# Power Control for Multi-Cell Networks with Non-Orthogonal Multiple Access

Zhaohui Yang, Cunhua Pan, , Wei Xu, , Yijin Pan, Ming Chen, , and Maged
Elkashlan, Z. Yang, W. Xu, Y. Pan and M. Chen are with the National Mobile
Communications Research Laboratory, Southeast University, Nanjing 211111,
China, Email: {yangzhaohui, wxu, panyijin<EMAIL_ADDRESS>Pan and M.
Elkashlan are with School of Electronic Engineering and Computer Science,
Queen Mary University of London, London E1 4NS, U.K., Email: {c.pan,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the problems of sum power minimization and sum
rate maximization for multi-cell networks with non-orthogonal multiple access.
Considering the sum power minimization, we obtain closed-form solutions to the
optimal power allocation strategy and then successfully transform the original
problem to a linear one with a much smaller size, which can be optimally
solved by using the standard interference function. To solve the nonconvex sum
rate maximization problem, we first prove that the power allocation problem
for a single cell is a convex problem. By analyzing the Karush-Kuhn-Tucker
conditions, the optimal power allocation for users in a single cell is derived
in closed form. Based on the optimal solution in each cell, a distributed
algorithm is accordingly proposed to acquire efficient solutions. Numerical
results verify our theoretical findings showing the superiority of our
solutions compared to the orthogonal frequency division multiple access and
broadcast channel.

###### Index Terms:

Non-orthogonal multiple access, power allocation, sum power minimization, sum
rate maximization, DC programming.

##  I Introduction

With the explosive growth of data traffic in mobile Internet [1], non-
orthogonal multiple access (NOMA) has been recently proposed [2, 3]. By using
superposition coding at the transmitter and successive interference
cancelation (SIC) at the receiver, NOMA can achieve higher spectral efficiency
than conventional orthogonal multiple access, such as orthogonal frequency
division multiple access (OFDMA). Besides, NOMA can support more connections
by letting more than one user simultaneously access the same frequency or time
resources [4, 5, 6, 7]. Therefore, NOMA has been deemed as a promising
multiple access scheme for the next generation mobile communication networks
[8, 9, 10].

NOMA can simultaneously serve multiple users with the same frequency band and
time slot by splitting them in the power domain [11]. The basic concept of
NOMA with SIC receiver was introduced in [12]. The ergodic sum rate and the
outage performance with fixed power allocation were analyzed in [13] for NOMA.
Specific impacts of power allocation on sum rate [14, 15, 16], fairness [17,
18, 19], and energy efficiency [20, 21] were investigated. Moreover, joint
sub-channel and power allocation was studied in [22, 23, 24, 25, 26]. However,
the above existing works [12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25,
26, 21] are limited to single-cell analysis, where there is no inter-cell
interference.

Recently, NOMA has been extended to multi-cell networks [27, 28, 29, 30, 31,
32, 33]. Different from multi-cell networks with OFDMA, users in the same cell
can receive intra-cell interference, which can be subtracted by using SIC in
multi-cell networks with NOMA. Different from single-cell networks with NOMA,
inter-cell interference should be considered in multi-cell networks with NOMA.
To harness the effect of inter-cell interference, joint processing (JP) and
coordinated scheduling (CS) technologies are usually adopted [34]. In NOMA-JP,
the users’ data symbols are available at more than one base station (BS) [27].
The designs of CS for NOMA differ from those of JP in that the users’ data are
not shared among the BSs [28]. However, the cooperating BSs in NOMA-CS still
need to exchange global channel state information and cooperative scheduling
information via a standardized interface named X2. In addition, the number of
transmission points is one for NOMA-CS [28], while the number of transmission
points is more than one for NOMA-JP in [27]. Moreover, the numbers of
supported users by NOMA-CS and NOMA-JP are different [29].

Based on NOMA-CS, the uplink and downlink power control problems for sum power
minimization in two-cell networks with NOMA were studied in [31] and [32],
respectively. However, the above existing works [30, 27, 32, 31] were merely
limited to power minimization in a two-cell network with only two users per
cell. The sum rate maximization problem in multiple-input multiple-output
(MIMO)-NOMA multi-cell networks was investigated in [33], where two users are
paired in a virtual cluster. Consequently, there is a lack of systematic
approach for sum power minimization and sum rate maximization problems via
power control in multi-cell networks with NOMA from a mathematical
optimization point of view.

The objective of this paper is to investigate the power control problems for
sum power minimization and sum rate maximization in multi-cell networks with
NOMA. The contributions of this paper are summarized as follows:

  1. 1.

The sum power minimization problem can be equivalently transformed into a
linear problem (LP) with smaller variables. Having obtained the total
transmission power of all BSs, the power allocation for each user can be
presented in a closed-form expression.

  2. 2.

To solve the sum rate maximization problem, we decouple it into two
subproblems, i.e., power allocation problem for users in a single cell and
power control problem in multiple cells. With the total transmission power of
all BSs fixed, we succeed in showing that power control problem in a single
cell is a convex problem. By analyzing the Karush-Kuhn-Tucker (KKT)
conditions, we observe that only the user with the highest channel gain
deserves additional power allocation, while the power for other users in the
same cell is merely determined to maintain their minimal rate demands. Closed-
form expressions of the optimal power allocation and optimal sum rate in a
single cell are further obtained.

  3. 3.

Based on the closed-form expression of the optimal sum rate in a single cell,
the original sum rate maximization problem can be equivalently transformed
into a simplified problem. Since the objective function for each BS can be
formulated as a difference of two convex functions (DC), a convex
approximation of the objective function is introduced. Furthermore, a
distributed algorithm is also provided to obtain a suboptimal solution.

The rest of the paper is organized as follows. In Section II, we introduce the
system model and power control formulation. Sum power minimization and sum
rate maximization for multi-cell networks with NOMA are addressed in Section
III and Section IV, respectively. The extension to MIMO-NOMA systems is
introduced in Section V. Some numerical results are shown in Section VI and
conclusions are finally drawn in Section VII.

##  II System Model and Problem Formulation

Consider a downlink multi-cell network with NOMA, where there are $I$ BSs and
$J$ users. Denote the set of BSs and users by
${\mathcal{I}}=\\{1,2,\cdots,I\\}$ and ${\mathcal{J}}=\\{1,2,\cdots,J\\}$,
respectively. The unique group of users served by BS $i\in{\mathcal{I}}$ is
denoted by set ${\cal{J}}_{i}=\\{J_{i-1}+1,J_{i-1}+2,\cdots,J_{i}\\}$, where
$J_{0}=0$, $J_{I}=J$, $J_{i}=\sum_{l=1}^{i}|{\cal{J}}_{l}|$, and $|\cdot|$ is
the cardinality of a set. We focus on the downlink where mutual interference
exists among cells.

By using NOMA, a BS serves multiple users by splitting them in the power
domain. Assume that each BS shares the same spectrum. For each BS, the total
bandwidth, $B_{\text{total}}$ is equally divided into $M$ subchannels, where
the bandwidth of each subchannel is $B=B_{\text{total}}/M$. Let
$\mathcal{M}=\\{1,2,\cdots,M\\}$ be the set of subchannels. The set of users
served by BS $i$ on subchannel $m$ is denoted by
$\mathcal{J}_{im}=\\{J_{i(m-1)}+1,J_{i(m-1)}+2,\cdots,J_{im}\\}$, where
$J_{i0}=J_{i-1}+1$, $J_{iM}=J_{i}$, $J_{im}=\sum_{l=1}^{m}|\mathcal{J}_{il}|$
and $|{\cal{J}}_{il}|\geq 2$. The channel gain between BS $i$ and user
$j\in\mathcal{J}_{im}$ on subchannel $m$ is denoted by $h_{ijm}$. Without loss
of generality, the channels are sorted as
$|h_{i(J_{i(m-1)}+1)m}|\leq|h_{i(J_{i(m-1)}+2)m}|\leq\cdots\leq|h_{iJ_{im}m}|$,
$\forall i\in\mathcal{I}$, $m\in\mathcal{M}$. According to the NOMA principle,
BS $i$ simultaneously transmits signal $s_{im}$ to all its served users in
$\mathcal{J}_{im}$. The transmitted signal $s_{im}$ can be expressed as

$s_{im}=\sum_{j=J_{i(m-1)}+1}^{J_{im}}\sqrt{p_{ijm}}s_{ijm},$ (1)

where $s_{ijm}$ and $p_{ijm}$ are the message and allocated power for user
$j\in\mathcal{J}_{im}$, respectively.

The observation at user $j\in\mathcal{J}_{im}$ on subchannel $m$ is given by

$\displaystyle y_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=h_{ijm}s_{im}+\sum_{k\in\mathcal{I}\setminus\\{i\\}}h_{kjm}s_{km}+n_{jm}$
(2)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\sum_{l=J_{i(m-1)}+1}^{J_{im}}h_{ijm}\sqrt{p_{ilm}}s_{ilm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum_{k\in\mathcal{I}\setminus\\{i\\}}\sum_{n=J_{k(m-1)}+1}^{J_{km}}h_{kjm}\sqrt{p_{knm}}s_{knm}+n_{jm},$

where $h_{kjm}$ is the cross channel gain between BS $k$ and user $j$ served
by BS $i$ on subchannel $m$, and $n_{jm}$ represents the additive zero-mean
Gaussian noise with variance $\sigma^{2}$. According to [12, 13, 14, 15], each
user should decode the messages of other users in the same cell with lower
channel gains before decoding its own message. Denoting the total transmission
power of BS $i$ on subchannel $m$ by
$q_{im}=\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}$, the achievable rate of user
$j\in\mathcal{J}_{im}$ to detect the message of user
$l\in\\{J_{i(m-1)}+1,\cdots,j\\}$ on subchannel $m$ is

$r_{ijlm}=B\log_{2}\left(1+\frac{|h_{ijm}|^{2}p_{ilm}}{|h_{ijm}|^{2}\sum_{n=l+1}^{J_{im}}p_{inm}+Z_{ijlm}}\right),$
(3)

where

$Z_{ijlm}=\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{kjm}|^{2}+\sigma^{2}.$
(4)

According to (3), strong user $j$ with high channel gain needs to decode the
message of weak user $l\leq j$ with low channel gain. To ensure successful
SIC, the achievable rate of user $j\in\mathcal{J}_{im}$ on subchannel $m$ can
be given by111As in [15], the decoding order is determined by the increasing
order of channel gains for users in the same cell. The optimal decoding order
for multi-cell networks is still an open problem[29], which is beyond the
scope of this paper.

