#  Multi-objective Resource Allocation for D2D and Enabled MC-NOMA Networks by
Tchebycheff Method

Siavash Bayat, Member, IEEE, Ata Khalili, Member, IEEE, Shayan Zargari,
Mohammad Robat Mili, Member, IEEE, and Zhu Han, Fellow, IEEE Siavash Bayat,
Ata Khalili, Shayan Zargari, and Mohammad Robat Mili are with the Electronics
Research Institute, Sharif University of Technology, Tehran, Iran. (e-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>mohammad.robatmili@ieee.org). Zhu Han is with the Electrical and Computer
Engineering Department, University of Houston, TX, 77004, USA. (e-mail:
hanzhu22@gmail.com).

###### Abstract

This paper considers a resource allocation problem in device-to-device (D2D)
communications sharing the same frequency spectrum. In particular, the
cellular users (CUs) utilize non-orthogonal multiple access (NOMA) while D2D
users (DUs) adopt the orthogonal frequency division multiple access (OFDMA). A
multi-objective optimization problem (MOOP) is formulated, which jointly
maximizes the sum rate of DUs and CUs in uplink communications while taking
into account the maximum transmit power budget and minimum data rate
requirement for DUs and CUs. This MOOP is handled by the weighted Tchebycheff
method, which converts it into a single-objective optimization (SOOP). Then,
the monotonic optimization approach is employed to solve this SOOP optimally.
Numerical results unveil an interesting tradeoff between D2D and CUs.

##  I Introduction

Device-to-Device (D2D) communication is promoted as an innovative paradigm to
improve the network performance as well as the system resources utilization in
the fifth-generation (5G) cellular networks and beyond. The power and channel
allocation for D2D communication need elaborate coordination with cellular
users (CUs), as D2D users (DUs) can impose interference to other users [1]. In
practice, D2D communication operates either in the overlay or underlay modes
along with existing CUs. In fact, the underlay mode is appealing as it follows
the system to achieve higher spectral efficiency in which the spectrum is
shared between DUs and CUs[2].

Non-orthogonal multiple access (NOMA) has been proposed as one of the
fundamental techniques for beyond 5G to achieve a better balance between
system spectral efficiency (SE) and user fairness [3]. As a result, the
integration between NOMA and D2D communication has significant attention in
order to improve user connectivity as well as SE [4, 5]. In this regard,
resource allocation is one challenging problem which can appropriately
mitigate interference, thereby improving the system SE. In [4], while using
the successive interference cancellation (SIC) to detect the multiplexing
signals, the resource block assignment and power allocation were optimized to
maximize the sum data rate of the D2D pairs. In [5], according to interference
status, different NOMA-aided spectrum-sharing modes (i.e., the D2D access
scheme) for the paired DUs and CUs were considered. Then, a connectivity-
maximization problem was formulated under mode selection, user pairing, and
power control while guaranteeing the decoding thresholds of CUs and DUs to
take advantage of the NOMA-and-D2D integrated structure. In [6], the authors
offered an innovative resource allocation policy to enhance the performance of
D2D communications underlying CUs in the downlink (DL). For a network
consisting of CUs and DUs in [7], the problem of power allocation and user
clustering was studied while the sum-rate of the NOMA-based network was
maximized. In [8], a joint power allocation and user scheduling for the
D2D-enabled HetNets with NOMA was proposed to maximize the ergodic sum rate of
the near users in the small cells, while guaranteeing the quality-of-service
requirements of the far users and the macro-cell users. The authors in [9]
offered a joint optimization framework for D2D-enabled NOMA networks, where
the performance of the D2D communication was maximized while considering the
SIC decoding order of the NOMA-based CU equipment. In [10], the mode selection
and resource allocation problem for D2D-enabled NOMA cellular networks was
considered. However, the inter-lay mode was introduced for D2D communications
in NOMA system to maximize the system sum rate, which exploited the SIC to
cancel the interference between D2D pairs and CUs. In [11], a NOMA-enhanced
D2D communication scheme was considered and then the system sum-rate was
maximized, while optimizing subchannel and power allocation. Then, a solution
was proposed to assign subchannels to D2D groups and allocate power to
receivers in each D2D group. The authors in [12] considered the resource
allocation problem for an uplink multi-carrier NOMA in D2D underlaid cellular
networks and then the maximization problem of the Nash product of each user
was investigated as a Nash bargaining game.

Despite the fruitful results in the literature, the performance of the D2D
communications can still be improved as the conflicting goals of D2D and
cellular create serious network performance bottlenecks. In the literature,
the framework of the multi-objective optimization problem (MOOP) was employed
to address the conflicting objectives in wireless systems [13]-[15]. Regarding
this, the authors in [13] proposed a MOOP for maximizing the signal-to-
interference-plus-noise ratio (SINR) to determine the optimal power allocation
for each D2D pair. A MOOP trade-off was analyzed in [14] to investigate the
trade-off between EE and SE in a D2D underlying system. This problem was
converted into a SOOP via the $\epsilon$-method, and a two-stage iterative
algorithm was propose. However, a non-trivial trade-off between the DUs and
the CUs would be expected. As a result, the spectrum sharing deployment under
such networks suffers from inter-cell interference originated from the DUs and
CUs in each cell which leads to an exciting optimization problem.

Nonetheless to the best of the authors’ knowledge the optimal subchannel
assignment and power allocation were not discussed in [4]-[12]. In particular,
designing the optimal resource management in D2D-enabled NOMA cellular
networks in underlay mode is challenging. Additionally, deriving achievable
rate regions for the DUs and CUs via employing the NOMA scheme leads to an
interesting trade-off problem that is not investigated in [4]-[12]. The
contributions of this letter are summarized as follows:

  * •

In contrast to the existing literature e.g., [4]-[12], we study the
performance trade-off between the DUs and the CUs in D2D networks underlying
NOMA CUs to maximize the sum data rate of the DUs and CUs simultaneously.

  * •

To study such a trade-off, we formulate a MOOP framework by jointly optimizing
the transmit powers and subcarrier allocation policies that can be obtained by
adjustable weighting parameters to execute the resource allocation policy.

  * •

To solve the MOOP at hand, we first apply the weighted Tchebycheff method,
which converts the MOOP into a SOOP. Then, a monotonic optimization method is
proposed to obtain the optimal resource allocation policy.

