# RadiOrchestra: Proactive Management of Millimeter-wave Self-backhauled Small
Cells via Joint Optimization of Beamforming, User Association, Rate Selection,
and Admission Control

Luis F. Abanto-Leon2, Arash Asadi2, Andres Garcia-Saavedra3, Gek Hong
(Allyson) Sim2, Matthias Hollick2

2{labanto, aasadi, asim<EMAIL_ADDRESS><EMAIL_ADDRESS>2Technische Universität Darmstadt, Germany,
3NEC Laboratories Europe GmbH

###### Abstract

Millimeter-wave self-backhauled small cells are a key component of next-
generation wireless networks. Their dense deployment will increase data rates,
reduce latency, and enable efficient data transport between the access and
backhaul networks, providing greater flexibility not previously possible with
optical fiber. Despite their high potential, operating dense self-backhauled
networks optimally is an open challenge, particularly for radio resource
management (RRM). This paper presents, RadiOrchestra, a holistic RRM framework
that models and optimizes beamforming, rate selection as well as user
association and admission control for self-backhauled networks. The framework
is designed to account for practical challenges such as hardware limitations
of base stations (e.g., computational capacity, discrete rates), the need for
adaptability of backhaul links, and the presence of interference. Our
framework is formulated as a nonconvex mixed-integer nonlinear program, which
is challenging to solve. To approach this problem, we propose three algorithms
that provide a trade-off between complexity and optimality. Furthermore, we
derive upper and lower bounds to characterize the performance limits of the
system. We evaluate the developed strategies in various scenarios, showing the
feasibility of deploying practical self-backhauling in future networks.

###### Index Terms:

radio resource management, self-backhauling, millimeter-wave, beamforming,
scheduling,

##  I Introduction

Network _densification_ , through the deployment of small cells, is
indispensable to meet the increasing user demands for emerging wireless
services [1]. Small cells are realized by low-cost radio access nodes, known
as small base stations (SBSs), that provide wireless connectivity to
undersized geographical areas [2]. SBSs are strategically installed in close
proximity to the end users, bolstering the quality of experience and improving
the radio access network (RAN) performance. _In this way, dense small cell
deployments are expected to increase data rates, maintain low latency, extend
coverage and support a large number of users, thereby enabling the rollout of
a wide range of new services._

As small cell deployments become denser, more efficient forms of backhauling
data traffic between SBSs and the core network will be needed [3]. Optical
fiber has been the predominant means for this task, but its installation and
maintenance are costly. Self-backhauling, standardized under the name of
_integrated access and backhaul_ (IAB) [4], is an innovative technology that
promises to reduce costs by sharing the wireless spectrum in
time/frequency/space between RAN and backhaul links [5]. Small cells with
self-backhauling capabilities benefit from a tight integration of access and
backhaul functions, leading to high reconfigurability and facilitating self-
adaptation to a wide range of cases.

Self-backhauled small cells require wide bandwidth to cope with the growing
access-backhaul traffic. The millimeter-wave spectrum offers the necessary
bandwidth to meet this requirement but it poses challenges, e.g., limited
transmission range. Fortunately, recent advances in beamforming [6] have
overcome the physical drawbacks of millimeter-waves by taking advantage of the
small antennas size that have enabled large antenna arrays. _Thus, millimeter-
wave self-backhauled small cell networks, realized by multi-antenna SBSs, will
play a key role in next-generation wireless networks. Their dense deployment
will reduce costs and enable efficient transport of massive data traffic
between access and backhaul networks. In addition, the flexibility of
millimeter-wave self-backhauled small cells will provide higher adaptability
in various topologies and network conditions, previously not possible with
fiber._

Despite consensus on the potential of millimeter-wave self-backhauling,
designing an optimal system remains an open research challenge [7], which
requires efficient radio resource management (RRM) across the access and
backhaul networks. To date, the body of work in this area often overlooks
practical challenges inherent to realistic wireless communications systems,
such as discrete modulations and coding schemes (MCSs), or low computing
capabilities of SBSs. _Our work is motivated by the absence of holistic RRM
frameworks providing a realistic model and a practical solution for
millimeter-wave self-backhauled small cells deployments._ In the following, we
introduce these challenges and put them in perspective with the literature.