$\displaystyle r_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\min_{l\in\\{j,\cdots,J_{im}\\}}r_{iljm}$
(5)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\min_{l\in\\{j,\cdots,J_{im}\\}}B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+\frac{Z_{iljm}}{|h_{ilm}|^{2}}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}}\right),$

where

$H_{ijm}=\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}.$
(6)

Denote $R_{ijm}>0$ as the minimal rate demand of user $j\in\mathcal{J}_{im}$
on subchannel $m$. Applying (5), $r_{ijm}\geq R_{ijm}$ is equivalent to the
following linear constraint:

$p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\\!\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right).$
(7)

Our objective is to optimize the power allocation in order to minimize the sum
power or maximize the sum rate under the total power constraints and
individual rate demands. Mathematically, the power control problem can be
formulated as

$\displaystyle\mathop{\min}_{\boldsymbol{p}\geq\boldsymbol{0},\boldsymbol{q}\geq\boldsymbol{0}}\;\quad$
$\displaystyle V(\boldsymbol{p},\boldsymbol{q})$ (8a) s.t. $\displaystyle
q_{im}=\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm},\quad\forall
i\in\mathcal{I},m\in\mathcal{M}$ (8b) $\displaystyle
p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\\!\right)\left(\sum_{n=j+1}^{J_{im}}p_{in}+H_{ijm}\right),$
$\displaystyle\qquad\forall
i\in\mathcal{I},m\in\mathcal{M},j\in\mathcal{J}_{im}$ (8c)
$\displaystyle\sum_{m=1}^{M}q_{im}\leq Q_{i},\quad\forall i\in\mathcal{I},$
(8d)

where $\boldsymbol{p}=[p_{111},\cdots,p_{1J_{1M}{M}},\cdots,p_{IJ_{IM}M}]^{T}$
is the transmission power vector, $\boldsymbol{q}=[q_{11},\cdots,$
$q_{1M},\cdots,q_{IM}]^{T}$ is the total transmission power vector, $H_{ijm}$
is defined in (6), and $Q_{i}$ is the maximum transmission power of the BS
$i$. Constraints (8c) reflect that the minimal rate demands of all users can
be satisfied. $V(\boldsymbol{p},\boldsymbol{q})$ is the objective function,
which can be sum power $\sum_{i=1}^{I}\sum_{m=1}^{M}q_{im}$ or negative sum
rate $-\sum_{i=1}^{I}\sum_{m=1}^{M}\sum_{j=J_{i(m-1)}+1}^{J_{im}}r_{ijm}$ with
$r_{ijm}$ defined in (5).

Obviously, the feasible set of Problem (8) is linear. For sum power
minimization, Problem (8) is a LP, of which the globally optimal solution can
be effectively obtained. In the following, we show that sum power minimization
problem can be equivalently transformed into a smaller LP. Since the problem
of sum rate maximization problem is nonconvex, obtaining global optimum
however is known to be difficult. To solve the sum rate maximization problem
efficiently, we first consider the power allocation problem for users in a
single cell with fixed total transmission power of all BSs. Then, based on the
optimal power allocation for users in a single cell, the primal multi-cell sum
rate maximization problem can also be simplified into an equivalent problem. A
distributed algorithm is proposed to obtain a suboptimal solution of the
simplified sum rate maximization problem.

##  III Sum Power Minimization for Multi-Cell Networks

In this section, we solve the sum power minimization Problem (8) with

$V(\boldsymbol{p},\boldsymbol{q})=\sum_{i=1}^{I}\sum_{m=1}^{M}q_{im}.$ (9)

Obviously, sum power minimization Problem (8) is a LP. According to (7), the
interference level received by each user is determined by total transmission
power $\boldsymbol{q}$. Once the total transmission power of other BSs is
given, Problem (8) with objective function (9) can be simplified to a single-
cell power minimization problem with fixed inter-cell interference, which
fortunately has closed-form solution. Substituting the closed-form solution
for each cell into Problem (8), we can obtain an equivalent LP with a much
smaller size, i.e., far fewer variables.

###  III-A Distributed Power Control Algorithm

###### Theorem 1

For sum power minimization Problem (8) with objective function (9), power
$\boldsymbol{p}$ can be optimally solved with closed-form expression as

$\displaystyle p_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\sum_{n=j}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right){2^{\sum_{s=j}^{n-1}\frac{R_{ism}}{B}}}H_{inm}}$
(10)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad-\sum_{n=j+1}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}\\!-\\!1\\!\right){2^{\sum_{s=j+1}^{n-1}\frac{R_{ism}}{B}}}H_{inm}},$

while the optimal solution to $\boldsymbol{q}$ (for determining $H_{inm}$) is
determined by solving a smaller LP in (11) with much lower complexity.

$\displaystyle\mathop{\min}_{\boldsymbol{q}\geq\boldsymbol{0}}\;\quad$
$\displaystyle\sum_{i=1}^{I}\sum_{m=1}^{M}q_{im}$ (11a) s.t. $\displaystyle
q_{im}\geq\sum_{j=J_{i(m-1)}+1}^{J_{im}}\left(2^{\frac{R_{ijm}}{B}}-1\right){2^{\sum_{s=J_{i(m-1)}+1}^{j-1}\frac{R_{ism}}{B}}}$
$\displaystyle\qquad\\!\\!\\!\\!\\!\\!\\!\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$
$\displaystyle\qquad\\!\\!\\!\\!\\!\\!\\!\>\;\triangleq
f_{im}(\boldsymbol{q}),\quad\forall i\in\mathcal{I},m\in\mathcal{M}$ (11b)
$\displaystyle\sum_{m=1}^{M}q_{im}\leq Q_{i},\quad\forall i\in\mathcal{I}.$
(11c)

Proof: Please refer to Appendix A. $\Box$

Note that the concept of strong user reflected from (3), (5) and (6) is
helpful in obtaining the closed-form expression in (10). It can be verified
that the optimal solution to LP (11) with given
$[q_{1m},\cdots,q_{(i-1)m},q_{(i+1)m},\cdots,q_{Im}]^{T}$ can be directly
obtained by $q_{im}=f_{im}(\boldsymbol{q})$. To solve LP (11) for multi-cell
power minimization, we provide a distributed algorithm in Algorithm 1.

Algorithm 1 Distributed Power Control for Sum Power Minimization (DPC-SPM)

1:Initialize $q_{im}^{(0)}=Q_{i}/M$, $\forall
i\in\mathcal{N},m\in\mathcal{M}$. Set $t=1$, and maximal iteration number
$T_{\max}$.

2:for $i=1,2,\cdots,I$ do

3: for $m=1,2,\cdots,M$ do

4: Calculate $q_{im}^{(t)}=f_{im}(\boldsymbol{q}^{(t-1)})$;

5: end for

6:end for

7:If $t>T_{\max}$ or objective function (11a) converges, output
$\boldsymbol{q}^{*}=\boldsymbol{q}^{(t)}$, and terminate. Otherwise,
$\boldsymbol{q}^{(t)}=[q_{11}^{(t)},\cdots,q_{1M}^{(t)},\cdots,q_{IM}^{(t)}]^{T}$,
set $t=t+1$ and go to step 2.

###  III-B Convergence and Global Optimality

To show the convergence and global optimality of DPC-SPM algorithm, we recap
the standard interference function introduced in [35].222 Consider an
arbitrary interference function
$\boldsymbol{D}(\boldsymbol{q})=[D_{11}(\boldsymbol{q}),\cdots,D_{IM}(\boldsymbol{q})]^{T}$,
we say $\boldsymbol{D}(\boldsymbol{q})$ is a standard interference function if
for all $\boldsymbol{q}\geq\boldsymbol{0}$, the following properties are
satisfied. 1) Positivity: $\boldsymbol{D}(\boldsymbol{q})>\boldsymbol{0}$. 2)
Monotonicity: If $\boldsymbol{q}^{(1)}\geq\boldsymbol{q}^{(2)}$, then
$\boldsymbol{D}(\boldsymbol{q}^{(1)})\geq\boldsymbol{D}(\boldsymbol{q}^{(2)})$.
3) Scalability: For all $\lambda>1$,
$\lambda\boldsymbol{D}(\boldsymbol{q})>\boldsymbol{D}(\lambda\boldsymbol{q})$.
Letting $\boldsymbol{f}=[f_{11},\cdots,f_{IM}]^{T}$, we have the following
theorem.

###### Theorem 2

$\boldsymbol{f}(\boldsymbol{q})$ is a standard interference function.

Proof: Please refer to Appendix B. $\Box$

Based on Theorem 2, we have the following corollaries.

###### Corollary 1

If there exists $\boldsymbol{q}$ such that
$\boldsymbol{q}\geq\boldsymbol{f}(\boldsymbol{q})$, the iterative fixed-point
method $\boldsymbol{q}^{(t+1)}=\boldsymbol{f}(\boldsymbol{q}^{(t)})$ will
converge to the unique fixed point
$\boldsymbol{q}^{*}=\boldsymbol{f}(\boldsymbol{q}^{*})$ with any initial point
$\boldsymbol{q}^{(0)}$.

Proof: Please refer to [35, Theorem 2]. $\Box$

###### Corollary 2

If Problem (11) is feasible, the optimal $\boldsymbol{q}^{*}$ of Problem (11)
is component-wise minimum in the sense that any other feasible solution
$\boldsymbol{q}^{\prime}$ that meets constraints (11b)-(11c) and
$\boldsymbol{q}^{\prime}\geq\boldsymbol{0}$ satisfying
$\boldsymbol{q}^{\prime}\geq\boldsymbol{q}^{*}$.

###### Corollary 3

Problem (11) is feasible, if and only if there exists
$\boldsymbol{q}^{*}=\boldsymbol{f}(\boldsymbol{q}^{*})$ and
$\sum_{m=1}^{M}q_{im}^{*}\leq Q_{i},\forall i\in\mathcal{I}$. If Problem (11)
is feasible, the optimal solution $\boldsymbol{q}^{*}$ to Problem (11) is
unique with satisfying
$\boldsymbol{q}^{*}=\boldsymbol{f}(\boldsymbol{q}^{*})$.

Since Corollary 2 and 3 can be easily proved by using the same method in [35,
Theorem 2], the proofs are omitted. From Corollary 1 to Corollary 3, the
convergence and global optimality of DPC-SPM algorithm can be verified.

###  III-C Further Discussion

Considering the special case where users in the same cell on the same
subchannel are with equal rate demands, i.e.,
$R_{i(J_{i(m-1)}+1)}=\cdots=R_{iJ_{im}}=R_{im}$, we show that user with poor
channel should be allocated with more power than user with better channel in
the same cell, i.e., $p_{i(J_{i(m-1)}+1)m}>\cdots>p_{iJ_{im}m}$. From (10),
for all $J_{i(m-1)}+2\leq j\leq J_{im}$, it is verified that:

$\displaystyle
p_{i(j-1)m}-p_{ijm}=\sum_{n=j-1}^{J_{im}}\left(2^{\frac{R_{im}}{B}}-1\right)2^{\frac{(n-j+1)R_{im}}{B}}H_{inm}$
$\displaystyle\quad+\sum_{n=j+1}^{J_{im}}\left(2^{\frac{R_{im}}{B}}-1\right)2^{\frac{(n-j-1)R_{im}}{B}}H_{inm}$
$\displaystyle\quad-2\sum_{n=j}^{J_{im}}\left(2^{\frac{R_{im}}{B}}-1\right)2^{\frac{(n-j)R_{im}}{B}}H_{inm}$
$\displaystyle=\\!\sum_{n=j}^{J_{im}}\left(2^{\frac{R_{im}}{B}}\\!-\\!1\right)2^{\frac{(n-j+1)R_{im}}{B}}H_{inm}\\!+\\!\left(2^{\frac{R_{im}}{B}}\\!-\\!1\right)H_{i(j-1)m}$
$\displaystyle\quad\\!-\\!\\!\left(\\!2^{\frac{R_{im}}{B}}\\!-\\!1\\!\right)\\!\\!2^{\frac{-R_{im}}{B}}\\!H_{ijm}\\!+\\!\sum_{n=j}^{J_{im}}\\!\left(\\!2^{\frac{R_{im}}{B}}\\!-\\!1\\!\right)\\!\\!2^{\frac{(n-j-1)R_{im}}{B}}\\!H_{inm}$
$\displaystyle\quad-2\sum_{n=j}^{J_{im}}\left(2^{\frac{R_{im}}{B}}-1\right)2^{\frac{(n-j)R_{im}}{B}}H_{inm}$
$\displaystyle>\sum_{n=j}^{J_{im}}{\left(2^{\frac{R_{im}}{B}}-1\right)2^{\frac{(n-j)R_{im}}{B}}}\left(2^{\frac{R_{im}}{B}}+2^{\frac{-R_{im}}{B}}-2\right)H_{inm}$
$\displaystyle>0,$

where the first inequality follows from

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{i(j-1)m}=\max_{l\in\\{j-1,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad\geq\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}=H_{ijm}$
(12)

and $2^{\frac{R_{im}}{B}}>1$, and the second inequality holds because
$x+x^{-1}>2$ for $x>1$ and $0<x<1$.