  * •

In the numerical results, we provide an interesting trade-off between CUs and
DUs and also demonstrate the superiority of the MC-NOMA scheme as compared to
multi-carrier orthogonal multiple access (MC-OMA) schemes. This also shows
that our proposed scheme outperforms the proposed algorithm in [12].

##  II System Model

In this paper, an uplink single-cell NOMA-based cellular network is considered
which compromises of one BS to serve $M$ CUs and $K$ D2D links (DUs) given by
$\mathcal{M}=\\{1,...,M\\}$ and $\mathcal{K}=\\{1,...,K\\}$, respectively. The
total system bandwidth of $B$ Hz is divided into a set of subchannels denoted
by $\mathcal{N}=\\{1,...,N\\}$, which is shared between CUs and DUs so that
each subchannel has bandwidth of $B_{c}=B/N$ Hz. We define $h^{n}_{t,r}$ as
the channel gain between transmitter $t$ and receiver $r$ over subcarrier $n$.
For simplify, the channel gain between the transmitter and BS is defined as
$h^{n}_{t}$. Denote the instantaneous channel power gains for the $k$-th DU to
the BS as $h_{k}^{n}$ and the link between the $m$-th CU and the BS as
$g_{m}^{n}.$ The transmitted power from the $k$-th DU over subchannel $n$ is
expressed by $p_{k}^{n}$ and for the $m$-th CU is given by $\hat{p}_{m}^{n}$.
Furthermore, the noise power spectral density is given by $N_{0}$. It should
be noted that the D2D devices are performed based on the OFDMA protocol due to
closing to each other, which indicates that the desired signal is more
considerable than the interference terms. In this network, the BS can employ
the SIC technique based on the descending order in the channel gain. The
channel gains between the users and the BS should satisfy $g_{m}>h_{k}>g_{i}$
then the BS will decode the signals $x_{m}$, $x_{k}$, and $x_{i}$ sequentially
by using the SIC technique. On the contrary, since the signal strength of
$x_{m}$ at the D2D receiver $x_{k}$ is the strongest, the D2D receiver will
directly decode its desired signal. This constraint ensures that the BS can
perform SIC properly. As a result, the instantaneous SINR for each CU $m$ on
subchannel $n$ is given by

${\color[rgb]{0,0,1}\gamma_{m,\text{Cellular}}^{n}=\frac{\psi_{m}^{n}\hat{p}_{m}^{n}|g_{m}^{n}|^{2}}{\sigma^{2}_{\text{CU}}+{\textstyle\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\backslash
m\\\
g_{i}^{n}<g_{m}^{n}\end{subarray}}}\psi_{i}^{n}\hat{p}_{i}^{n}|g_{i}^{n}|^{2}+{\textstyle\sum\limits_{\begin{subarray}{c}j\in\mathcal{K}\\\
h_{j}^{n}<g_{m}^{n}\end{subarray}}}\varphi_{j}^{n}p_{j}^{n}|h_{j}^{n}|^{2}}},$
(1)

where $\psi_{i}^{n}$ and $\varphi_{j}^{n}$ are the binary variables for
subchannel allocation for the CU and DU, respectively. If $\psi_{m}^{n}=1$,
subchannel $n$ is allocated to the $m$-th CU and $\psi_{m}^{n}=0,$ otherwise.
In a similar manner, if $\varphi_{j}^{n}=1$, subchannel $n$ is assigned to the
$j$-th DU and ${\small\varphi_{j}^{n}=0},$ otherwise. In $(\ref{m11})$, the
term ${\textstyle\sum_{\begin{subarray}{c}(i\neq
m)\in\mathcal{M},\;g_{i}^{n}<g_{m}^{n}\end{subarray}}}\psi_{i}^{n}\hat{p}_{i}^{n}|g_{i}^{n}|^{2}$
is the interference term from the CUs due to operating on the same subchannel
based on the NOMA scheme, and the term
${\small{\textstyle\sum_{\begin{subarray}{c}j\in\mathcal{K},\;h_{j}^{n}<g_{m}^{n}\end{subarray}}}\varphi_{j}^{n}p_{j}^{n}|h_{j}^{n}|^{2}}$
corresponds to the interference from the DUs. In addition, the instantaneous
received SINR at the $k$-th DU on subchannel $n$ can be written as

${\color[rgb]{0,0,1}\gamma_{k,\text{D2D}}^{n}=\frac{\varphi_{k}^{n}p_{k}^{n}|h_{k}^{n}|^{2}}{\sigma^{2}_{\text{D2D}}+\sum\limits_{m\in\mathcal{M}}\psi_{m}^{n}\hat{p}_{m}^{n}|g_{m,k}^{n}|^{2}+{\textstyle\sum\limits_{j\in\mathcal{K}\backslash
k}}\varphi_{j}^{n}p_{j}^{n}|h_{j,k}^{n}|^{2}}},$ (2)

where $g_{m,k}^{n}$ is the channel power gain between the $m$-th CU and the
$k$-th DU receiver over subchannel $n$, and
${\textstyle}\sum_{m\in\mathcal{M}}\psi_{m}^{n}\hat{p}_{m}^{n}|g_{m,k}^{n}|^{2}$
is the interference term arising from the CUs. Besides,
${\textstyle\sum_{{j\in\mathcal{K}\backslash
k}}}\varphi_{j}^{n}p_{j}^{n}|h_{j,k}^{n}|^{2}$ denotes the interference term
resulting from the other D2D pairs, where $h_{j,k}^{n}$ is the instantaneous
channel power gain from the $j$-th DU transmitter to the $k$-th DU receiver on
subchannel $n$.