Challenge 1: Scalable self-backhauling design. The majority of prior works
relies on point-to-point links, e.g., [8, 9], between macro base station (MBS)
and SBSs, which is unscalable in dense SBS deployments. The scalability issue
is addressed in a handful of works, e.g., [10, 11] assume that SBSs are
capable of multi-layer successive interference cancellation (SIC). While this
assumption simplifies traffic transport, it involves heavy computational tasks
(i.e., SIC) not suited for SBSs. _Thus, to keep SBS economical for the
operators, it is necessary to reduce the computational burden of SBSs by
developing practical backhauling mechanisms._

Challenge 2: Adaptive backhaul capacity. Although self-backhauling relies on
wireless media, whose capacity is inherently highly variable due to noise and
interference, the assumption of unlimited or fixed capacity prevails in many
prior works, e.g., [9, 12, 13, 14]. _However, it is necessary to consider the
capacity limitation of backhaul links as well as their variability in real
systems._

Challenge 3: User association. It is conventionally assumed that users are
served by a single SBS [15, 13] or by all SBSs within a given range [9, 12].
While these assumptions simplify the problem formulation and solution, they
are neither realistic nor optimal. _Thus, a general scheme is needed where
users are associated to multiple SBSs in a flexible manner without considering
extremes cases._

Challenge 4: Admission control. Many works assume that all users can be served
simultaneously [14, 16, 17], which is unrealistic due to limitations in power,
number of antennas or RF chains. _Admission control (or user scheduling) is
crucial to guarantee the quality of service requirements for at least a subset
of admitted users, thereby circumventing unfeasibility issues._

Challenge 5: Discrete data rates. It is usually assumed that data rates are
continuous-valued, e.g., [10, 18, 17, 8, 19]. However, in practice they are
limited to a number of possible choices, i.e., finite set of MCSs. _It is
critical to consider the discreteness of rates since results obtained from
solving problems for continuous values cannot be easily applied to real
systems and are not expected to work properly._

Table I: Categorization of related work

Approach |  Solution |  Spectrum |  Network | Access network | Backhaul network
---|---|---|---|---|---
Topology | Beamforming |  User association |  Rate selection |  Admission control | Topology | Link | Medium | Beamforming |  Rate selection
[20, 21] | Joint | Sub-6GHz | Single-SBS | Unicast | ✓ | ✗ | ✓ | ✓ | N/A | N/A | N/A | N/A | N/A
[22, 23, 24, 25] | Joint | Sub-6GHz | Multi-SBS | Unicast | ✓ | Many | ✗ | ✗ | N/A | N/A | N/A | N/A | N/A
[26, 27] | Decoupled | Sub-6GHz | Single-SBS | Multicast | ✓ | ✗ | ✗ | ✓ | N/A | N/A | N/A | N/A | N/A
[28] | Joint | Millimeter-wave | Multi-SBS | Unicast | 3D | Many | ✗ | ✗ | N/A | N/A | N/A | N/A | N/A
[29, 8] | Joint | Millimeter-wave | Multi-SBS | N/A | N/A | N/A | N/A | N/A | Unicast | Adaptive | Wireless | 2D | ✗
[30] | Joint | Sub-6GHz | Multi-SBS | Multicast | 2D | Many | ✗ | ✗ | Unicast | Fixed | Wired | ✗ | ✗
[31, 32] | Joint | Sub-6GHz | Multi-SBS | Both | 2D | Many | ✗ | ✗ | Unicast | Fixed | Wired | ✗ | ✗
[9, 12] | Decoupled | Sub-6GHz | Multi-SBS | Unicast | 2D | ✗ | ✗ | ✓ | Unicast | Unbounded | Wired | ✗ | ✗
[13] | Joint | Sub-6GHz | Multi-SBS | Unicast | 2D | One | ✗ | ✓ | Unicast | Fixed | Wireless | ✗ | ✗
[17] | Decoupled | Millimeter-wave | Multi-SBS | Unicast | 3D | Many | ✗ | ✗ | Unicast | Adaptive | Wireless/TDM | 3D | ✗
[18] | Decoupled | Sub-6GHz | Multi-SBS | Unicast | 2D | One | ✗ | ✓ | Unicast | Adaptive | Wireless/SDM | 2D | ✗
[10] | Decoupled | Sub-6GHz | Multi-SBS | Unicast | 2D | Many | ✗ | ✗ | Multicast | Adaptive/SIC | Wireless/SDM | 2D | ✗
[15] | Joint | Sub-6GHz | Multi-SBS | Unicast | 2D | One | ✗ | ✓ | Unicast | Adaptive | Wireless/SDM | 2D | ✗
[14, 16] | Joint | Sub-6GHz | Multi-SBS | Multicast | 2D | ✗ | ✗ | ✗ | Unicast | Fixed | Wireless | ✗ | ✗
[33] | Joint | Sub-6GHz | Multi-SBS | Unicast | 2D | One | ✗ | ✓ | Unicast | Adaptive | Wireless/SDM | 2D | ✗
[19] | Joint | Sub-6GHz | Multi-SBS | Unicast | 2D | Many | ✗ | ✗ | Multicast | Adaptive | Wireless/TDM | 2D | ✗
[11] | Joint | Sub-6GHz | Multi-SBS | Unicast | 2D | ✗ | ✗ | ✗ | Unicast | Adaptive/SIC | Wireless/SDM | 2D | ✗
Proposed | Joint | Millimeter-wave | Multi-SBS | Unicast | 3D | Many | ✓ | ✓ | Multicast | Adaptive | Wireless/SDM | 3D | ✓