To implement the NOMA scheme, each BS broadcasts the channel gain orders and
the cancellation schemes to all served users. The weakest user directly
decodes its own message. With the help of SIC, the strong user decodes the
messages in two stages. In the first stage, the strong user needs to decode
the messages of weak users served by the same BS on the same subchannel. In
the next stage, the strong user subtracts the decoded messages and then
decodes its own message. For OFDMA and broadcast channel (BC), each user only
needs to decode its own message, i.e., all users do not need to conduct SIC.
Thus, for NOMA, the BSs need to broadcast additional information to assist SIC
of the users and the receivers are complicated compared with OFDMA and BC.

##  IV Sum Rate Maximization for Multi-Cell Networks

In this section, we solve the sum rate maximization Problem (8) with

$V(\boldsymbol{p},\boldsymbol{q})=-\sum_{i=1}^{I}\sum_{m=1}^{M}\sum_{j=J_{i(m-1)}+1}^{J_{im}}r_{ijm}.$
(13)

For sum rate maximization Problem (8), we show that it can be decoupled into
two subproblems, i.e., power allocation problem in a single cell, and power
control problem in multiple cells. Given total transmission power
$\boldsymbol{q}$, inter-cell interference can be evaluated as constant value,
hence the sum rate maximization Problem (8) can be further decoupled into
multiple single-cell power allocation problems. The power allocation problem
in a single cell can be proved to be convex by checking the convexity of the
objective function. By solving the KKT conditions, the closed-form expression
of power allocation problem in a single cell can be obtained. Based on the
results of power allocation problem in a single cell, the original sum rate
maximization Problem (8) can be transformed into an equivalent problem with
fewer variables. A distributed algorithm is proposed to solve the transformed
problem.

With given total transmission power $\boldsymbol{q}$, sum rate maximization
Problem (8) with objective function (13) becomes the following problem,

$\displaystyle\mathop{\min}_{\boldsymbol{p}\geq\boldsymbol{0}}\;$
$\displaystyle-\sum_{i=1}^{I}\\!\sum_{m=1}^{M}\\!\sum_{j=J_{i(m-1)}+1}^{J_{im}}\\!B\log_{2}\left(\\!1\\!+\\!\frac{p_{ijm}}{\sum\limits_{n=j+1}^{J_{im}}\\!p_{inm}\\!+\\!H_{ijm}}\\!\right)$
(14a) s.t.
$\displaystyle\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}=q_{im},\quad\forall
i\in\mathcal{I},m\in\mathcal{M}$ (14b) $\displaystyle
p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right),$
$\displaystyle\qquad\forall
i\in\mathcal{I},m\in\mathcal{M},j\in\mathcal{J}_{im}.$ (14c)

Since Problem (14) has a decoupling objective function and decoupling
constraints, Problem (14) can be decoupled into $IM$ individual problems.
Having solved the $IM$ individual problems, we can substitute those $IM$
optimal solutions into Problem (8), which results in the power control problem
in multiple cells. Thus, the original Problem (8) can be decoupled into power
allocation problem in a single cell and power control problem in multiple
cells.

###  IV-A Power Allocation in A Single Cell

For Problem (14), power allocation problem for BS $i$ on subchannel $m$ is
formulated as

$\displaystyle\mathop{\min}_{\boldsymbol{p}_{im}\geq\boldsymbol{0}}\;$
$\displaystyle-\sum_{j=J_{i(m-1)}+1}^{J_{im}}B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}}\right)$
(15a) s.t. $\displaystyle\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}=q_{im}$ (15b)
$\displaystyle
p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right),$
$\displaystyle\qquad\forall j\in\mathcal{J}_{im},$ (15c)

where $\boldsymbol{p}_{im}=[p_{i(J_{i(m-1)}+1)m},\cdots,p_{iJ_{im}m}]^{T}$.

Since the convexity of objective function (15a) cannot be easily checked, the
sum rate maximization problem was regarded as a nonconvex multivariate
optimization problem via nonlinear programming approaches [20, 22]. To solve
Problem (15), we show that Problem (15) is indeed convex by checking the
convexity of objective function (15a).

###### Theorem 3

Problem (15) is a convex problem, and there exists at least one feasible
solution to Problem (15) if and only if the total transmission power of BS $i$
and rate constraints of users served by BS $i$ satisfy the following
condition:

$\sum_{j=J_{i(m-1)}+1}^{J_{im}}{\left(2^{\frac{R_{ijm}}{B}}-1\right)}{2^{\sum_{s=J_{i(m-1)}+1}^{j-1}\frac{R_{ism}}{B}}}H_{ijm}\leq
q_{im}.$ (16)

Proof: Please refer to Appendix C. $\Box$

Due to that Problem (15) is convex, we can obtain the globally optimal
solution by solving the KKT conditions [36, 37], and the following theorem is
hence provided.

###### Theorem 4

The optimal power allocation strategy for each BS is to allocate additional
power to the user with the best channel gain, while other users served by this
BS are allocated with minimal power to maintain their minimal rate demands.
More specifically, the globally optimal power allocation for Problem (15)
equals

$\displaystyle\\!\\!\\!\\!\\!p_{ijm}\\!\\!=\\!\\!\begin{cases}\frac{q_{im}\\!\left(2^{\frac{R_{ijm}}{B}}\\!-\\!1\right)}{2^{\sum_{n\\!=\\!J_{i(m-1)}\\!+\\!1}^{j}\\!\frac{R_{inm}}{B}}}\\!+\\!\frac{\left(\\!2^{\frac{R_{ijm}}{B}}\\!-\\!1\\!\right)\\!H_{ijm}}{2^{\frac{R_{ijm}}{B}}}-\\\
\sum_{n=J_{i(m-1)}+1}^{j-1}\\!\frac{\left(\\!2^{\frac{R_{ijm}}{B}}\\!-\\!1\\!\right)\\!\\!\left(\\!2^{\frac{R_{inm}}{B}}\\!-\\!1\\!\right)\\!H_{inm}}{2^{\sum_{l=n}^{j}\frac{R_{ilm}}{B}}},&\\!\\!\\!\\!\\!\\!\mbox{
if $j<J_{im}$}\\\
\frac{q_{im}}{2^{\sum_{n=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{inm}}{B}}}-\\\
\sum_{n=J_{i(m-1)}+1}^{J_{im}-1}\frac{\left(2^{\frac{R_{inm}}{B}}-1\right)H_{inm}}{2^{\sum_{l=n}^{J_{im}-1}\frac{R_{ilm}}{B}}},&\\!\\!\\!\\!\\!\\!\mbox{
if $j=J_{im}$}\end{cases},\\!\\!\\!\\!\\!$ (17)

and the corresponding optimal value of Problem (15) is

$\displaystyle-B\log_{2}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(1+\frac{q_{im}}{2^{\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{ijm}}{B}}H_{iJ_{im}m}}-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\right.$
(18)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)H_{ijm}}{2^{\sum_{l=j}^{J_{im}-1}\frac{R_{ilm}}{B}}H_{iJ_{im}m}}\right)-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}R_{ijm},$

where the first term is the negative rate of user $J_{im}$ with the largest
channel gain among users served by BS $i$ on subchannel $m$, and the second
term is the negative sum rate of all other users served by BS $i$ on
subchannel $m$.

Proof: Please refer to Appendix D. $\Box$

Similar to (10), the concept of strong user reflected from (3), (5) and (6) is
helpful in obtaining the closed-form expression in (17). Especially, the
decreasing order of $H_{ijm}$ in (III-C) obtained from (6) is crucial in
proving that Problem (15) is convex, which fortunately has closed-form
solution. According to (18), we can find that the optimal sum rate is mainly
determined by the rate of user $J_{im}$ with best channel gain among users
served by BS $i$ on subchannel $m$ and the optimal sum rate increases with the
total transmission power of BS $i$ on subchannel $m$ on a logarithmic scale.
Note that we obtain the similar conclusion as in [16]. The difference is that
the inter-cell interference is not considered in [16].

###  IV-B Power Control in Multiple Cells

Based on Theorem 3 and Theorem 4, we can readily transform sum rate
maximization Problem (8) with objective function (13) into an equivalent
problem.

###### Theorem 5

Sum rate maximization Problem (8) with objective function (13) is equivalent
to the following problem:

$\displaystyle\mathop{\min}_{\boldsymbol{q}\geq\boldsymbol{0},\boldsymbol{x}\geq\boldsymbol{0}}\;\quad$
$\displaystyle\\!-\\!\sum_{i=1}^{I}\sum_{m=1}^{M}B\left(\\!1\\!+\\!\frac{q_{im}}{2^{\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{ijm}}{B}}x_{iJ_{im}m}}\right.$
$\displaystyle\left.\quad-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)x_{ijm}}{2^{\sum_{l=j}^{J_{im}-1}\frac{R_{ilm}}{B}}x_{iJ_{im}m}}\\!\right)$
(19a) s.t. $\displaystyle
q_{im}\geq\sum_{j=J_{i(m-1)}+1}^{J_{im}}{\left(2^{\frac{R_{ijm}}{B}}-1\right)}$
$\displaystyle\qquad{2^{\sum_{s=J_{i(m-1)}+1}^{j-1}\frac{R_{ism}}{B}}}x_{ijm},\quad\forall
i\in\mathcal{I},m\in\mathcal{M}$ (19b) $\displaystyle
x_{ijm}\geq\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}},$
$\displaystyle\qquad\forall
i\in\mathcal{I},m\in\mathcal{M},j\in\mathcal{J}_{im},{l\in\\{j,\cdots,J_{im}\\}}$
(19c) $\displaystyle\sum_{m=1}^{M}q_{im}\leq Q_{i},\quad\forall
i\in\mathcal{I},$ (19d)

where $\boldsymbol{x}=[x_{111},\cdots,x_{1J_{1M}{M}},\cdots,x_{IJ_{IM}M}]^{T}$
and the power for each user is given by (17).