##  III Problem Formulation

In this section, to strike a tradeoff between the system performance of DUs
and CUs, we formulate a MOOP in which jointly the sum rate of the DUs,
$\text{R}_{\text{DU}}$, and CUs, $\text{R}_{\text{CU}}$, are maximized. The
proposed MOOP framework aims to obtain power allocation as well as the
subchannel assignment strategy to study the performance tradeoff between
conflicting system objectives. This joint optimization can be formulated
through the following MOOP:

$\displaystyle\max_{\left\\{{\normalsize\mathbf{p}},\mathbf{\hat{p}},\boldsymbol{\varphi},\boldsymbol{\psi}\right\\}}\
\ \text{R}_{\text{DU}}=\sum_{k=1}^{K}\text{R}_{\text{DU,}k}$ (3a)
$\displaystyle\max_{\left\\{{\normalsize\mathbf{p}},\mathbf{\hat{p}},\boldsymbol{\varphi},\boldsymbol{\psi}\right\\}}\text{
\ R}_{\text{CU}}=\sum_{m=1}^{M}\text{R}_{\text{CU,}m}$ (3b)
$\displaystyle~{}~{}\mathrm{s.t.}~{}~{}$
$\displaystyle\sum_{n=1}^{N}\varphi_{k}^{n}p_{k}^{n}\leq
p^{k}_{\max,\text{D2D}},\>\>\>\sum_{n=1}^{N}\psi_{m}^{n}\hat{p}_{m}^{n}\leq
p^{m}_{\max,\text{Cellular}},$ (3c)
$\displaystyle\varphi_{k}^{n}\in\\{0,1\\},\text{\ }\forall k,n,\text{\ \ \ \ \
\ }\psi_{m}^{n}\in\\{0,1\\},\text{\ }\forall m,n,\text{{}}$ (3d)
$\displaystyle\sum_{n=1}^{N}\varphi_{k}^{n}\leq 1,\text{\ }\forall k,\text{\
}\text{ \ \ \ \ \ \ }\sum_{n=1}^{N}\psi_{m}^{n}\leq 1,\text{ \ }\forall
m,\text{\ }$ (3e)
$\displaystyle\sum_{k=1}^{K}\varphi_{k}^{n}+\sum_{m=1}^{M}\psi_{m}^{n}\leq
L_{\max},\text{ \ }\forall n,\text{\ }$ (3f)
$\displaystyle\text{R}_{\text{CU,}m}\geq R^{m}_{\text{CU},\min},~{}\forall
m~{}~{}\text{R}_{\text{DU,}k}\geq R^{k}_{\text{DU},\min},~{}\forall k,$ (3g)

where
$\text{R}_{\text{CU,}m}=\sum_{n=1}^{N}\ln(1+\gamma_{m,\text{Cellular}}^{n})$
and $\text{R}_{\text{DU,}k}=\sum_{n=1}^{N}\ln(1+\gamma_{k,\text{D2D}}^{n})$.
To facilitate the system design, we define
$\boldsymbol{\varphi}\in\mathbb{Z}^{{\normalsize KN\times 1}}$ and$\
\boldsymbol{\psi}\in\mathbb{Z}^{{\normalsize MN\times 1}}$ as the vectors of
subchannel assignment variables in D2D and cellular networks, respectively.
Furthermore, variables $\mathbf{p}$ $\in$ $\mathbb{R}^{{\normalsize KN\times
1}}$ and $\ \mathbf{\hat{p}}$ $\in$ $\mathbb{R}^{{\normalsize MN\times 1}}$
are the collections of power allocation variables in D2D and cellular
networks, respectively. Variables $P_{\max,\text{D2D}}$ and
$P_{\max,\text{Cellular}}$ are the maximum total power for the DUs and CUs,
respectively. $L_{\max}$ denotes the maximum number of CUs that can be paired
on a subchannel under spectrum sharing scheme111It can be perceived that at
most $(L_{\max}-1)$ CUs share on the same subchannel.. Note that the
optimization problem (3) is a mixed-integer non-linear programming (MINLP)
because of the interference in the rate function, and the presence of the
binary constraints.

##  IV Proposed Solution

One approach to solve a MOOP is the weighted Tchebycheff technique [15, 16],
which offers an auxiliary optimization variable, $\chi$ for (3) as follows

$\displaystyle\underset{\left\\{{\normalsize\mathbf{p}},\mathbf{\hat{p}},\boldsymbol{\varphi},\boldsymbol{\psi,}\chi\right\\}}{\text{minimize}}\text{
\ \ \ }\chi$ $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle(\text{3c})-(\text{3g}),$ (4a) $\displaystyle\text{ \
}\frac{\alpha}{\text{R}_{\text{DU,max}}}\left(\text{R}_{\text{DU,max}}-\text{R}_{\text{DU}}\right)-\chi\leq
0,$ (4b) $\displaystyle\text{ \
}\frac{(1-\alpha)}{\text{R}_{\text{CU,max}}}\left(\text{R}_{\text{CU,max}}-\text{R}_{\text{CU}}\right)-\chi\leq
0,$ (4c)

where $\alpha$ and $1-\alpha$ are the weighting coefficients222The non-
negative weight $\alpha$ denotes the priorities of CUs and DUs in the resource
allocation policy specified in the media access control (MAC) layer to achieve
a certain notion of fairness, especially for users who suffer from poor
channel conditions.indicating the impact of the different objectives. The
weighted Tchebycheff method ensures to produce a set of Pareto-optimal
solutions when R${}_{\text{DU,max}}$ and R${}_{\text{CU,max}}$ are the utopian
objective points as the maximum of each objective, respectively [16]. In the
following, to solve the highly nonconvex optimization problem in (4) globally,
we apply a global optimization approach known as the monotonic optimization
method. By exploiting monotonicity or hidden monotonicity in the objective
function as well as constraints, this method guarantees the convergence [17].

Note that (4) is not a monotonic optimization problem in canonical form since
(4b) and (4c) are not monotonic. To facilitate the presentation, we rewrite
these constraints as follows:

$\displaystyle\sum_{k=1}^{K}\sum_{n=1}^{N}\ln\bigg{(}1+\frac{\bar{q}^{n}_{k}|h_{k}^{n}|^{2}}{\sigma^{2}_{\text{D2D}}+\sum\limits_{m\in\mathcal{M}}\tilde{p}^{n}_{m}|g_{m,k}^{n}|^{2}+{\textstyle\sum\limits_{j\in\mathcal{K}\backslash
k}}\bar{q}^{n}_{j}|h_{j,k}^{n}|^{2}}\bigg{)}$ $\displaystyle\geq
R_{\text{DU,max}}-\frac{\chi}{\alpha},$ (5)
$\displaystyle\sum_{m=1}^{M}\sum_{n=1}^{N}\ln\bigg{(}1+\frac{\tilde{p}^{n}_{m}|g_{m}^{n}|^{2}}{\sigma^{2}_{\text{CU}}+{\textstyle\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\backslash
m\\\
g_{i}^{n}<g_{m}^{n}\end{subarray}}}\tilde{p}^{n}_{i}|g_{i}^{n}|^{2}+{\textstyle\sum\limits_{\begin{subarray}{c}j\in\mathcal{K}\\\
h_{j}^{n}<g_{m}^{n}\end{subarray}}}\bar{q}^{n}_{j}|h_{j}^{n}|^{2}}\bigg{)}$
$\displaystyle\geq R_{\text{CU,max}}-\frac{\chi}{1-\alpha},$ (6)

where $\tilde{p}^{n}_{m}=\psi_{m}^{n}\hat{p}_{m}^{n}$ and
$\bar{q}^{n}_{k}=\varphi_{k}^{n}p_{k}^{n}$. Note that optimization problem
(4a) is not monotonic as a result of constraints (4b) and (4c). First, the
optimization problem in (4a) can be rewritten as a monotonic optimization
problem, and then we adopt the polyblock algorithm [17] to obtain a globally
optimal solution. To doing so, let us define
${\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}}=\\{p^{k}_{\max,\text{D2D}}\\},~{}\forall
k$ and
${\normalsize\tilde{{\mathbf{{q}}}}_{\text{Max}}}=\\{q^{m}_{\max,\text{Cellular}}\\},~{}\forall
m$ as the maximum transmit power for each user over all subchannels. Moreover,
the relations in (IV), (IV), and (3g) can be equivalently stated as the
following single constraints:

$\displaystyle\underbrace{\sum_{i=1}^{M}\bigg{(}c_{i}^{+}({\normalsize\tilde{\mathbf{{p}}}})\bigg{)}}_{c^{+}(\tilde{{\normalsize{\mathbf{{p}}}}})}-\underbrace{\sum_{i=1}^{M}\bigg{(}c_{i}^{-}({\normalsize\tilde{\mathbf{{p}}}})}_{c^{-}(\tilde{{\normalsize{\mathbf{{p}}}}})}\bigg{)}+R_{\text{DU,max}}-\frac{\chi
R_{\text{DU,max}}}{\alpha}\geq 0,$ (7)
$\displaystyle\underbrace{\sum_{j=1}^{K}\bigg{(}c_{j}^{+}({\normalsize\bar{\mathbf{{q}}}})\bigg{)}}_{\hat{c}^{+}(\bar{{\normalsize{\mathbf{{q}}}}})}-\underbrace{\sum_{j=1}^{K}\bigg{(}c_{j}^{-}({\normalsize\bar{\mathbf{{q}}}})}_{\hat{c}^{-}(\bar{{\normalsize{\mathbf{{q}}}}})}\bigg{)}+R_{\text{CU,max}}-\frac{\chi
R_{\text{CU,max}}}{1-\alpha}\geq 0,$ (8)
$\displaystyle\underbrace{\min_{i\in\\{1,...,K\\}}\left[c_{k}^{+}({\normalsize\tilde{\mathbf{{p}}}})+\sum\limits_{i\in\mathcal{K}\backslash
k}c_{i}^{-}({\normalsize\tilde{\mathbf{{p}}}})\right]}_{r^{+}({\normalsize\tilde{\mathbf{{p}}}})}-\underbrace{\sum\limits_{i\in\mathcal{K}}c_{i}^{-}({\normalsize\tilde{\mathbf{{p}}}})}_{r^{-}({\normalsize\tilde{\mathbf{{p}}}})}-R^{i}_{\text{DU,min}}\geq
0,$ (9)
$\displaystyle\underbrace{\min_{j\in\\{1,...,M\\}}\left[c_{m}^{+}({\normalsize\bar{\mathbf{{q}}}})+\sum\limits_{j\in\mathcal{M}\backslash
m}c_{j}^{-}({\normalsize\bar{\mathbf{{q}}}})\right]}_{\hat{r}^{+}({\normalsize\bar{\mathbf{{q}}}})}-\underbrace{\sum\limits_{j\in\mathcal{M}}c_{j}^{-}({\normalsize\bar{\mathbf{{q}}}})}_{\hat{r}^{+}({\normalsize\bar{\mathbf{{q}}}})}-R^{j}_{\text{CU,min}}\geq
0,$ (10)

where $c_{i}^{+}({\normalsize\tilde{\mathbf{{p}}}})$,
$c_{i}^{-}({\normalsize\tilde{\mathbf{{p}}}})$,
$c_{j}^{+}({\normalsize\bar{\mathbf{{q}}}})$, and
$c_{j}^{-}({\normalsize\bar{\mathbf{{q}}}})$ are increasing in
${\normalsize\tilde{\mathbf{{p}}}}$ and ${\normalsize\bar{\mathbf{{q}}}}$
given at the top of the next page.

$\displaystyle
c_{i}^{+}({\normalsize\tilde{\mathbf{{p}}}})=\sum_{n=1}^{N}\ln\bigg{(}\sigma^{2}_{\text{D2D}}+{\textstyle\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\backslash
m\\\
g_{i}^{n}<g_{m}^{n}\end{subarray}}}\tilde{p}^{n}_{i}|g_{i}^{n}|^{2}+{\textstyle\sum\limits_{\begin{subarray}{c}j\in\mathcal{K}\\\
h_{j}^{n}<g_{m}^{n}\end{subarray}}}\bar{q}^{n}_{j}|h_{j}^{n}|^{2}+\tilde{p}_{m}^{n}|g_{m}^{n}|^{2}\bigg{)}$
(11) $\displaystyle
c_{i}^{-}({\normalsize\tilde{\mathbf{{p}}}})=\sum_{n=1}^{N}\ln\bigg{(}\sigma^{2}_{\text{D2D}}+{\textstyle\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\backslash
m\\\
g_{i}^{n}<g_{m}^{n}\end{subarray}}}\tilde{p}^{n}_{i}|g_{i}^{n}|^{2}+{\textstyle\sum\limits_{\begin{subarray}{c}j\in\mathcal{K},\\\
h_{j}^{n}<g_{m}^{n}\end{subarray}}}\bar{q}^{n}_{j}|h_{j}^{n}|^{2}\bigg{)},~{}c_{j}^{-}({\normalsize\bar{\mathbf{{q}}}})=\sum_{n=1}^{N}\ln\bigg{(}\sigma^{2}_{\text{CU}}+\sum\limits_{m\in\mathcal{M}}\tilde{p}^{n}_{m}|g_{m}^{n}|^{2}+{\textstyle\sum\limits_{j\in\mathcal{K}\backslash
k}}\bar{q}^{n}_{j}|h_{j,k}^{n}|^{2}\bigg{)},$ (12) $\displaystyle
c_{j}^{+}({\normalsize\bar{\mathbf{{q}}}})=\sum_{n=1}^{N}\ln\bigg{(}\sigma^{2}_{\text{CU}}+\sum\limits_{m\in\mathcal{M}}\tilde{p}^{n}_{m}|g_{m}^{n}|^{2}+{\textstyle\sum\limits_{j\in\mathcal{K}\backslash
k}}\bar{q}^{n}_{j}|h_{j,k}^{n}|^{2}+\bar{q}^{n}_{k}|h_{k}^{n}|^{2}\bigg{)},$
(13)