The connection between the MBS and SBSs is called backhaul link, which is a
convention in small cells literature. However, in a cloud-RAN context, MBSs
are called central processors or BBUs, SBSs are called RRHs, and the
connection between MBS and SBSs are called fronthaul links. In Table I, we
have considered both kinds of nomenclatures since the problems originated from
these two contexts are essentially the same.

In contrast to prior art, we propose a comprehensive RRM framework that
includes the challenges mentioned above, allowing us to more realistically
validate millimeter-wave self-backhauled small cell deployments. Our approach
makes the following novel contributions.

Contribution 1: In Section II, we address _Challenge 1_ by proposing a simple
yet effective _clustering mechanism for SBSs and users_ that results in
multiple non-overlapping _virtual cells_ or _clusters_. This allows us to
exploit multigroup multicast beamforming for backhaul traffic transmissions.
Our clustering approach simplifies the backhaul design and reduces
hardware/computational requirements at the sending and receiving nodes.

Contribution 2: In Section II-A and Section II-B we model _Challenge 2_ ,
_Challenge 3_ , _Challenge 4_ , _Challenge 5_ considering the access-backhaul
interdependencies between MBS, SBSs and users. In Section II-C, we include
these challenges in our formulation to jointly optimize _beamforming, user
association, rate selection, admission control in the access network_ and
_beamforming, rate selection in the backhaul network_ for maximizing the
access network downlink weighted sum-rate. We cast the problem as a nonconvex
mixed-integer nonlinear program (MINLP), which to the best of our knowledge,
has not been investigated before.

Contribution 3: To tackle the nonconvex MINLP, we propose three formulations
and their corresponding algorithms. In Section IV, we recast the nonconvex
MINLP as a mixed-integer second-order cone program (MISOCP), which can be
solved optimally. Due to the large number of integral variables, the cost of
solving the MISOCP via branch-and-cut (BnC) techniques is prohibitive. To cope
with this issue, in Section VI we propose a formulation solved via an
iterative algorithm that tackles a SOCP at every instance. In Section VII, a
much simpler SOCP formulation further decreases the complexity by reducing the
number of variables, and optimizing only the beamformers gains. In particular,
the complexity of the latter algorithm with respect to the former decreases
roughly by a factor equal to the third power of the number of antennas at the
SBS.

Contribution 4: In Section V, we derive an upper bound to provide insights on
the performance gaps and trade-offs of RadiOrchestra. We also provide a simple
lower bound marking the oerformance. We note that the upper bound is a novel
problem itself that has not been investigated before.

Contribution 5: In Section VIII, we examine RadiOrchestra exhaustively under
several scenarios including transmit power, number of clusters, and channel
estimation errors.