Note that there always exists $l\in\\{j,\cdots,J_{im}\\}$ such that
$x_{ijm}=\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$
for all $i\in\mathcal{I},m\in\mathcal{M},j\in\mathcal{J}_{im}$ as the
objective function increases with $x_{ijm}$, which shows that
$x_{ijm}=H_{ijm}$ defined in (6). Since Theorem 5 can be easily proved through
substituting (17) into Problem (8) with objective function (13), the proof of
Theorem 5 is omitted. It can be found that the objective function in Problem
(19) is still nonconvex and the constraints of Problem (19) are all linear
with respect to $\boldsymbol{q}$ and $\boldsymbol{x}$. In the following, one
distributed algorithm with low complexity is proposed. Denote
$\boldsymbol{q}_{i}=[q_{i1},\cdots,q_{iM}]^{T}$ and
$\boldsymbol{x}_{i}=[x_{i(J_{i0}+1)1},\cdots,x_{iJ_{iM}M}]^{T}$. With given
$\boldsymbol{q}_{-i}=[\boldsymbol{q}_{1}^{T},\cdots,\boldsymbol{q}_{i-1}^{T},\boldsymbol{q}_{i+1}^{T},\cdots,\boldsymbol{q}_{I}^{T}]^{T}$
and
$\boldsymbol{x}_{-i}=[\boldsymbol{x}_{1}^{T},\cdots,\boldsymbol{x}_{i-1}^{T},\boldsymbol{x}_{i+1}^{T},\cdots,\boldsymbol{x}_{I}^{T}]^{T}$,
Problem (19) becomes the following optimization problem for BS $i$,

$\displaystyle\mathop{\min}_{\boldsymbol{q}_{i}\geq\boldsymbol{0},\boldsymbol{x}_{i}\geq\boldsymbol{0}}\;\quad$
$\displaystyle-\sum_{m=1}^{M}B\left(1+\frac{q_{im}}{2^{\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{ijm}}{B}}x_{iJ_{im}m}}\right.$
$\displaystyle\left.\quad-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)x_{ijm}}{2^{\sum_{l=j}^{J_{im}-1}\frac{R_{ilm}}{B}}x_{iJ_{im}m}}\right)$
(20a) s.t. $\displaystyle
q_{im}\geq\sum_{j=J_{i(m-1)}+1}^{J_{im}}{\left(2^{\frac{R_{ijm}}{B}}-1\right)}$
$\displaystyle\qquad{2^{\sum_{s=J_{i(m-1)}+1}^{j-1}\frac{R_{ism}}{B}}}x_{ijm},\quad
m\in\mathcal{M}$ (20b) $\displaystyle x_{ijm}\geq H_{ijm},\quad\forall
m\in\mathcal{M},j\in\mathcal{J}_{im}$ (20c) $\displaystyle q_{im}\leq
Q_{im},\quad\forall m\in\mathcal{M}$ (20d)
$\displaystyle\sum_{m=1}^{M}q_{im}\leq Q_{i},$ (20e)

where $H_{ijm}$ is defined in (6), and constraints (20d) follow from (19c)
with

$\displaystyle Q_{im}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\min_{n\in\mathcal{N}\setminus\\{i\\},j\in\mathcal{J}_{nm},l\in\\{j,\cdots,J_{nm}\\}}$
(21)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{|h_{nlm}|^{2}x_{njm}-\sum_{k\in\mathcal{\setminus}\\{i,n\\}}q_{km}|h_{klm}|^{2}-\sigma^{2}}{|h_{ilm}|^{2}}.$

The representation of (20a) is similar to the DC problem representation [38,
23, 24]. Thus, objective function (20a) can be rewritten as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\!\sum_{m=1}^{M}\\!B\log_{2}\\!\left(\\!x_{iJ_{im}m}\\!+\\!\frac{q_{im}}{2^{\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{ijm}}{B}}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)x_{ijm}}{2^{\sum_{l=j}^{J_{im}-1}\frac{R_{ilm}}{B}}}\right)+\sum_{m=1}^{M}B\log_{2}(x_{iJ_{im}m}).$

Letting

$\displaystyle F_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\sum_{m=1}^{M}B\log_{2}\left(x_{iJ_{im}m}+\frac{q_{im}}{2^{\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{R_{ijm}}{B}}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\sum_{j=J_{i(m-1)}+1}^{J_{im}-1}\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)x_{ijm}}{2^{\sum_{l=j}^{J_{im}-1}\frac{R_{ilm}}{B}}}\\!\right),$

and

$G_{i}(\boldsymbol{x}_{i})=-\sum_{m=1}^{M}B\log_{2}(x_{iJ_{im}m}),$

Problem (20) can be written as

$\displaystyle\mathop{\min}_{\boldsymbol{q}_{i}\geq\boldsymbol{0},\boldsymbol{x}_{i}\geq\boldsymbol{0}}\;\quad$
$\displaystyle
K_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})=F_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})-G_{i}(\boldsymbol{x}_{i})$
(22a) s.t.
$\displaystyle(\ref{ratemaxPC1_11}b),(\ref{ratemaxPC1_11}c),(\ref{ratemaxPC1_11}d),(\ref{ratemaxPC1_11}e).$
(22b)

Define function $v(y,z)=-\log_{2}(y-z)$. Since

$\displaystyle\bigtriangledown^{2}v$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\begin{pmatrix}\frac{\partial^{2}v}{\partial
y^{2}}&\frac{\partial^{2}v}{\partial y\partial z}\\\
\frac{\partial^{2}v}{\partial y\partial z}&\frac{\partial^{2}v}{\partial
z^{2}}\end{pmatrix}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{(\ln
2)(y-z)^{2}}\begin{pmatrix}1&-1\\\ -1&1\end{pmatrix}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{(\ln
2)(y-z)^{2}}(1,-1)^{T}(1,-1)\succeq\boldsymbol{0},$

both $F_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})$ and
$G_{i}(\boldsymbol{x}_{i})$ can be proved convex according to the nonnegative
weighted sums operation and composition operation with an affine mapping that
preserve convexity [36, Page 79]. Therefore, the DC programming approach can
be applied to realize multi-cell power control [38]. From [38], nonconvex
Problem (22) can be solved suboptimally by converting a nonconvex problem to
convex subproblems with replacing the term $-G_{i}(\boldsymbol{x}_{i})$ in the
objective function (22a) with its convex majorant
$-G_{i}(\tilde{\boldsymbol{x}}_{i})-\triangledown
G_{i}^{T}(\tilde{\boldsymbol{x}}_{i})(\boldsymbol{x}_{i}-\tilde{\boldsymbol{x}}_{i})$.
$\triangledown G_{i}(\tilde{\boldsymbol{x}}_{i})$ is the gradient of
$G_{i}(\boldsymbol{x}_{i})$ at point $\tilde{\boldsymbol{x}}_{i}$, and
$\triangledown G_{i}(\tilde{\boldsymbol{x}}_{i})$ can be calculated by

$\displaystyle\triangledown G_{i}(\tilde{\boldsymbol{x}}_{i})=\frac{B}{\ln
2}\left[0,\cdots,-\frac{1}{\tilde{x}_{iJ_{i1}1}},\cdots,-\frac{1}{\tilde{x}_{iJ_{iM}M}}\right]^{T}.$
(23)

The convex optimization problem in (25) can be solved by the interior point
method [36, 37].

In the following, we provide a distributed power control algorithm to solve
sum rate maximization Problem (19) in Algorithm 2.

Algorithm 2 Distributed Power Control for Sum Rate Maximization (DPC-SRM)

1:Initialize $\boldsymbol{q}^{(0)}$, $\boldsymbol{x}^{(0)}$, the iteration
number $t=0$, and the tolerance $\epsilon$.

2:for $i=1,2,\cdots,I$ do

3: $\tilde{\boldsymbol{x}}_{i}=\boldsymbol{x}_{i}^{(t)}$.

4: repeat

5: Define a convex approximate function of
$\boldsymbol{H}_{i}(\boldsymbol{x}_{i})$ at point $\tilde{\boldsymbol{x}}_{i}$
as
$\tilde{K}_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})=F_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})-G_{i}(\tilde{\boldsymbol{x}}_{i})-\triangledown
G^{T}(\tilde{\boldsymbol{x}}_{i})(\boldsymbol{x}_{i}-\tilde{\boldsymbol{x}}_{i}).$
(24)

6: Solve the convex optimization problem
$(\tilde{\boldsymbol{q}}_{i},\tilde{\boldsymbol{x}}_{i})=\left\\{\begin{array}[]{ll}\\!\\!\arg\mathop{\min}\limits_{\boldsymbol{q}_{i}\geq\boldsymbol{0},\boldsymbol{x}_{i}\geq\boldsymbol{0}}&\\!\\!\tilde{K}_{i}(\boldsymbol{q}_{i},\boldsymbol{x}_{i})\\\
\quad\textrm{s.t.}&\\!\\!(\ref{ratemaxPC1_11}b),(\ref{ratemaxPC1_11}c),(\ref{ratemaxPC1_11}d),(\ref{ratemaxPC1_11}e).\end{array}\right.$
(25)

7: until the objective function (22a) converges

8: $q_{i}^{(t+1)}=\tilde{\boldsymbol{q}}_{i}$,
$\boldsymbol{x}_{i}^{(t+t)}=\tilde{\boldsymbol{x}}_{i}$.

9:end for

10:If
$|\sum_{i=1}^{I}K_{i}(\boldsymbol{q}_{i}^{(t)},\boldsymbol{x}_{i}^{(t)})-\sum_{i=1}^{I}K_{i}(\boldsymbol{q}_{i}^{(t+1)},\boldsymbol{x}_{i}^{(t+1)})|\leq\epsilon$,
terminate. Otherwise, set $t=t+1$ and go to step 2.

###  IV-C Convergence and Complexity Analysis

According to [38], the DC programming steps (i.e., Step 4 to Step 7) in DPC-
SRM algorithm always converge to a suboptimal stationary point, i.e.,
$K_{i}(\boldsymbol{q}_{i}^{(t)},\boldsymbol{x}_{i}^{(t)})\geq
K_{i}(\boldsymbol{q}_{i}^{(t+1)},\boldsymbol{x}_{i}^{(t+1)})$, which shows
that the objective value (19a) is nonincreasing when sequence
($\boldsymbol{q},\boldsymbol{x}$) is updated. Furthermore, the objective value
(19a) can be found upper-bounded. As a result, DPC-SRM algorithm must
converge.

For DPC-SRM algorithm, the major complexity in each iteration lies in solving
convex optimization problem (25). Assume that the number of users on every
subchannel in each cell is $N$. Considering that the dimension of variables in
problem (25) is $M(N+1)$, the complexity of solving problem (25) by using the
standard interior point method is $\mathcal{O}(M^{3}N^{3})$ [36, Page 487,
569]. Hence, the total complexity of DPC-SRM is
$\mathcal{O}(L_{\text{SR}}L_{\text{DC}}IM^{3}N^{3})$, where $L_{\text{SR}}$
and $L_{\text{DC}}$ denote the total number of iterations of the out layer of
DPC-SR algorithm and the DC programming, respectively.