Bear in mind that equations (7) and (8) are the difference of two increasing
functions. However, the constraint (3d) is a binary constraint which is
intractable. To tackle it, we rewrite (3d) in the equivalent form as:

$\displaystyle 0\leq\varphi_{k}^{n}\leq
1,~{}\sum_{k=1}^{K}\sum_{n=1}^{N}\varphi_{k}^{n}-(\varphi_{k}^{n})^{2}\leq 0,$
(14) $\displaystyle 0\leq\psi_{m}^{n}\leq
1,~{}\sum_{m=1}^{M}\sum_{n=1}^{N}\psi_{m}^{n}-(\psi_{m}^{n})^{2}\leq 0.$ (15)

It can be perceived that right hand side of (14) and (15) are non-convex as
well as non-monotonic. To tackle this issue, we introduce two slack variables
$\nu$ and $\mu$ and rewrite as:

$\displaystyle\sum_{k=1}^{K}\sum_{n=1}^{N}(\varphi_{k}^{n})^{2}+\nu\geq\mathcal{R}_{1},~{}\sum_{k=1}^{K}\sum_{n=1}^{N}(\varphi_{k}^{n})+\nu\leq\mathcal{R}_{1},$
(16)
$\displaystyle\sum_{m=1}^{M}\sum_{n=1}^{N}(\psi_{m}^{n})^{2}+\mu\geq\mathcal{R}_{2},~{}\sum_{m=1}^{M}\sum_{n=1}^{N}\psi_{m}^{n}+\mu\leq\mathcal{R}_{2},$
(17)

where $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are constant. We note that the
left hand sides of (14) and (15) are monotonically increasing with respect to
$\nu$ and $\mu$, respectively. Consequently, by introducing the auxiliary
variable $s_{1}$, $s_{2}$, $s_{3}$, and $s_{4}$ the problem in (4) can be then
reformulated as following:

$\displaystyle\underset{\left\\{{\normalsize\tilde{\mathbf{p}},\bar{\mathbf{q}}},s_{1},s_{2},s_{3},s_{4},\chi\right\\}}{\text{minimize}}\text{
}\chi$ (18a) $\displaystyle\mathrm{s.t.}~{}\ 0\leq s_{1}\leq
c^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}})-c^{-}({\normalsize\mathbf{0}}),$
(18b) $\displaystyle\quad\quad
c^{-}({\normalsize\tilde{{\mathbf{{p}}}}})+s_{1}\leq
c^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}}),$ (18c)
$\displaystyle\quad\quad c^{+}({\normalsize\tilde{{\mathbf{{p}}}}})+s_{1}\geq
c^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}}),$ (18d)
$\displaystyle\quad\quad 0\leq
s_{2}\leq\hat{c}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}})-\hat{c}^{-}({\normalsize\mathbf{0}}),$
(18e)
$\displaystyle\quad\quad\hat{c}^{-}({\normalsize\bar{{\mathbf{{q}}}}})+s_{2}\leq\hat{c}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}}),$
(18f)
$\displaystyle\quad\quad\hat{c}^{+}({\normalsize\bar{{\mathbf{{q}}}}})+s_{2}\geq\hat{c}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}}),$
(18g) $\displaystyle\quad\quad\tilde{p}^{n}_{m}\geq 0,\bar{q}^{n}_{k}\geq
0,\forall m,k,n,$ (18h)
$\displaystyle\quad\quad\sum_{n=1}^{N}\bar{q}^{n}_{k}\leq
P_{\max,\text{D2D}},\>\>\>\sum_{n=1}^{N}\tilde{p}^{n}_{m}\leq
P_{\max,\text{Cellular}},$ (18i) $\displaystyle\quad\quad
0\leq\varphi_{k}^{n}\leq 1,~{}0\leq\psi_{m}^{n}\leq 1,$ (18j)
$\displaystyle\quad\quad\sum_{k=1}^{K}\sum_{n=1}^{N}(\varphi_{k}^{n})^{2}+\nu\geq\mathcal{R}_{1},~{}\sum_{m=1}^{M}\sum_{n=1}^{N}(\psi_{m}^{n})^{2}+\mu\geq\mathcal{R}_{2},$
(18k)
$\displaystyle\quad\quad\sum_{k=1}^{K}\sum_{n=1}^{N}(\varphi_{k}^{n})+\nu\leq\mathcal{R}_{1},~{}\sum_{m=1}^{M}\sum_{n=1}^{N}\psi_{m}^{n}+\mu\leq\mathcal{R}_{2},$
(18l) $\displaystyle\quad\quad\text{(\ref{sS11})-(\ref{s11})},$ (18m)
$\displaystyle\quad\quad\ 0\leq s_{3}\leq
r^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}})-r^{-}({\normalsize\mathbf{0}}),$
(18n) $\displaystyle\quad\quad
r^{-}({\normalsize\tilde{{\mathbf{{p}}}}})+s_{3}\leq
r^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}}),$ (18o)
$\displaystyle\quad\quad r^{+}({\normalsize\tilde{{\mathbf{{p}}}}})+s_{3}\geq
r^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}}),$ (18p)
$\displaystyle\quad\quad\ 0\leq
s_{4}\leq\hat{r}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}})-\hat{r}^{-}({\normalsize\mathbf{0}}),$
(18q)
$\displaystyle\quad\quad\hat{r}^{-}({\normalsize\bar{{\mathbf{{q}}}}})+s_{4}\leq\hat{r}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}}),$
(18r)
$\displaystyle\quad\quad\hat{r}^{+}({\normalsize\bar{{\mathbf{{q}}}}})+s_{4}\geq\hat{r}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}}).$
(18s)