There is a plethora of literature on self-backhauling for sub-6GHz spectrum,
e.g., [33, 19, 32], which assume signals properties that do not work for
millimeter-wave. Many works have focused on the design of either the backhaul,
e.g., [29, 8, 34, 35] or the access network, e.g., [28, 27] alone. However,
the growth that mobile networks are experiencing calls for heterogeneous
networks with wireless backhauling, which require joint optimization.
Considering linear antenna arrays, many works have optimized beamforming,
e.g., [18, 33]. However, planar arrays are capable of 3D beamforming and hence
are more suitable for dense deployments. The joint optimization of beamforming
and user association _(Challenge 3)_ , admission _(Challenge 4)_ , or rate
selection _(Challenge 5)_ , generally requires solving complex nonconvex
MINLPs. Thus, many works facing these challenges split the problem into stages
and solve them separately. For instance, the integer variables are eliminated
first by assuming a given set of scheduled users, e.g., [18, 17]. Then, the
nonconvex functions are linearized and the problem is solved in the continuous
domain. Although simpler to solve, variable decoupling affects optimality due
to interdependencies removal. To meet the continuously growing demands,
resources have to be exploited more optimally. Therefore, RRM problems need to
be solved as a whole, without relying on variable partitioning which
translates to inefficient radio resource usage.

After a scrupulous study of the state of the art, we found that the works most
related to ours are [10, 19]. Like us, the authors of [10] assumed a multicast
topology in the backhaul network, with a MBS transmitting multiple signals to
various SBSs using multigroup multicasting beamforming (each signal carrying
the data of a user). Since a single SBS may serve several users, SBSs are
therefore required to decode many signal layers via SIC, which entails heavy
computational burden for low-cost SBSs. Further, the decoding order of signals
is known to affect the performance, leading to potential high decoding errors
and making SIC impractical, which was not evaluated in [10]. The authors of
[19] considered multiple SBS groups served in a multicast manner using time
division multiplexing (TDM), i.e., each group at a time. However, as the
number of clusters grows, the multiplexing time generates longer latency that
is unavoidable as SBSs need to transmit coordinately to users, making it less
practical. In addition, these works do not consider discrete rates, admission
control, millimeter-wave spectrum and 3D beamforming. For completeness, we
summarize in Table I the related literature on RRM for small cells.

Figure 1: Overview of the steps to formulate and solve the problem in
RadiOrchestra.

Overview: Not surprisingly, the inherent couplings among all the different
parameters of the system result in a complex problem which is difficult to
address. However, our framework helps to realize the true potential of self-
backhauled mobile networks, in particular in the presence of real-world
constraints. To the best of our knowledge, this is the first work that has
modeled an integrated access-backhaul system with such practical constraints
and proposed solutions to assess its performance. The investigated problem is
unique and hence existing solutions are not applicable to it. In the
following, we provide an overview of the steps taken to solve our problem,
from a systems design perspective as well as the mathematical treatment.

_Systems aspect._ 3GPP specifications for 5G leave several design choices to
the operators such as spectrum allocation of backhaul and access. We leveraged
these degrees of freedom to reduce the complexity of the problem while
maintaining a realistic setup. The wireless nature of the access and backhaul
links, coupled with the dense deployment of SBSs and users, creates a very
complex interference environment. In RadiOrchestra, we choose an out-of-band
system where backhaul and access links use different frequency bands, thus
disentangling the interference between the two networks. Conventionally, the
MBS sends individual backhaul signals to each SBS thus producing interference,
which is handled via (point-to-point) unicast beamforming. In dense
deployments this solution does not scale well due to the need to multiplex
various data streams. Thus, we propose a clustering strategy where the MBS
divides the SBSs into _clusters_ , which are served simultaneously via (point-
to-multipoint) multigroup multicast beamforming. This has three advantages:
$(i)$ Enhancing the scalability of self-backhauling by avoiding point-to-point
transmissions which cause higher interference; $(ii)$ Eliminating the need for
heavy signal processing (e.g., SIC operation) at SBSs [10, 11]; $(iii)$
Reducing hardware requirements and costs since MBS becomes more cost-efficient
only requiring as many RF chains as SBS clusters, which is far less than the
point-to-point topology (i.e., dedicated RF-chain per link).