###  IV-D Implementation Method

To implement the proposed DPC-SRM algorithm, each BS $i$ needs to update power
vector $\boldsymbol{q}_{i}$ and auxiliary vector $\boldsymbol{x}_{i}$ by
solving Problem (20). Solving Problem (20) involves $R_{ijm}$, $H_{ijm}$,
$Q_{im}$, $\forall m\in\mathcal{M}$, $j\in\mathcal{J}_{im}$. Assume that the
minimal rate demands for users served by BS $i$ are available at BS $i$. Since
$H_{ijm}=\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$,
numerator $|h_{ilm}|$ is the channel gain between BS $i$ and its served user
$l$ on subchannel $m$, which can be estimated by channel reciprocity. Beside,
$\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}$ is the
total interference power of user $l$ served by BS $i$ on subchannel $m$, and
the value of interference power can be measured by user $l$. Due to the fact
that

$\displaystyle Q_{im}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\min_{n\in\mathcal{N}\setminus\\{i\\},j\in\mathcal{J}_{nm},l\in\\{j,\cdots,J_{nm}\\}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{|h_{nlm}|^{2}x_{njm}-\sum_{k\in\mathcal{\setminus}\\{n\\}}q_{km}|h_{klm}|^{2}-\sigma^{2}+q_{im}|h_{ilm}|^{2}}{|h_{ilm}|^{2}},$

numerator $|h_{ilm}|$ is the cross channel gain between BS $i$ and user $l$
served by BS $n$ $(n\neq i)$ on subchannel $m$, which can be estimated at BS
$i$ for receiving the pilot from user $l$ according to channel reciprocity.
The denominator of calculating $Q_{im}$ contains three parts. In the first
part, i.e., $|h_{nlm}|^{2}x_{njm}$, $|h_{nlm}|$ is the channel gain between BS
$n$ and its served user $l$ on subchannel $m$ and $x_{njm}$ is the strategy of
BS $n$. Hence, $|h_{nlm}|^{2}x_{njm}$ is known at BS $n$. In the second part,
$\sum_{k\in\mathcal{\setminus}\\{n\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}$ is the
total interference power of user $l$ served by BS $n$ on subchannel $m$. In
the third part $q_{im}|h_{ilm}|^{2}$, $q_{im}$ is the transmission power of BS
$i$ on subchannel $m$. To calculate $Q_{im}$, user $l$ served by BS $n$ sends
its overall received interference and noise to BS $n$. Then, having obtained
the messages from its served users, BS $n$ calculates
$|h_{nlm}|^{2}x_{njm}-\sum_{k\in\mathcal{\setminus}\\{n\\}}q_{km}|h_{klm}|^{2}-\sigma^{2}$
and sends these calculated values to BS $i$, which helps BS $i$ calculate
$Q_{im}$. As a result, BS $i$ calculates the optimal $\boldsymbol{q}_{i}$ and
$\boldsymbol{x}_{i}$ by solving Problem (20). Each BS updates its power vector
and auxiliary vector until the total interference power of each user
converges.

##  V Extension to MIMO-NOMA Systems

Consider a downlink multi-cell network with NOMA, where there are $I$ BSs with
$M$ antennas each and $J$ users with $N$ antennas each. For MIMO-NOMA,
superposition coding is employed at each BS. The transmit signal at BS $i$ is
given by [10]

${\boldsymbol{s}}_{i}=\left[\begin{array}[]{c}\sqrt{p_{i(J_{i-1}+1)1}}s_{i(J_{i-1}+1)1}+\cdots+\sqrt{p_{iJ_{i}1}}s_{iJ_{i}1}\\\
\vdots\\\
\sqrt{p_{i(J_{i-1}+1)M}}s_{i(J_{i-1}+1)M}+\cdots+\sqrt{p_{iJ_{i}M}}s_{iJ_{i}M}\\\
\end{array}\right],$ (26)

where $s_{ijm}$ denotes the information bearing signal to be transmitted to
user $j$ in cluster $m$ served by BS $i$, and $p_{ijm}$ is the NOMA power
allocation coefficient. Obviously, $p_{ijm}>0$ shows that user $j$ is assigned
in cluster $m$. It is assumed that all users are already clustered and each
user is only assigned to one cluster. Let $\mathcal{M}=\\{1,2,\cdots,M\\}$ be
the set of clusters. The set of users in cluster $m$ is denoted by
$\mathcal{J}_{im}=\\{J_{i(m-1)}+1,J_{i(m-1)}+2,\cdots,J_{im}\\}$, where
$J_{i0}=J_{i-1}+1$, $J_{iM}=J_{i}$, $J_{im}=\sum_{l=1}^{m}|\mathcal{J}_{il}|$.
The observation at user $j\in\mathcal{J}_{im}$ in cluster $m$ is given by

$\displaystyle\\!\\!\\!\\!\\!\\!\boldsymbol{y}_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\boldsymbol{H}_{ij}\boldsymbol{s}_{i}+\sum_{k\in\mathcal{I}\setminus\\{i\\}}\boldsymbol{H}_{kj}\boldsymbol{s}_{k}+\boldsymbol{n}_{jm}$
(27)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\boldsymbol{h}_{ijm}\sqrt{p_{ijm}}s_{ijm}+\underbrace{\sum_{l\in\mathcal{J}_{im}\setminus\\{j\\}}\boldsymbol{h}_{ijm}\sqrt{p_{ilm}}s_{ilm}}_{\text{intra-
cell intra-cluster interference}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad+\underbrace{\sum_{t\in\mathcal{M}\setminus\\{m\\}}\sum_{l=J_{i(t-1)}+1}^{J_{it}}\boldsymbol{h}_{ijt}\sqrt{p_{ilt}}s_{ilt}}_{\text{intra-
cell inter-cluster interference}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad+\underbrace{\sum_{k\in\mathcal{I}\setminus\\{i\\}}\sum_{t=1}^{M}\sum_{n=J_{k(t-1)}+1}^{J_{kt}}\boldsymbol{h}_{kjt}\sqrt{p_{knt}}s_{knt}}_{\text{inter-
cell interference}}+\boldsymbol{n}_{jm},$

where $\boldsymbol{H}_{ij}$ is the channel gain between BS $i$ and user $j$,
$\boldsymbol{h}_{ijm}$ is the channel gain between BS $i$ and user $j$ in
cluster $m$, and $\boldsymbol{n}_{jm}$ represents the additive zero-mean
Gaussian noise vector with variance $\sigma^{2}\boldsymbol{I}$.

Denote by $\boldsymbol{v}_{ijm}$ the detection vector used by user $j$ served
by BS $i$ in cluster $m$. After applying this vector into (27), the signal
model can be rewritten as follows:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\boldsymbol{v}_{ijm}^{H}\boldsymbol{y}_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{ijm}\sqrt{p_{ijm}}s_{ijm}+\boldsymbol{v}_{ijm}^{H}\boldsymbol{n}_{jm}$
(28)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\underbrace{\sum_{l\in\mathcal{J}_{im}\setminus\\{j\\}}\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{ijm}\sqrt{p_{ilm}}s_{ilm}}_{\text{intra-
cell intra-cluster interference}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\underbrace{\sum_{t\in\mathcal{M}\setminus\\{m\\}}\sum_{l=J_{i(t-1)}+1}^{J_{it}}\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{ijt}\sqrt{p_{ilt}}s_{ilt}}_{\text{intra-
cell inter-cluster interference}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\underbrace{\sum_{k\in\mathcal{I}\setminus\\{i\\}}\sum_{t=1}^{M}\sum_{n=J_{k(t-1)}+1}^{J_{kt}}\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{kjt}\sqrt{p_{knt}}s_{knt}}_{\text{inter-
cell interference}}.$

The channel conditions are crucial to the implementation of NOMA. As in [10],
it is assumed that the channel gains are sorted as follows:

$|\boldsymbol{v}_{i(J_{i(m-1)}+1)m}^{H}\boldsymbol{h}_{i(J_{i(m-1)}+1)m}|^{2}\leq\cdots\leq|\boldsymbol{v}_{iJ_{im}m}^{H}\boldsymbol{h}_{iJ_{im}m}|^{2}.$
(29)

With removing intra-cell inter-cluster interference, the detection vector can
be obtained as in (13) in [10]. In order to remove intra-cell inter-cluster
interference, the number of users’ antennas is larger than or equal to that of
the BS [10]. With detection vector $\boldsymbol{v}_{ijm}$ fixed and without
intra-cell inter-cluster interference, the achievable rate of user
$j\in\mathcal{J}_{im}$ to detect the message of user
$l\in\\{J_{i(m-1)}+1,\cdots,j\\}$ in cluster $m$ is

$r_{ijlm}=B\log_{2}\left(1+\frac{|\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{ijm}|^{2}p_{ilm}}{|\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{ijm}|^{2}\sum_{n=l+1}^{J_{im}}p_{inm}+Y_{ijlm}}\right),$
(30)

where
$Y_{ijlm}=\sum_{t=1}^{M}\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{kt}|\boldsymbol{v}_{ijm}^{H}\boldsymbol{h}_{kjt}|^{2}+\sigma^{2}|\boldsymbol{v}_{ijm}|^{2}$,
and $q_{km}=\sum_{j=J_{k(m-1)}+1}^{J_{km}}p_{kjm}$ is the total transmission
power of BS $k$ in cluster $m$. According to (30), strong user $j$ with high
effective channel gain needs to decode the message of weak user $l\leq j$ with
low effective channel gain. To ensure successful SIC, the achievable rate of
user $j\in\mathcal{J}_{im}$ in cluster $m$ can be given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!r_{ijm}=\min_{l\in\\{j,\cdots,J_{im}\\}}r_{iljm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!=\min_{l\in\\{j,\cdots,J_{im}\\}}B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+\frac{Y_{iljm}}{|\boldsymbol{v}_{ilm}^{H}\boldsymbol{h}_{ilm}|^{2}}}\\!\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!=B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}}\right),$
(31)

where

$H_{ijm}\\!=\\!\\!\max_{l\in\\{j,\cdots,J_{im}\\}}\\!\\!\frac{\sum_{t=1}^{M}\\!\sum_{k\in\mathcal{I}\setminus\\{i\\}}\\!\\!q_{kt}|\boldsymbol{v}_{ilm}^{H}\boldsymbol{h}_{klt}|^{2}\\!+\\!\sigma^{2}|\boldsymbol{v}_{ilm}|^{2}}{|\boldsymbol{v}_{ilm}^{H}\boldsymbol{h}_{ilm}|^{2}}.$
(32)

Since the rate formulations (31) and (32) respectively have similar structures
as equations (5) and (6), the power control methods in Section III and Section
IV can be applied to MIMO-NOMA systems.

##  VI Numerical Results

In this section, numerical results are presented to evaluate the performance
of the proposed schemes for multi-cell networks with NOMA. In the simulations,
we consider a three-site 3GPP LTE network with an inter-site distance of 800
m, adopting a wrap-around technique [39]. The simulated system operates at 2
GHz, the number of subchannels is $M=10$ and the bandwidth of each subchannel
is $B=1$ MHz. The three-sector antenna pattern is used for each site and the
gain for the three-sector, of which 3dB beamwidth in degrees is 70 degrees, is
14dBi [40]. In the propagation model, we use the large-scale path loss
$L(d)=128.1+37.6\log(d)$, $d$ is in km, and the standard deviation of shadow
fading is set as $8$ dB [40].

The total number of BSs $I$ is set as 15. To reduce the receiver complexity
and error propagation due to SIC, it is reasonable for each subchannel to be
multiplexed by two or three users [41]. In the simulations, the number of
users in each cell is set as 20 and two users are paired on each subchannel.
We set the tolerance in Algorithm 1 and Algorithm 2 $\epsilon=0.001$ and noise
power $\sigma^{2}=-114$ dBm. We assume equal rate demands for all users (i.e.,
$R_{ij}=R=0.3$ Mbps, $\forall i\in\mathcal{I},j\in\mathcal{J}_{i}$) and equal
maximum transmission power for all BSs (i.e., $Q_{i}=Q$, $\forall
i\in\mathcal{I}$).