Consequently, the feasible set of problem (18) can be expressed as the
intersection of two following sets:

$\displaystyle\mathcal{G}=\\{{\left\\{{\normalsize\tilde{{\mathbf{p}}},\bar{{\mathbf{q}}}},s_{1},s_{2},s_{3},s_{4},\chi\right\\}}:\tilde{{\mathbf{p}}}\preceq\tilde{{\mathbf{p}}}_{\text{
Max}},\bar{{\mathbf{q}}}\preceq\bar{{\mathbf{q}}}_{\text{
Max}},\text{(\ref{13c})},$
$\displaystyle\text{(\ref{13f})},\text{(\ref{13i})},\text{(\ref{16j})},\text{(\ref{16l})},\text{(\ref{13j})},\text{(\ref{13cc})},\text{(\ref{13ccc})}\\},$
(19)
$\displaystyle\mathcal{H}=\\{{\left\\{{\normalsize\tilde{{\mathbf{p}}},\bar{{\mathbf{q}}}},s_{1},s_{2},s_{3},s_{4},\chi\right\\}}:\tilde{{\mathbf{p}}}\succeq\mathbf{0},\bar{{\mathbf{q}}}\succeq\mathbf{0},\text{(\ref{13d})}$
$\displaystyle,\text{(\ref{13g})},\text{(\ref{13h})},\text{(\ref{16k})},\text{(\ref{13dd})},\text{(\ref{13ddd})}\\},$
(20)

where $\mathcal{G}$ and $\mathcal{H}$ are normal and co-normal sets,
respectively, in the hyper-rectangle [17]

$\displaystyle\big{[}0,c^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max}}})-c^{-}({\normalsize\mathbf{0}})\big{]}\times\big{[}0,\hat{c}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max}}})-\hat{c}^{-}({\normalsize\mathbf{0}})\big{]}\times\big{[}0,\tilde{{\mathbf{{p}}}}_{\text{Max}}\big{]}$
$\displaystyle\times\big{[}0,\bar{{\mathbf{{q}}}}_{\text{Max}}\big{]}\times\big{[}0,c_{i}^{-}({\normalsize\tilde{{\mathbf{{p}}}}_{\text{Max},k}})-c_{i}^{-}({\normalsize\mathbf{0}})\big{]}$
$\displaystyle\times\big{[}0,\hat{c}_{j}^{-}({\normalsize\bar{{\mathbf{{q}}}}_{\text{Max},m}})-\hat{c}_{j}^{-}({\normalsize\mathbf{0}})\big{]}\times\big{[}0,\tilde{{\mathbf{{p}}}}_{\text{Max},k}\big{]}\times\big{[}0,\bar{{\mathbf{{q}}}}_{\text{Max},m}\big{]}.$
(21)

Finally, it can be shown that problem (18) is a monotonic problem. Thus, the
optimal solutions can be found at the upper boundary of the feasible set by
applying the polyblock algorithm. However, approaching the upper boundary is
not possible since it is not precisely known. Hence, we employ the outer
polyblock approximation to construct a poly-block. Next, a new poly-block
$\mathbf{\Omega}^{{(l+1)}}$ can be constructed from the old one
$\mathbf{\Omega}^{{(l)}}$ based on the cutting a cone from the old one, i.e.,
$\mathbf{\Omega}^{{(l)}}\backslash\mathcal{K}^{+}_{\mathbf{x}}$ where the
definition of the cone is given in the following.

###### Definition 1

Assume that $\mathcal{G}$ is a normal set in $\mathcal{R}^{n}_{+}$, and
$\mathbf{y}\in\mathcal{R}^{n}_{+}\backslash\mathcal{G}$. If
$\hat{\mathbf{x}}\in\partial^{+}\mathcal{G}$ such that
$\hat{\mathbf{x}}<\mathbf{y}$, where $\partial^{+}\mathcal{G}$ is the upper
boundary of $\mathcal{G}$, then the cone
$\mathcal{K}^{+}_{\hat{\mathbf{x}}}:=\\{\mathbf{R}^{n}_{+}\>|\>\mathbf{x}>\hat{\mathbf{x}}\\}$
isolates $\mathbf{y}$ strictly from $\mathcal{G}$.

Let us define vertex set of polyblock $\boldsymbol{\Omega}$ as
$\boldsymbol{\omega}$, then
$\boldsymbol{w}^{(\ast)}=\\{\mathbf{z}\in\boldsymbol{w}\>|\>\mathbf{z}>\mathbf{x}\\}$
is the subset of $\boldsymbol{\Omega}$ that involves all vertices in cone
$\mathcal{K}^{+}_{\mathbf{x}}$. The following equation,
${\boldsymbol{z}}^{(i)}=\boldsymbol{z}+\big{(}{x}^{(i)}-z^{(i)}\big{)}\boldsymbol{e}_{i},\forall
i\in\\{1,...,n\\}$ can be adopted to obtain each vertex
$\mathbf{z}\in\boldsymbol{\Omega}^{*}$. Note that ${\boldsymbol{z}}^{(i)}$ is
obtained by substituting the $i$-th entry of ${\boldsymbol{z}}$ by the $i$-th
entry of $\mathbf{x}$. Furthermore, $\boldsymbol{e}_{i}$ is a unit vector
whose elements are equal to one. Let $\boldsymbol{\pi}^{(l)}$ indicate the
vertex of $\boldsymbol{\Omega}^{(l)}$ such that maximizes the objective
function over $\boldsymbol{\Omega}^{(l)}$. The projection operation is given
by $\phi(\boldsymbol{\pi}^{(l)})$ which requires solving a one-dimensional
problem as
$\text{max}\\{\lambda>0|\lambda\boldsymbol{\pi}^{(l)}\in\mathcal{G}\\}$. As a
result of the normality of $\mathcal{G}$, $\lambda$ can be attained by the
bisection search algorithm illustrated in Algorithm 1. Let
$\hat{\mathbf{{x}}}^{(l)}$ express the most trustworthy feasible solution at
the $l$-th iteration, and $V^{(l)}=f(\hat{\mathbf{{x}}}^{(l)})$ be the current
most suitable value. At the next iteration $(l+1)$, if
$\phi(\boldsymbol{\pi}^{l+1})\in\mathcal{G}\bigcap\mathcal{H}$ and
$f(\phi(\boldsymbol{\pi}^{l+1}))\geq V^{(l)}$, we have
$\hat{\mathbf{{x}}}^{l+1}=\phi(\boldsymbol{\pi}^{l+1})$ and
$V^{l+1}=f(\phi(\boldsymbol{\pi}^{l+1}))$. Otherwise, let
$\hat{\mathbf{{x}}}^{l+1}=\hat{\mathbf{{x}}}^{l}$ and $V^{l+1}=V^{l}$. The
algorithm stops if $|f(\boldsymbol{\pi}^{(l)})-V^{(l)}|\leq\epsilon$
satisfies, where $\epsilon\geq 0$ is a given tolerance. Moreover,
$\hat{\mathbf{{x}}}^{l}$ is said to be an $\epsilon$-optimal solution if
$f({\mathbf{{x}}}^{\ast})-\epsilon\leq f(\hat{\mathbf{{x}}}^{l+1})\leq
f({\mathbf{{x}}}^{\ast})$. We summarize the polyblock approach in Algorithm 2.