_Problem formulation and solution._ Considering our design choices above, we
model the system and propose solutions in a series of steps that are
demonstrated in Fig. 1. We formulate a RRM problem for integrated access-
backhaul networks considering real-world constraints, which results in a
nonconvex MINLP with entangled variables (see Section II). We adopt a series
of procedures to simplify the structure of the nonconvex MINLP without
altering its optimality. Thus, we _(i)_ improve its tractability by
eliminating additive binary couplings and multiplicative mixed-integer
couplings, and _(ii)_ reduce the search space by adding cuts. Although the
problem structure is greatly simplified after these procedures, it still
remains a nonconvex MINLP. However, its more amenable layout allows us to
tailor algorithms for its solution (see Section III). We transform some of the
nonconvex constraints into equivalent (convex) SOC constraints and remodel
others as convex inner SOC approximations. As a result, we recast the
nonconvex MINLP into a MISOCP, which can be solved optimally (see Section IV).
Although solving the proposed MISOCP guarantees an optimal solution, it
requires a considerable amount of time due to the numerous integral variables.
To deal with that, which translates to more branches evaluations by the BnC
method, we propose a reformulation based on relaxation and penalization of the
integral variables that only requires to solve iteratively a SOCP, and is
guaranteed to attain a local optimum (see Section VI). To further simplify the
computational burden and expedite the solving time, we offer a much simpler
reformulation that reduces the number of continuous variables, where we
predesign the access and backhaul beamforming vectors and only optimize their
gains. As a result, we only solve a low-complexity SOCP problem iteratively
(see Section VII). Finally, we derive an upper bound for the problem, which we
use to characterize the performance of the developed algorithms (see Section
V).

##  II System Model and Problem Formulation

We consider that data is transported from the core network to the user
equipments (UEs) via a MBS and a deployment of SBSs as shown in Fig. 3. The
SBSs are connected to the MBS through wireless backhaul links. We assume an
out-of-band full-duplex access-backhaul system, i.e., the backhaul network
(connecting SBSs to the MBS) and the access network (connecting UEs to SBSs)
operate simultaneously employing orthogonal bands. In the following, we detail
the modeling assumptions.

Figure 2: Self-backhauled SBSs grouped into clusters. The backhaul exploits
multigroup multicast beamforming for data sharing whereas the access network
is based on distributed unicast beamforming.

Figure 3: SBS clustering allows to merge data of all the served users into one
stream, minimizing interference and simplifying data decoding at the SBSs.

Backhaul model: We rely on an advantageous clustering approach, where we
divide the SBSs into $L$ non-overlapping _virtual cells or clusters_ , each
formed by $B$ SBSs (as in distributed antenna systems). In this way, data
streams sent from the MBS to a SBS cluster contain the aggregate content for
all the served UEs in that cluster, as shown in Fig. 3. The SBSs are deployed
in a planned fashion and grouped based on their proximity. The antenna arrays
are oriented towards the cluster center, as shown in Fig. 4.

Figure 4: SBS distribution and clustering with a MBS transmitting multicast
streams to three different clusters.

Access model: Each UE is pre-associated to a SBS cluster, based on the
geographical distance or a given operator policy. Without loss of generality,
we assume that each cluster has $U$ UEs. Thus, SBSs in a cluster transmit
collaboratively to UEs only within that cluster. However, not all SBSs are
necessarily involved in serving a particular UE, and not all UEs may be
served. The information for all the served UEs is co-processed by all SBSs,
thus allowing to handle interference more efficiently.

Channel model: The backhaul links operate over a bandwidth
$W_{\mathrm{BW}}^{\mathrm{backhaul}}$ and we assume line-of-sight (LOS)
connectivity since the MBS and SBSs are usually strategically installed in the
planning phase. Besides, the access network operates over a bandwidth
$W_{\mathrm{BW}}^{\mathrm{access}}$ and its channels (i.e., between SBSs and
UEs) exhibit multipath scattering containing both line-of-sight (LOS) and Non-
line-of-sight (NLOS) components. Both access and backhaul channels are modeled
according to [36].

Optimization model: In line with the related literature, we assume that the
MBS has knowledge of the access channels between the SBSs and UEs. In
particular, 3GPP specifies channel training procedures in the access network
that we can rely upon. In addition, the MBS also knows the backhaul channels,
i.e., between itself and the SBSs. This knowledge is even simpler to acquire
than the access channels since backhaul links are rather static with small
variability. In summary, the MBS collects knowledge of all the wireless
channels and, accordingly, optimizes all the radio resources of the system.