We compare the NOMA system with two systems: the OFDMA system, where multiple
users on the same subchannel are allocated with orthogonal time fractions, and
the BC system, where multiple users on the same subchannel suffer from both
intra-cell (without performing SIC) and inter-cell interference. For sum power
minimization, we compare the proposed power minimization scheme for NOMA
systems (labeled as ‘NOMA-PM’) by using Algorithm 1 with sum power
minimization problem for OFDMA systems (labeled as ‘OFDMA-SP’), which can be
optimally solved by using the optimal power vector algorithm in [39, Section
V], and sum power minimization problem for BC systems (labeled as ‘BC-SP’),
which can be optimally solved by using the simplex method [36]. For sum rate
maximization, we compare the proposed sum rate maximization scheme for NOMA
systems (labeled as ‘NOMA-RM’) through using Algorithm 2, with sum rate
maximization problem for OFDMA systems (labeled as ‘OFDMA-SR’), which can be
suboptimally solved by using the distributed power control and time allocation
algorithm in [42, Section V], and sum rate maximization problem for BC systems
(labeled as ‘BC-RM’), which can be suboptimally solved by using the weighted
mean-square error approach [43, Section II].

Figure 1: This figure illustrates pair selection in a typical cell of 8 users.

We study the influence of user pairing by considering three different user-
pairing methods [44] and [45]. Fig. 1 illustrates pair selection in a typical
cell of 8 users, where users are sorted in increasing order of channel gains,
i.e., user 8 enjoys the strongest channel gain while user 1 is of the weakest
channel gain. In strong-strong (SS) pair selection, the user with the
strongest channel condition is paired with the one with the second strongest,
and so on. In strong-weak (SW) pair selection, the user with the strongest
channel condition is paired with the user with the weakest, and the user with
the second strongest is paired with one with the second weakest, and so on. In
strong-middle (SM) pair selection, the user with the strongest channel
condition is paired with the user with the middle strongest user, i.e., user 8
is paired with user 4 in Fig. 1(c), and so on.

Figure 2: Sum power of the system for different user-pairing methods. Figure
3: Sum rate of the system for different user-pairing methods.

In Fig. 2 and Fig. 3, we show the sum power and sum rate of the system for
different user-pairing methods, respectively. From Fig. 2, it is observed that
SW outperforms the other two methods in terms of power consumption for NOMA-
PM. Besides, we can also find that SW achieves the best sum rate among three
user-pairing methods for NOMA-RM according to Fig. 3. Combing Fig. 2 and Fig.
3, we can conclude that it tends to pair users with distinctive gains for both
sum power minimization and sum rate maximization, which coincides with
previous findings in [44]. Due to the superiority of SW, the following
simulations are based on SW pair selection.

Figure 4: Convergence behavior of NOMA-PM. Figure 5: Convergence behavior of
NOMA-RM.

The convergence behaviors of NOMA-PM and NOMA-RM are illustrated in Fig. 4 and
Fig. 5, respectively. It can be seen that both NOMA-PM and NOMA-RM converge
rapidly, which makes our proposed algorithms suitable for practical
applications. From Fig. 5, the sum rate of NOMA-RM monotonically increases,
which confirms the convergence analysis in Section IV-C.

Figure 6: Performance comparison of NOMA-RM and NOMA-RM-EXH. Figure 7: Sum
power of the system versus the maximal transmission power.

We try multiple starting points in the simulations to exhaustively obtain a
near globally optimal solution. We test 20 randomly generated channels shown
in Fig. 6, where NOMA-RM-EXH refers to the NOMA-RM algorithm with 1000
starting points for each channel realization. It can be seen that the sum rate
of NOMA-RM is almost the same as that of NOMA-RM-EXH, which indicates that the
proposed NOMA-RM approaches the near globally optimal solution.

Fig. 7 shows the sum power versus the maximal transmission power under various
algorithms. It is observed that the sum power of NOMA/OFDMA/BC-RM increases
with the transmission power constraint. This is because increasing the overall
transmission power is always beneficial in enhancing the sum rate of the
system. It is also found that the sum power keeps the same for NOMA/OFDMA/BC-
PM. This is due to that the maximal transmission power for each BS is set as
the same and the sum power does not change value with the maximal transmission
power for power minimization. From Fig. 7, the NOMA-PM is better than
OFDMA/BC-PM in terms of the sum power consumption. The reason is that NOMA
applies SIC to utilize intra-cell interference and each user can occupy the
total available bandwidth, which results in lower sum power than OFDMA and BC.

Figure 8: Sum rate of the system versus the maximal transmission power. Figure
9: Energy efficiency of the system versus the maximal transmission power.

We illustrate the sum rate and energy efficiency (the ratio of sum rate and
sum power) versus the maximal transmission power under various algorithms in
Fig. 8 and Fig. 9, respectively. It is seen from Fig. 8 that the sum rate is
the same for NOMA/OFDMA/BC-PM. Due to power minimization, each user is set as
satisfying the minimal rate demand. Therefore, the sum rate of NOMA/OFDMA/BC-
PM keeps a constant, i.e., $2IMR$, as shown in Fig. 8. From Fig. 8, it is
observed that NOMA-RM outperforms OFDMA/BC-RM in terms of sum rate. One reason
is that each user in NOMA networks can be allocated with higher fraction of
bandwidth than in OFDMA networks, where the bandwidth is orthogonally
distributed to different users in the same cell. The other reason is that NOMA
can efficiently utilize intra-cell interference by using SIC, which results in
higher rate than that of BC. From Fig. 9, it is interesting to observe that
NOMA-PM achieves the best energy efficiency among all algorithms. It is also
found that the energy efficiency of NOMA/OFDMA/BC-RM monotonically decreases
with the maximal transmission power. This is because sum rate maximization
algorithms tend to transmit with large power from Fig. 7, which results in
large intra/inter-interference and low energy efficiency. NOMA needs to
broadcast additional information about the decoding orders and the receivers
at the users are complicated compared with OFDMA/BC-PM. Thus, we can conclude
that NOMA achieves some performance gains at the cost of some additional
information broadcasting of the BS and extra computations of the users from
Fig. 7 and Fig. 8.

##  VII Conclusion

In this paper, we aim at sum power minimization and sum rate maximization
through power control for multi-cell networks with NOMA. Both sum power
minimization and sum rate maximization problems can be transformed into
correspondingly equivalent problems with smaller variables. For sum power
minimization, users with poor channel gains tend to be allocated with more
power. Sum rate maximization problem can be decoupled into two subproblems,
i.e., power allocation problem in a single cell and power control problem in
multiple cells. The power allocation problem in a single cell is proved to be
convex and its globally optimal solution can be obtained in the closed-form
expression. Based on the optimal solution to power allocation problem in a
single cell, only user in each cell with the best channel gain deserves
additional power from its served BS to maximize sum rate in multi-cell
networks with NOMA. Through simulation results, it tends to pair users with
distinctive channel gains for both sum power minimization and sum rate
maximization. It is shown that the proposed power control methods can be
applied to MIMO systems. It is also verified that NOMA outperforms OFDMA and
BC in terms of sum power minimization and sum rate maximization at the cost of
some additional information broadcasting of the BSs and computations of the
users. The users’ mobility issue for multi-cell NOMA systems is left for our
future work.

##  Appendix A Proof of Theorem 1

To prove Theorem 1, we find that the objective function (9) is only a function
of $\boldsymbol{q}$ and variable $\boldsymbol{p}$ only exists in constraints
(8b), (8c) and $\boldsymbol{p}\geq\boldsymbol{0}$. Hence, $\boldsymbol{p}$ can
be viewed as an intermediate variable. With this observation, sum power
minimization Problem (8) with objective function (9) can be simplified by
removing $\boldsymbol{p}$ without loss of optimality. Specifically, the
constraints (8b), (8c) and $\boldsymbol{p}\geq\boldsymbol{0}$ about variable
$\boldsymbol{p}$ can be equivalently transformed into constraints about
$\boldsymbol{q}$.

Given total transmission power of other BSs, inter-cell interference is fixed,
hence Problem (8) with objective function (9) can be readily simplified into a
single-cell power allocation problem. For BS $i$, the power allocation problem
with given
$\boldsymbol{q}_{-i}=[q_{11},\cdots,q_{(i-1)M},q_{(i+1)1},\cdots,q_{IM}]^{T}$
can be formulated as

$\displaystyle\mathop{\min}_{\boldsymbol{p}_{i}\geq\boldsymbol{0},\boldsymbol{q}_{i}\geq\boldsymbol{0}}\;\quad$
$\displaystyle\sum_{m=1}^{M}q_{im}$ (A.1a) s.t.
$\displaystyle\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}=q_{im},\quad\forall
m\in\mathcal{M}$ (A.1b) $\displaystyle
p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right),$
$\displaystyle\qquad\quad\forall m\in\mathcal{M},j\in\mathcal{J}_{im},$ (A.1c)

where $\boldsymbol{p}_{i}=[p_{i(J_{i0}+1)1},\cdots,p_{iJ_{iM}M}]^{T}$, and
$\boldsymbol{q}_{i}=[q_{i1},\cdots,q_{iM}]^{T}$.

Observing that both the objective function and constraints of Problem (A.1)
can be decoupled, Problem (A.1) can be further decoupled into multiple single-
subchannel problems. We consider the following optimization problem on
subchannel $m$:

$\displaystyle\mathop{\min}_{\boldsymbol{p}_{im}\geq\boldsymbol{0}}\;\quad$
$\displaystyle\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}$ (A.2a) s.t.
$\displaystyle
p_{ijm}\geq\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right),$
$\displaystyle\qquad\quad\forall j\in\mathcal{J}_{im},$ (A.2b)

where $\boldsymbol{p}_{im}=[p_{i(J_{i(m-1)}+1)m},\cdots,p_{iJ_{im}m}]^{T}$.
Combining (A.2b) and $\boldsymbol{p}_{im}\geq\boldsymbol{0}$, we find that
constraints (A.2b) hold with equality for any optimal solution to Problem
(A.2), as otherwise (A.2a) can be further improved, contradicting that the
solution is optimal. Setting constraints (A.2b) with equality, we obtain

$\displaystyle
2^{\frac{R_{ijm}}{B}}\sum_{n=j+1}^{J_{im}}p_{inm}+\left(2^{\frac{R_{ijm}}{B}}-1\right)H_{ijm}=\sum_{n=j}^{J_{im}}p_{inm},$
(A.3)

for all $j\in\mathcal{J}_{im}$. Define

$b_{ijm}=\sum_{n=j}^{J_{im}}p_{inm},\quad j\in\mathcal{J}_{im},$ (A.4)

which represents the summation of the transmission power from user $j$ to user
$J_{im}$. Based on (A.3) and (A.4), we can obtain

${b_{ijm}}={2^{\frac{R_{ijm}}{B}}}b_{i({j+1})m}+{\left(2^{\frac{R_{ijm}}{B}}-1\right)H_{ijm}},\quad\forall
j\in\mathcal{J}_{i}.$ (A.5)

Defining $b_{i(J_{im}+1)m}=0$ and
${2^{\sum_{s=J_{im}}^{J_{im}-1}\frac{R_{ism}}{B}}}=2^{0}$, we can rewrite
(A.5) as follows:

$b_{ijm}=\sum_{n=j}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right)}{2^{\sum_{s=j}^{n-1}\frac{R_{ism}}{B}}}H_{inm},\quad
j\in\mathcal{J}_{im}.$ (A.6)

According to (A.4), we know $p_{ijm}=b_{ijm}-b_{i(j+1)m},\forall
j\in\mathcal{J}_{im}$. Substituting (A.6) into $p_{ijm}=b_{ijm}-b_{i(j+1)m}$
yields

$\displaystyle p_{ijm}$
$\displaystyle=\sum_{n=j}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right)}{2^{\sum_{s=j}^{n-1}\frac{R_{ism}}{B}}}H_{inm}-$
(A.7)
$\displaystyle\sum_{n=j+1}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right)}{2^{\sum_{s=j+1}^{n-1}\frac{R_{ism}}{B}}}H_{inm},\quad\forall
j\in\mathcal{J}_{im},$

which is the optimal solution to Problem (A.2).