Algorithm 1 Bisection Projection Search Algorithm

0: $\boldsymbol{\pi}^{(l)}$ and $\mathcal{G}$Output $\lambda$ such that
$\lambda=\text{
argmax}\\{\lambda>0\>|\>\lambda\boldsymbol{\pi}^{(l)}\in\mathcal{G}\\}$

1: Set $\lambda_{\text{min}}=0$, $\lambda_{\text{max}}=1$, and the error
tolerance $\delta\ll 1$.

2: repeat

3: Let ${\bar{\lambda}}=\frac{\lambda_{\text{min}}+\lambda_{\text{max}}}{2}$

4: Solve the feasibility problem $(\ref{monnn})$.

5: If

6: Check if $\bar{\lambda}$ is feasible, i.e.,
$\bar{\lambda}\boldsymbol{\pi}^{(l)}\in\mathcal{G}$, then set
$\lambda_{\text{min}}={\bar{\lambda}}$

7: else

8: Set $\lambda_{\text{max}}={\bar{\lambda}}$

9: until $\lambda_{\text{max}}-\lambda_{\text{min}}<\delta$.

10: Return ${\lambda}=\lambda_{\text{min}}$.

Taking into account that the projection of $\boldsymbol{\pi}^{(l)}$ in the
$l$-th iteration over set $\mathcal{G}$ is required. Therefore, we can obtain
the projection by
$\phi(\boldsymbol{\pi}^{(l)})=\lambda\boldsymbol{\pi}^{(l)}$, where $\lambda$
is the projection parameter. Moreover, ${\lambda}$ is given by
${\lambda}=\text{max}\\{\alpha\>|\>\alpha\boldsymbol{\pi}^{(l)}\in\mathcal{G}\\}$,
where ${\lambda}\in[0,1]$. In particular, the bisection search technique can
be applied to obtain ${\lambda}$. For a given $\lambda$ and vertex
$\boldsymbol{\pi}^{(l)}$ in the $l$-th iteration, the following feasibility
problem needs to be solved:

$\displaystyle\max_{\textbf{y}}~{}1$ (22a) s.t.
$\displaystyle~{}\mathbf{y}\in\mathcal{G}.$ (22b)

We can obtain the optimal value for subcarrier allocations ($\phi^{n}_{k}$,
$\psi^{n}_{m}$) via comparing the values of the entries of the power
allocations ($\tilde{{\mathbf{{p}}}}$, $\tilde{{\mathbf{{q}}}}$) with zero. If
the values of ($\tilde{{\mathbf{{p}}}}$, $\tilde{{\mathbf{{q}}}}$) be greater
than 0, it means that the corresponding subcarrier allocations
($\phi^{n}_{k}$, $\psi^{n}_{m}$) would be zero. The details of the projection
bisection search algorithm are given in Algorithm 2.

##  V Computational Complexity

The polyblock algorithm is influenced by the configuration of the objective
function and the constraints that create the normal set. First, the most
suitable vertex by its projection on the normal set is determined. Then, the
projection of the picked vertex is obtained. Eventually, we obtained the new
vertex set by eliminating the inappropriate vertices. In particular, the
dimension of problem (P1), the number of iterations required for convergence,
and the number of iterations expected for the projection of each vertex are
assumed to be $B_{1}$, $B_{2}$ and $B_{3}$, respectively. In summary, the
complexity order can be stated as $\mathcal{O}(B_{2}(B_{2}\times
B_{1}+B_{3}))$ [17, 18].

Algorithm 2 Outer Poly-block Approximation Algorithm

0: An function $f(\cdot)$, a compact normal set $\mathcal{G}$ and a closed
conormal set $\mathcal{H}$, such that $\mathcal{G}\bigcap\mathcal{H}\neq{\O}$

1: Output: an $\epsilon$-optimal solution $\mathbf{x}^{*}$

2: Initialize iteration index $l=0$, poly-block $\mathbf{\Omega}^{(1)}$ be box
$[0,\mathbf{b}]$ that encloses $\mathcal{G}\bigcap\mathcal{H}$ with vertex set
$\boldsymbol{\omega}^{(1)}=\\{\mathbf{b}\\}$. $\epsilon$ is a small positive
number. $V^{(l)}=-\infty$

3: Repeat

4: From $\boldsymbol{\omega}^{(l)}$, select,
$\boldsymbol{\pi}^{(l)}\in\text{argmax}\\{f(\boldsymbol{\pi})|\boldsymbol{\pi}\in\boldsymbol{\omega}^{(l)}\\}$
.

