnormal{s}}}^{a_{n}}.$

Then we have

(A.61) $\displaystyle
a_{n}\cdot(\nabla_{p_{n}}r_{n,\textnormal{s}})\cdot(p_{n}+p_{n}^{\textnormal{cir}})={r_{n,\textnormal{s}}}.$

From
$r_{n,\textnormal{s}}(p_{n},B_{n}):=B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})-r_{n,\textnormal{e}}$,
we have

(A.62)
$\displaystyle\nabla_{p_{n}}r_{n,\textnormal{s}}=\frac{\frac{g_{n}}{{\sigma_{n}}^{2}}}{(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})\ln
2},$

which is used in (A.61) so that

(A.63) $\displaystyle
a_{n}\cdot\frac{\frac{g_{n}}{{\sigma_{n}}^{2}}}{(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})\ln
2}\cdot(p_{n}+p_{n}^{\textnormal{cir}})=B_{n}\log_{2}(1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}})-r_{n,\textnormal{e}}.$

With $\chi_{n}:=1+\frac{g_{n}p_{n}}{{\sigma_{n}}^{2}B_{n}}$, we further obtain

(A.64) $\displaystyle
a_{n}\cdot\frac{\frac{g_{n}}{{\sigma_{n}}^{2}}}{\chi_{n}}((\chi_{n}-1)\frac{{\sigma_{n}}^{2}B_{n}}{g_{n}}+p_{n}^{\textnormal{cir}})+r_{n,\textnormal{e}}\ln
2=B_{n}\ln\chi_{n},$

which is simplified as

(A.65) $\displaystyle a_{n}+\frac{r_{n,\textnormal{e}}\ln
2}{B_{n}}+(\frac{a_{n}g_{n}p_{n}^{\textnormal{cir}}}{{\sigma_{n}}^{2}B_{n}}-a_{n})\frac{1}{\chi_{n}}=\ln\chi_{n}.$

Then it follows that

(A.66)
$\displaystyle\chi_{n}=\textstyle{\exp\big{(}a_{n}+\frac{r_{n,\textnormal{e}}\ln
2}{B_{n}}+(\frac{a_{n}g_{n}p_{n}^{\textnormal{cir}}}{{\sigma_{n}}^{2}B_{n}}-a_{n})\exp(-a_{n}-\frac{r_{n,\textnormal{e}}\ln
2}{B_{n}})\big{)}}.$

Using Result (ii) of Lemma C.1, the optimal $p_{n}$ given $B_{n}$ for the
weighted sum-UEE optimization is given by the maximum of the following two
numbers: $p_{n}^{\min}$ of Eq. (A.32), and
$(\chi_{n}-1)\frac{{\sigma_{n}}^{2}B_{n}}{g_{n}}$, where $\chi_{n}$ is given
by the above Eq. (A.66). ∎

##  Appendix E Additional Simulation Results

We provide more simulation results here. Note that our theoretical analysis
and simulation results apply to uplink communications as well as downlink
communications. For downlink communications, the circuit power
$p_{n}^{\textnormal{cir}}$ in the denominator $p_{n}+p_{n}^{\textnormal{cir}}$
of (3) is the additional power that the server consumes to transmit signals
with power $p_{n}$ to user $U_{n}$, and it is possible that
$p_{n}^{\textnormal{cir}}$ can be the same for different $n$ for such downlink
communications. As noted in Section 7.3, the simulations set the circuit power
$p_{n}^{cir}$ as 2 dBm (i.e., 1.6 milliwatts) for each $n$.

###  E.1. Impact by the number of users

We compare the UEE under different numbers of users $N$: 10, 20, 30, 40, and
50 under three different user scenarios (i.e., three utility functions in Fig.
2(a)). Fig. 6(a) shows how the sum-UEE changes with the increase of the number
of users $N$. It could be seen that as $N$ becomes larger, the sum-UEE also
increases. They are positively correlated. In contrast, the average UEE tends
to decrease as the number of users $N$ increases, which could be seen in Fig.
6(b). That is due to a reduction in the bandwidth allocated to each user.

Figure 6. (a). Sum-UEE under different $N$. (b). Average UEE under different
$N$.  Figure 7. Sum-UEE of each user group with different priorities and
utility functions. The utility functions are obtained from the table results
in Section 7.2 with the SSV360 dataset (ssv360, ). User group 1 uses
$f_{n}(r_{n,\textnormal{s}})=0.5424\ln(1+37.2965r_{n,\textnormal{s}})$. User
group 2 uses
$f_{n}(r_{n,\textnormal{s}})=2.9351(1-e^{-2.1224r_{n,\textnormal{s}}})$. User
group 3 uses
$f_{n}(r_{n,\textnormal{s}})=3.2956{(r_{n,\textnormal{s}}/15.94)}^{0.2733}$.

###  E.2. Heterogeneous types of utility functions among the users

We use Figure 7 to show that our studied system and proposed algorithm allow
heterogeneous types of utility functions among the users. We consider that