From (A.4) and (A.6), the optimal objective value of Problem (A.2) is

$\displaystyle q_{im}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\sum_{n=J_{i(m-1)}+1}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right)}{2^{\sum_{s=J_{i(m-1)}+1}^{n-1}\frac{R_{ism}}{B}}}H_{inm},$

which is minimal sum power of $\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}$
satisfying (A.2b) and $\boldsymbol{p}_{im}\geq\boldsymbol{0}$. Applying (A)
into Problem (8) with objective function (9) yields equivalent Problem (11),
where the inequality shown in (11b) is due to the fact that (A) is the minimal
value of $q_{im}$.

##  Appendix B Proof of Theorem 2

We prove each of the three properties required for standard function below.

Positivity: Since
$\max\limits_{l\in\\{j,\cdots,J_{i}\\}}\frac{\sigma^{2}}{|h_{ilm}|^{2}}>0$ for
$j\in\mathcal{J}_{im}$, we have $f_{im}(\boldsymbol{q})>0$ from (11b).

Monotonicity: Let total transmission power vector
$\boldsymbol{q}^{(1)}=[q_{11}^{(1)},\cdots,q_{IM}^{(1)}]^{T}$ and
$\boldsymbol{q}^{(2)}=[q_{11}^{(2)},\cdots,q_{IM}^{(2)}]^{T}$ be such that
$q_{im}^{(1)}\geq q_{im}^{(2)}$, $\forall i\in\mathcal{I},m\in\mathcal{M}$.
Then, we have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\max_{l\in\\{j,\cdots,J_{i}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}^{(1)}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\geq\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}^{(2)}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}.$
(B.1)

According to (B) and (11b), we have $f_{im}(\boldsymbol{q}^{(1)})\geq
f_{im}(\boldsymbol{q}^{(2)})$.

Scalability: Letting $\boldsymbol{q}\geq\boldsymbol{0}$ and $\lambda>1$, we
have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\max_{l\in\\{j,\cdots,J_{im}\\}}\lambda\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!>\max_{l\in\\{j,\cdots,J_{im}\\}}\frac{\sum_{k\in\mathcal{I}\setminus\\{i\\}}\lambda
q_{km}|h_{klm}|^{2}+\sigma^{2}}{|h_{ilm}|^{2}},$ (B.2)

where the inequality follows from that $\sigma^{2}>0$. Based on (B) and (11b),
we have $\lambda f_{im}(\boldsymbol{q})>f_{im}(\lambda\boldsymbol{q})$.

##  Appendix C Proof of Theorem 3

Since the constraints of Problem (15) are all linear, we only need to prove
that the objective function (15a) is convex. We first rewrite (15a) as

$\displaystyle R_{im}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!-B\log_{2}\left({\sum_{n=J_{i(m-1)}+1}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+1)m}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum_{j=J_{i(m-1)}+2}^{J_{im}}B\left(\log_{2}\left({\sum_{n=j}^{J_{im}}p_{inm}+H_{i(j-1)m}}\right)\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\log_{2}\left({\sum_{n=j}^{J_{im}}p_{inm}+H_{ijm}}\right)\right)+B\log_{2}(H_{iJ_{im}m}),$

where $R_{im}$ is negative sum rate of all users served by BS $i$ on
subchannel $m$. The second-order derivative of $R_{im}$ equals

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial^{2}R_{im}}{\partial
p_{ijm}^{2}}=\sum_{l=J_{i(m-1)}+2}^{j}\frac{1}{\ln
2}\left(\frac{B}{\left({\sum_{n=l}^{J_{im}}p_{inm}+H_{ijm}}\right)^{2}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\frac{B}{\left({\sum_{n=l}^{J_{im}}p_{inm}+H_{i(j-1)m}}\right)^{2}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{B}{(\ln
2)\left(\sum_{n=J_{i(m-1)}+1}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+1)m}\right)^{2}}$
(C.1)

for all $j=J_{i-1}+1,J_{i-1}+2,\cdots,J_{i}$, and

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial^{2}R}{\partial
p_{ijm}\partial p_{ilm}}=\sum_{l=J_{i(m-1)}+2}^{j}\frac{1}{\ln
2}\left(\frac{B}{\left({\sum_{n=l}^{J_{im}}p_{inm}+H_{ijm}}\right)^{2}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\frac{B}{\left({\sum_{n=l}^{J_{im}}p_{inm}+H_{i(j-1)m}}\right)^{2}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{B}{(\ln
2)\left(\sum_{n=J_{i(m-1)}+1}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+1)m}\right)^{2}}$
(C.2)

for all $J_{i-1}+\leq j<l\leq J_{i}$. Comparing (C.1) and (C.2), we find that
$\frac{\partial^{2}R_{im}}{\partial p_{ijm}\partial
p_{ilm}}=\frac{\partial^{2}R_{im}}{\partial p_{ijm}^{2}}$ for any
$J_{i(m-1)}+1\leq j\leq l\leq J_{im}$. Therefore, denoting
$a_{ijm}=\frac{\partial^{2}R_{im}}{\partial p_{ijm}^{2}}$ for notational
simplicity, the Hessian matrix $\boldsymbol{A}_{im}$ of (15a) has the
following structure:

$\boldsymbol{A}_{im}\\!\\!=\\!\\!\\!\begin{pmatrix}\\!a_{i(J_{i(m-1)}+1)m}\\!&\\!\\!\\!a_{i(J_{i(m-1)}+1)m}\\!\\!\\!&\\!\cdots\\!&\\!\\!\\!a_{i(J_{i(m-1)}+1)m}\\!\\\
\\!a_{i(J_{i(m-1)}+1)m}\\!&\\!\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\!\\!&\\!\cdots\\!&\\!\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\\
\vdots&\vdots\ &&\vdots\\\
\\!a_{i(J_{i(m-1)}+1)m}\\!&\\!\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\!\\!&\\!\cdots\\!&\\!\\!\\!a_{iJ_{im}m}\\!\end{pmatrix}\\!\\!.$
(C.3)

Based on (C.3), the $t$-th order principal minor of matrix
$\boldsymbol{A}_{im}$ can be expressed as

$\displaystyle A_{imt}\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\!\begin{vmatrix}\\!a_{i(J_{i(m-1)}+1)m}\\!\\!&\\!\\!a_{i(J_{i(m-1)}+1)m}\\!\\!&\\!\cdots\\!&\\!\\!a_{i(J_{i(m-1)}+1)m}\\!\\\
\\!a_{i(J_{i(m-1)}+1)m}\\!\\!&\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\!&\\!\cdots\\!&\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\\
\vdots&\vdots&&\vdots\\\
\\!a_{i(J_{i(m-1)}+1)m}\\!\\!&\\!\\!a_{i(J_{i(m-1)}+2)m}\\!\\!&\\!\cdots\\!&\\!\\!a_{i(J_{i(m-1)}+t)m}\\!\\\
\end{vmatrix}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=a_{i(J_{i(m-1)}+1)m}\prod_{l=2}^{t}(a_{i(J_{i(m-1)}+l)m}-a_{i(J_{i(m-1)}+l-1)m}).$

Since

$\displaystyle a_{i(J_{i(m-1)}+1)m}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\!\frac{B/(\ln
2)}{\left(\sum_{n=J_{i(m-1)}+1}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+1)m}\right)^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!>0,$

and for $J_{i(m-1)}+2\leq l\leq J_{im}$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad
a_{i(J_{i(m-1)}+l)m}-a_{i(J_{i(m-1)}+l-1)m}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad=\frac{1}{\ln
2}\left(\frac{B}{\left({\sum_{n=J_{i(m-1)}+l}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+l)m}}\right)^{2}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\qquad\left.-\frac{B}{\left({\sum_{n=J_{i(m-1)}+l}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+l-1)m}}\right)^{2}}\right)\overset{(\text{a})}{\geq}0,$

where inequality (a) holds based on (III-C), we have from (LABEL:hessianT)
that $A_{imt}\geq 0$ for $t=1,\cdots,J_{im}-J_{i(m-1)}$. According to [46,
Page 558], a function whose Hessian is positive semi-definite throughout a
convex set is convex. Besides, if the principal minors of a matrix are all
nonnegative, this matrix is positive semi-definite [46, Page 558]. Thus,
matrix $\boldsymbol{A}_{im}$ is positive semi-definite, which implies that
objective function (15a) is convex. As a result, Problem (15) is convex.

Then, we prove the feasibility condition for Problem (15). To prove this, we
denote

$\tilde{q}_{im}=\left\\{\begin{array}[]{ll}\\!\\!\mathop{\min}\limits_{\boldsymbol{p}_{im}\geq\boldsymbol{0}}&\\!\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}\\\
\>\textrm{s.t.}&\\!p_{ijm}\\!\geq\\!\\!\left(\\!2^{\frac{R_{ijm}}{B}}\\!-\\!1\\!\right)\\!\left(\\!\sum_{n=j+1}^{J_{im}}\\!p_{inm}\\!+\\!H_{ijm}\\!\right),\\\
&\qquad\forall j\in\mathcal{J}_{im}.\end{array}\right.$

From (15b), (15c), and $\boldsymbol{p}_{im}\geq\boldsymbol{0}$, we can find
that Problem (15) is feasible if and only if $\tilde{q}_{im}\leq q_{im}$. To
obtain $\tilde{q}_{im}$, we observe that constraints (15c) hold with equality
for all $j\in\mathcal{J}_{im}$, as otherwise $\tilde{q}_{im}$ can be further
improved. By solving these $J_{im}-J_{i(m-1)}$ linear equations, we have
$\tilde{q}_{im}=\sum_{n=J_{i(m-1)}+1}^{J_{im}}{\left(2^{\frac{R_{inm}}{B}}-1\right)}{2^{\sum_{s=J_{i(m-1)}+1}^{n-1}\frac{R_{ism}}{B}}}H_{inm}$
from (A.3), (A.4) and (A). Hence, the feasibility condition for Problem (15)
is achieved as (16).