_For the sake of clarity, variables and parameters used in the following
sections are summarized in Table II._

Table II: Parameters and variables of the system Parameters and Variables | Notation
---|---
Number of transmit antennas at the MBS and SBSs |  $N^{\mathrm{MBS}}_{\mathrm{tx}}$, $N^{\mathrm{SBS}}_{\mathrm{tx}}$
Maximum transmit power at the MBS and SBSs |  $P^{\mathrm{MBS}}_{\mathrm{tx}}$, $P^{\mathrm{SBS}}_{\mathrm{tx}}$
Number of clusters in the system | $L$
Number of UEs per cluster | $U$
Number of SBSs per cluster | $B$
Number of predefined rate/SINR values |  $J^{\mathrm{UE}}$, $J^{\mathrm{SBS}}$
Bandwidth of the access and backhaul networks |  $W_{\mathrm{BW}}^{\mathrm{access}}$, $W_{\mathrm{BW}}^{\mathrm{backhaul}}$
Set of clusters | $\mathcal{L}=\left\\{1,\cdots,L\right\\}$
Set of SBSs | $\mathcal{B}=\bigcup_{l\in\mathcal{L}}\mathcal{B}_{l}$
Set of UEs | $\mathcal{U}=\bigcup_{l\in\mathcal{L}}\mathcal{U}_{l}$
Set of predefined rate/SINR values at SBSs | $\mathcal{J}^{\mathrm{SBS}}$
Set of predefined rate/SINR values at UEs | $\mathcal{J}^{\mathrm{UE}}$
Set of UEs in the $l$-th cluster | $\mathcal{U}_{l}$
Set of SBSs in the $l$-th cluster | $\mathcal{B}_{l}$
Channel between the MBS and SBS $b$ | $\mathbf{g}_{b}$
Channel between SBS $b$ and UE $u$ | $\mathbf{h}_{b,u}$
Multicast precoder from the MBS to SBS cluster $\mathcal{B}_{l}$ | $\mathbf{m}_{l}$
Unicast precoder from SBS $b$ to UE $u$ | $\mathbf{w}_{b,u}$
Binary variable for UE rate/SINR selection | $\alpha_{u,j}$
Binary variable for SBS rate/SINR selection | $\beta_{l,j}$
Binary variable for UE association | $\kappa_{b,u}$

###  II-A Backhaul Network: Multicast Transmissions from MBS to SBSs

In the backhaul network, two important aspects are dealt with. First, _rate
selection_ , i.e. choosing appropriate data rates at which the MBS transmits
information to the SBSs. Second, _beamforming_ , i.e. adjusting the amplitude
and phases of the signals at the MBS to guarantee the selected rates.

Beamforming: The MBS is equipped with a planar array of
$N^{\mathrm{MBS}}_{\mathrm{tx}}$ transmit antennas operating on Band 1 used
for communication with the SBSs, which have $N^{\mathrm{SBS}}_{\mathrm{rx}}=1$
receive antenna. The MBS transmits as many streams as clusters. Every stream
contains the aggregate data for the served UEs in their respective clusters
(see Fig. 3). The instantaneous multicast symbol for the SBSs in cluster
$\mathcal{B}_{l}$ is denoted by $z_{l}$, with $\mathbb{E}\left[z_{l}\right]=0$
and $\mathbb{E}\left[\left|z_{l}\right|^{2}_{2}\right]=1$. The beamforming
vector conveying $z_{l}$ is denoted by $\mathbf{m}_{l}$. The composite signal
transmitted from the MBS to all SBS clusters is given by
$\mathbf{x}^{\mathrm{MBS}}=\sum_{l\in\mathcal{L}}\mathbf{m}_{l}z_{l}$. The
received signal at SBS $b\in\mathcal{B}_{l}$ is expressed as

$\displaystyle y_{b}^{\mathrm{SBS}}$
$\displaystyle=\mathbf{g}_{b}^{H}\mathbf{x}^{\mathrm{MBS}}+n_{b}$
$\displaystyle=\underbrace{\mathbf{g}_{b}^{H}\mathbf{m}_{l}z_{l}}_{\text{signal
for SBS
$b$}}+\underbrace{\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L},l^{\prime}\neq
l\end{subarray}}\mathbf{g}_{b}^{H}\mathbf{m}_{l^{\prime}}z_{l^{\prime}}}_{\text{interference}}+\underbrace{n_{b}}_{\text{noise}},$
(1)

where $\mathbf{g}_{b}$ is the channel between SBS $b\in\mathcal{B}_{l}$ and
the MBS whereas
$n_{b}\sim\mathcal{CN}\left(0,\sigma_{\mathrm{SBS}}^{2}\right)$ symbolizes
circularly symmetric Gaussian noise. The signal-to-interference-plus-noise
ratio (SINR) at SBS $b$ is