5: obtain the projection of $\boldsymbol{\pi}^{(l)}$, i.e.,
$\phi(\boldsymbol{\pi}^{(l)})$ on the upper boundary of $\mathcal{G}$

6: if $\phi(\boldsymbol{\pi}^{(l)})=\boldsymbol{\pi}^{(l)}$, i.e.,
$\boldsymbol{\pi}^{(l)}\in\mathcal{G}$ then

7: $\hat{\mathbf{x}}^{(l)}=\boldsymbol{\pi}^{(l)}$ and
$V^{(l)}=f(\boldsymbol{\pi}^{(l)})$

8: else

9: if $\phi(\boldsymbol{\pi}^{(l)})\in\mathcal{G}\bigcap\mathcal{H}\neq{\O}$
and $f(\phi(\boldsymbol{\pi}^{(l)}))\geq V^{l-1}$, then let then best value be
$\hat{\mathbf{{x}}}^{(l)}=\phi(\boldsymbol{\pi}^{(l)})$ and ${\small
V^{l}=f(\phi(\boldsymbol{\pi}^{(l)}))}$. Otherwise,
${\small{\hat{\mathbf{{x}}}^{(l)}=\hat{\mathbf{{x}}}^{l-1}}}$ and ${\small
V^{l}=V^{l-1}}$.

10: Let ${\mathbf{{x}}}=\phi(\boldsymbol{\pi}^{(l)})$ and
${\small\boldsymbol{w}^{(l+1)}=(\boldsymbol{w}^{(l)}\backslash\boldsymbol{w}^{(\ast)})\bigcup\big{\\{}{\boldsymbol{z}}^{(i)}=\boldsymbol{z}+\big{(}{x}^{(i)}-z^{(i)}\big{)}\boldsymbol{e}_{i}|}$
$\boldsymbol{z}\in\boldsymbol{w}^{(\ast)},\forall i\in\\{1,...,n\\}\big{\\}}$,
where $\boldsymbol{w}^{(\ast)}=\\{\mathbf{z}\in$
$\boldsymbol{w}^{l}\>|\>\mathbf{z}>\mathbf{x}\\}$.

11: Remove from $\boldsymbol{w}^{(l+1)}$ the improper vertices and the
vertices
$\\{\mathbf{z}\in\boldsymbol{w}^{l+1}\>|\>\mathbf{z}\notin\mathcal{H}\\}$.

12: end

13: Set $l$=$l$+1.

14: Until $|f(\boldsymbol{\pi}^{(l)})-V^{(l)}|\leq\epsilon$.

15: Return $\mathbf{x}^{\ast}=\hat{\mathbf{x}}^{(l)}$.

##  VI Simulation Results

In this section, we evaluate the performance of the proposed algorithm by
numerical simulation. For each subchannel, a Rayleigh flat fading involving
the path-loss model is considered. The simulation parameters are given in
Table I, unless otherwise specified.

TABLE I: simulation parameters Parameter | Value
---|---
Cell diameter |  $250$ m
Distance between D2D link |  $30$m
Number of CUs ($M$) | $6$
Number of DUs ($K$) | $3$
Number of sub-carriers ($N$) | $4$
Noise power |  $-120$ dBm
Sub-carrier bandwidth |  $180$ kHz
Path-loss model for cellular links | $128.1+37.6\log(d)$
Path-loss model for D2D links | $148.1+40\log(d)$
Maximum transmit power of the DUs |  $25$ dBm
Minimum data rate requirement ($\overline{R}_{\textnormal{min}}$) |  $1$ bps$/$Hz
Figure 1: Average sum rate versus maximum transmit power of CUs. Figure 2:
Average sum rate versus maximum number of D2D pairs.

Fig. 1 shows the system sum-rate versus maximum transmit power of CUs for an
equal weight of CUs and DUs i.e., $\alpha=0.5$. It can be seen that by
increasing $p_{\max,\text{Cellular}}$, not only the CUs more easily meet their
minimum requirement but also the system throughput is improved. In fact with
the increase $p_{\max,\text{Cellular}}$, the system throughput increases
monotonically due to enhancing the achievable rate for CUs which results in an
enhancement of the total throughput of the system. This figure also shows that
large value of $p_{\max,\text{Cellular}}$ leads to releasing the potential of
DUs that enables more D2D pairs to access the network. For comparison, we also
consider three baseline schemes for the system sum rate. For baseline scheme
1, we adopt a subcarrier assignment randomly where the power allocation is
obtained based on our proposed monotonic approach. For baseline scheme 2, we
consider MC-OMA in which each subcarrier is allocated to at most one user. For
baseline scheme 3, we considered the proposed method in [12] in which an
iterative algorithm is adopted to find the resource allocation policy. It can
be perceived that our proposed algorithm outperforms the proposed algorithm in
[12] due to performing a monotonic approach which gives us optimal resource
allocation policy. Furthermore, we observe that the system sum rate for MC-OMA
achieves a lower system sum rate compared to MC-NOMA due to underutilizing the
orthogonal subcarrier assignment.

Fig. 2 illustrates the sum-rate versus the number of D2D pairs. It can be
perceived that our proposed solution significantly ameliorates the system sum
rate as compared to baseline scheme 3. This is due to the fact that in our
proposed algorithm the spectrum resources are profoundly reused and our
proposed algorithm outperforms the baseline scheme 3 with SIC scheme.

The trade-off between the total data rate of DUs and CUs is investigated in
Fig. 3. This figure is obtained by solving problem (18) for different values
of $\alpha$ $\in[0,1]$, with a step size of $0.1$. It can be seen that a non-
trivial trade-off between the total data rate of DUs and CUs exists. In
particular, the total DUs data rate is a decreasing function versus the total
CUs data rate.In other words, maximizing the total DUs data rate leads to a
reduction in total CUs data rate due to conflicting objectives functions. In
fact, by changing the weight factor, we can provide fairness between the
cellular and DUs. This figure also demonstrates the superiority of the NOMA
scheme as compared to the conventional OMA method.

##  VII Conclusion

In this paper, we investigated the tradeoff between DUs and CUs in uplink
underlaying CUs-enabled NOMA networks. We formulated a MOOP framework which
jointly maximizes the throughput of DUs and CUs simultaneously, to obtain
power allocation strategy and subchannel assignment. The MOOP was converted
into a SOOP using a weighted Tchebychef method and then solved via monotonic
optimization to obtain an optimal solution. Simulation results not only
unveiled an interesting tradeoff between the studied competing objective
functions but also investigated the superiority of our proposed scheme as
compared to MC-OMA.

Figure 3: Throughput trade-off region of CUs and DUs.

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