##  Appendix D Proof of Theorem 4

The Lagrangian function of Problem (15) can be written by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{L}(\boldsymbol{p}_{im},\alpha_{im},\boldsymbol{\beta}_{im},\boldsymbol{\gamma}_{im})=\alpha_{im}\\!\left(\\!\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}-q_{im}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-\sum_{j=J_{i(m-1)}+1}^{J_{im}}B\log_{2}\left(1+\frac{p_{ijm}}{\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!+\sum_{j=J_{i(m-1)}+1}^{J_{im}}\beta_{ijm}\left(\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right)\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-p_{ijm}\Bigg{)}-\sum_{j=J_{i(m-1)}+1}^{J_{im}}\gamma_{ijm}p_{ijm},$

where $\alpha_{im}$,
$\boldsymbol{\beta}_{im}=[\beta_{i(J_{i(m-1)}+1)m},\cdots,\beta_{iJ_{im}m}]^{T}\geq\boldsymbol{0}$
and
$\boldsymbol{\gamma}_{im}=[\gamma_{i(J_{i(m-1)}+1)m},\cdots,\gamma_{iJ_{im}m}]^{T}\geq\boldsymbol{0}$
are the Lagrange multipliers associated with the corresponding constraints of
Problem (15). The KKT conditions of Problem (15) are:

$\displaystyle\frac{\partial\mathcal{L}}{\partial p_{ijm}}=-\frac{B}{(\ln
2)\left(\sum_{n=J_{i(m-1)}+1}^{J_{im}}p_{inm}+H_{i(J_{i(m-1)}+1)m}\right)}$
$\displaystyle\qquad+\alpha_{im}+\sum_{n=J_{i(m-1)}+1}^{j-1}\left(2^{\frac{R_{inm}}{B}}-1\right)\beta_{inm}-\beta_{ijm}$
$\displaystyle\qquad-\gamma_{ijm}-\sum_{l=J_{i(m-1)}+2}^{j}\frac{1}{\ln
2}\left(\frac{B}{{\sum_{n=l}^{J_{im}}p_{inm}+H_{ijm}}}\right.$
$\displaystyle\qquad\left.-\frac{B}{{\sum_{n=l}^{J_{im}}p_{inm}+H_{i(j-1)m}}}\right),\quad\forall
j\in\mathcal{J}_{im}$ (D.1a)
$\displaystyle\beta_{ijm}\left(\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right)-p_{ijm}\right)=0,$
$\displaystyle\qquad\quad\forall j\in\mathcal{J}_{im}$ (D.1b)
$\displaystyle\gamma_{ijm}p_{ijm}=0,\quad\forall j\in\mathcal{J}_{im}$ (D.1c)
$\displaystyle\sum_{j=J_{i(m-1)}+1}^{J_{im}}p_{ijm}-q_{im}=0$ (D.1d)
$\displaystyle\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right)-p_{ijm}\leq
0,$ $\displaystyle\qquad\forall j\in\mathcal{J}_{im}$ (D.1e)
$\displaystyle\boldsymbol{\beta}_{im},\boldsymbol{\gamma}_{im},\boldsymbol{p}_{im}\geq\boldsymbol{0}.$
(D.1f)

According to (D.1e) and $R_{ijm}>0$, we can obtain $p_{ijm}>0$, $\forall
j\in\mathcal{J}_{im}$. Hence, further combing (D.1c), we have

$\gamma_{ijm}=0,\quad\forall j\in\mathcal{J}_{im}.$ (D.2)

Assume that

${H_{i(J_{i(m-1)}+1)m}}>{H_{i(J_{i(m-1)}+2)m}}>\cdots>{H_{iJ_{im}m}}.$ (D.3)

The special case with ${H_{i(j-1)m}}={H_{ijm}}$ for at least one
$j\in\\{J_{i(m-1)}+2,\cdots,J_{im}\\}$ is considered later. From (D.1a) and
(D.2), we obtain

$\displaystyle\frac{\partial\mathcal{L}}{\partial
p_{ijm}}-\frac{\partial\mathcal{L}}{\partial p_{i(j-1)m}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{\ln
2}\left(\frac{B}{{\sum_{n=j}^{J_{im}}p_{inm}+H_{i(j-1)m}}}\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-\frac{B}{{\sum_{n=j}^{J_{im}}p_{inm}+H_{ijm}}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\beta_{ijm}+\beta_{i(j-1)m}2^{\frac{R_{i(j-1)m}}{B}}=0$

for $j=J_{i(m-1)}+2,\cdots,J_{im}$. Considering (D.3), we have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\beta_{ijm}-\beta_{i(j-1)m}2^{\frac{R_{i(j-1)m}}{B}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\!\frac{1}{\ln
2}\\!\left(\\!\frac{B}{{\sum_{n=j}^{J_{im}}p_{inm}\\!+\\!{H_{i(j-1)m}}}}\\!-\\!\frac{B}{\\!{\sum_{n=j}^{J_{im}}p_{inm}\\!+\\!{H_{ijm}}}}\right)<0$

for $j=J_{i(m-1)}+2,\cdots,J_{im}$. Since $\beta_{ijm}\geq 0$, we have
$\beta_{i(j-1)m}2^{\frac{R_{i(j-1)m}}{B}}>\beta_{ijm}\geq 0$,
$j=J_{i(m-1)}+2,\cdots,J_{im}$. Thus, we can obtain $\beta_{i(J_{im}-1)m}>0$,
$\cdots$, $\beta_{i(J_{i(m-1)}+1)m}>0$. Hence, we only need to consider the
following two cases of $\beta_{iJ_{im}m}$ for user $J_{im}$.

1) If $\beta_{iJ_{im}m}>0$, constraints in (D.1b) are satisfied via
$\left(2^{\frac{R_{ijm}}{B}}-1\right)\left(\sum_{n=j+1}^{J_{im}}p_{inm}+H_{ijm}\right)-p_{ijm}=0$
for all $j\in\mathcal{J}_{im}$, which implies that the minimal rate
constraints (D.1e) hold with equality for all users. Thus, the optimal value
of Problem (15) is $-\sum_{j=J_{i(m-1)}+1}^{J_{im}}R_{ijm}$, and the optimal
solution to Problem (15) can be obtained as in (10) by solving constraints
(D.1e) with equality for all users.

2) If $\beta_{iJ_{im}m}=0$, we find that constraints (D.1e) hold with equality
except for the user $J_{im}$.

Due to that Problem (15) can be easily solved for the case
$\beta_{iJ_{im}m}>0$, we only need to consider the case $\beta_{iJ_{im}m}=0$
in the following. Since minimal rate constraints (D.1e) hold with equality for
$j\in\mathcal{J}_{im}\setminus\\{J_{im}\\}$, we find that the additional power
is allocated to the user with the highest channel gain and other users served
by BS $i$ on subchannel $m$ are allocated with minimal transmission power to
meet the minimal rate demands. Now, it remains optimal to solve constraints
(D.1e) with equality for $j\in\mathcal{J}_{im}\setminus\\{J_{im}\\}$. Thus, we
have

$\displaystyle
2^{\frac{R_{ijm}}{B}}\sum_{n=j+1}^{J_{im}}p_{inm}+\left(2^{\frac{R_{ijm}}{B}}-1\right)H_{ijm}=\sum_{n=j}^{J_{im}}p_{inm},$
(D.4)

for all $j=J_{i(m-1)}+1,\cdots,J_{im}-1$. Define

$b_{ijm}=\sum_{n=j}^{J_{im}}p_{ijm},\quad j=J_{i(m-1)}+1,\cdots,J_{im}.$ (D.5)

Substituting (D.5) into (D.1d) yields

$b_{i(J_{i(m-1)}+1)}=q_{im}.$ (D.6)

Based on (D.4) and (D.5), we can obtain

$b_{i({j+1})m}=\frac{b_{ijm}}{2^{\frac{R_{ijm}}{B}}}-\frac{\left(2^{\frac{R_{ijm}}{B}}-1\right)H_{ij}}{2^{\frac{R_{ijm}}{B}}},$
(D.7)

for all $j=J_{i(m-1)}+1,\cdots,J_{im}-1$. By further using (D.6), we have

$\displaystyle b_{i(j+1)m}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{q_{im}}{2^{\sum_{n=J_{i(m-1)}+1}^{j}\frac{R_{inm}}{B}}}$
(D.8)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-\sum_{n=J_{i(m-1)}+1}^{j}\frac{\left(2^{\frac{R_{inm}}{B}}-1\right)H_{inm}}{2^{\sum_{l=n}^{j}\frac{R_{ilm}}{B}}},$

for all $j=J_{i(m-1)}+1,\cdots,J_{im}-1$. From (D.5), we can obtain

$\displaystyle
p_{ijm}\\!=\\!\begin{cases}b_{ijm}-b_{i(j+1)m},&\\!\\!\\!\\!\\!\mbox{ if
$j=J_{i(m-1)}+1,\cdots,J_{im}-1$}\\\ b_{ijm},&\\!\\!\\!\\!\\!\mbox{ if
$j=J_{im}$}\end{cases}.$ (D.9)

By inserting (D.8) into (D.9), we can obtain closed-form expression of
$p_{ijm}$ as (17). Substituting (17) into objective function (15a), we can
obtain the optimal sum rate of Problem (15) as (18).

Now, we consider the special case remained to be discussed. Assume that there
are two users served by BS $i$ with satisfying ${H_{i(j-1)m}}={H_{ijm}}$. In
this case, we can define a new user $j^{\prime}$ with
$R_{ij^{\prime}m}=R_{i(j-1)m}+R_{ijm}$. Calculate the optimal power allocation
strategy $[p_{i(J_{i(m-1)}+1)m}^{*},\cdots,$
$p_{i(j-2)m}^{*},p_{ij^{\prime}m}^{*},p_{i(j+1)m}^{*},\cdots,p_{iJ_{im}m}^{*}]^{T}$
for users $J_{i(m-1)}+1,\cdots,j-2,j^{\prime},j+1,\cdots,J_{im}$ according to
(17). Based on (5), we have

$\displaystyle r_{i(j-1)m}+r_{ijm}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=B\log_{2}\left(\frac{\sum_{n=j-1}^{J_{im}}p_{inm}\\!+\\!{H_{i(j-1)m}}}{\sum_{n=j}^{J_{im}}p_{inm}\\!+\\!{H_{i(j-1)m}}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad+B\log_{2}\left(\frac{\sum_{n=j}^{J_{im}}p_{inm}\\!+\\!{H_{ijm}}}{\sum_{n=j+1}^{J_{im}}p_{inm}\\!+\\!{H_{ijm}}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=B\log_{2}\left(\frac{\sum_{n=j-1}^{J_{im}}p_{inm}+{H_{ijm}}}{\sum_{n=j+1}^{J_{im}}p_{inm}+{H_{ijm}}}\right),$

which means that the sum rate of user $j-1$ and user $j$ is determined by the
sum power $p_{i(j-1)m}+p_{ijm}$. In the optimal power allocation strategy for
user $j-1$ and user $j$, we can arbitrarily allocate power $p_{i(j-1)m}^{*}$
and $p_{ijm}^{*}$ with $p_{i(j-1)m}^{*}+p_{ijm}^{*}=p_{ij^{\prime}m}^{*}$
fixed and the minimal rate constraints $r_{i(j-1)m}^{*}\geq R_{i(j-1)m}$ and
$r_{ijm}^{*}\geq R_{ijm}$ satisfied. If $j\neq J_{im}$, we can observe that
$r_{ij^{\prime}m}^{*}=R_{ij^{\prime}m}=R_{i(j-1)m}+R_{ijm}$ according to (D.4)
for the optimal power allocation strategy. Then, $r_{i(j-1)m}^{*}=R_{i(j-1)m}$
and $r_{ijm}^{*}=R_{ijm}$, which indicates that the optimal power for user
$j-1$ and user $j$ can be presented as (17). If $j=J_{im}$, we can observe
that $r_{ij^{\prime}m}^{*}\geq R_{ij^{\prime}m}$ according to (D.1e) for the
optimal power allocation strategy. If we set $r_{i(j-1)m}^{*}=R_{i(j-1)m}$ and
$r_{ijm}^{*}=r_{ij^{\prime}m}^{*}-R_{ijm}$, the optimal power for user $j-1$
and user $j$ can also be presented as (17). Thus, we can still obtain the
optimal sum rate of Problem (15) as (18).

As a result, Theorem 4 is proved.

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