$\displaystyle\mathrm{SINR}_{b}^{\mathrm{SBS}}$
$\displaystyle=\frac{\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right|^{2}}{\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L},l^{\prime}\neq
l\end{subarray}}\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l^{\prime}}\right|^{2}+\sigma_{\mathrm{SBS}}^{2}}.$
(2)

Since all SBSs within a cluster receive the same common information (i.e.
aggregate UE content), the effective rate/SINR per cluster is determined by
the SBS with the worst conditions. As a result, a more sensible means of
quantifying the maximal SINR per cluster is the following
$\widetilde{\mathrm{SINR}}_{l}^{\mathrm{SBS}}=\min_{b\in\mathcal{B}_{l}}\left\\{\mathrm{SINR}_{b}^{\mathrm{SBS}}\right\\},\forall
l\in\mathcal{L}$.

Remark: This system is known as multigroup multicast beamforming [37] and has
been studied for transmissions from a MBS/SBS to multiple clusters of UEs. We
exploit that same idea to transmit data streams from the MBS to the SBSs. We
assume that the number of streams that the MBS can handle is sufficient to
serve all SBS clusters, i.e. $N^{\mathrm{MBS}}_{\mathrm{streams}}\geq L$.

Rate Selection: In practical wireless communications systems, the set of
eligible data rates is finite [38, pp. 64]. These predefined rates are
uniquely identified by their associated CQI index, and each corresponds to a
specific MCS. In addition, for each rate, a minimum received SINR is required
in order to ensure a target block error rate (BLER) [39]. While the rates and
MCSs are standardized, the corresponding target SINRs are usually vendor- and
equipment-specific. We consider the target SINRs in [40], which are shown in
Table III (in linear scale) and approximately exhibit increments of twice the
previous rate starting from $R^{\mathrm{SBS}}_{1}=0.2344$ bps/Hz.

Table III: Rates and target SINR values Coding rate |  Rate $R^{\mathrm{SBS}}_{j}$ [bps/Hz] |  SINR $\Gamma^{\mathrm{SBS}}_{j}$
---|---|---
120/1024 (QPSK) | 0.2344 | 0.2159
308/1024 (QPSK) | 0.6016 | 0.6610
602/1024 (QPSK) | 1.1758 | 1.7474
466/1024 (QAM) | 2.7305 | 10.6316
948/1024 (QAM) | 5.5547 | 95.6974

In order to assign $R^{\mathrm{SBS}}_{j}$ to the $l$-th SBS cluster, it is
required that
$\widetilde{\mathrm{SINR}}_{l}^{\mathrm{SBS}}\geq\Gamma^{\mathrm{SBS}}_{j}$,
$j\in\mathcal{J}^{\mathrm{SBS}}$, where $\mathcal{J}^{\mathrm{SBS}}$ represent
the set of possible rates. To represent the rate assignment, we introduce the
binary variables $\beta_{l,j}\in\left\\{0,1\right\\}$ with $\beta_{l,j}=1$
denoting that the SBSs in $\mathcal{B}_{l}$ are allocated
$R^{\mathrm{SBS}}_{j}$. We assume that all SBS clusters are served, which is
ensured through $\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}=1,\forall
l\in\mathcal{L}$ and $N^{\mathrm{MBS}}_{\mathrm{streams}}\geq L$. Thus, to
guarantee the predefined target BLER for cluster $\mathcal{B}_{l}$, it must
hold that
$\widetilde{\mathrm{SINR}}_{l}^{\mathrm{SBS}}\geq\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}\Gamma_{j}^{\mathrm{SBS}}$.

###  II-B Access Network: Distributed Unicast Transmissions from SBSs to UEs

In the access network, four pivotal aspects are addressed. First, _admission
control_ , i.e. deciding which UEs are served. Second, _rate selection_ , i.e.
choosing data rates for the served UEs. Third, _user association_ , i.e.
determining which subset of SBSs transmit to a served UE. Fourth,
_beamforming_.

Beamforming and User Association: Each SBS is equipped with a planar array of
$N^{\mathrm{SBS}}_{\mathrm{tx}}$ transmit antennas operating on Band 2 and
used for communication with the UEs, which have
$N^{\mathrm{UE}}_{\mathrm{rx}}=1$ receive antenna. A SBS $b\in\mathcal{B}_{l}$
serving a subset of UEs in $\mathcal{U}_{l}$ transmits multiple unicast
signals simultaneously, each signal targeting a specific UE. The instantaneous
unicast symbol for UE $u\in\mathcal{U}_{l}$ is denoted by $s_{l,u}$, with
$\mathbb{E}\left[s_{l,u}\right]=0$ and
$\mathbb{E}\left[\left|s_{l,u}\right|^{2}_{2}\right]=1$. In addition, the
beamforming vector from SBS $b\in\mathcal{B}_{l}$ transmitting $s_{l,u}$ to UE
$u\in\mathcal{U}_{l}$ is denoted by $\mathbf{w}_{b,u}$. Therefore, the
composite signal that SBS $b$ in $\mathcal{B}_{l}$ sends to the UEs in
$\mathcal{U}_{l}$ is represented by
$\mathbf{x}_{b}^{\mathrm{SBS}}=\sum_{u\in\mathcal{U}_{l}}\mathbf{w}_{b,u}s_{l,u}\kappa_{b,u}$,
where $\kappa_{b,u}$ is a binary variable that is $1$ when SBS
$b\in\mathcal{B}_{l}$ serves UE $u\in\mathcal{U}_{l}$ and $0$ otherwise. A
served UE $u\in\mathcal{U}_{l}$ receives its information from at least
$B_{\mathrm{min}}=1$ and at most $B_{\mathrm{max}}=B$ SBSs in
$\mathcal{B}_{l}$. The signal received by UE $u$ in $\mathcal{U}_{l}$ is given
by (3), where $n_{u}\sim\mathcal{CN}\left(0,\sigma_{\mathrm{UE}}^{2}\right)$
and $\mathbf{h}_{b,u}$ represents the channel between SBS $b$ and UE $u$.
Every UE perceives interference from within its own cluster and from
neighboring clusters. The SINR at UE $u$ in $\mathcal{U}_{l}$ is defined by
(4). When $\kappa_{b,u}=0$, no information is sent to the UE. The effective
beamforming vector is $\kappa_{b,u}\cdot\mathbf{w}_{b,u}$, which becomes a
zero-vector for unserved UEs, thus accomplishing the association between UEs
and SBSs.

$y_{u}^{\mathrm{UE}}=\underbrace{{\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}s_{l,u}\kappa_{b,u}}}_{\text{signal
for UE $u$ in cluster
$\mathcal{U}_{l}$}}+\underbrace{{\sum_{b\in\mathcal{B}_{l}}\sum_{\begin{subarray}{c}u^{\prime}\in\mathcal{U}_{l}\\\
u^{\prime}\neq
u\end{subarray}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u^{\prime}}s_{l,u^{\prime}}\kappa_{b,u^{\prime}}}}_{\text{interference
originated in cluster
$\mathcal{U}_{l}$}}+\underbrace{{\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\\\
l^{\prime}\neq
l\end{subarray}}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}s_{l^{\prime},u^{\prime}}\kappa_{b^{\prime},u^{\prime}}}}_{\text{aggregate
interference originated in clusters $\mathcal{U}_{l^{\prime}\neq
l}$}}+\underbrace{n_{u}}_{\text{noise}}\\\ $ (3)

$\mathrm{SINR}^{\mathrm{UE}}_{u}=\frac{{\left|\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\kappa_{b,u}\right|^{2}}}{{\sum_{\begin{subarray}{c}u^{\prime}\in\mathcal{U}_{l}\\\
u^{\prime}\neq
u\end{subarray}}\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u^{\prime}}\kappa_{b,u^{\prime}}\Big{|}^{2}}+{\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\\\
l^{\prime}\neq
l\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\kappa_{b^{\prime},u^{\prime}}\Big{|}^{2}}+\sigma_{\mathrm{UE}}^{2}}.$
(4)

$\displaystyle\mathcal{P}^{\prime}:$
$\displaystyle\max_{\begin{subarray}{c}\mathbf{m}_{l},\mathbf{w}_{b,u},\alpha_{u,j},\beta_{l,j},\kappa_{b,u}\end{subarray}}$
$\displaystyle
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\equiv\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\omega_{u}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}R_{j}^{\mathrm{UE}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\mathrm{C_{1}}:\leavevmode\nobreak\ $
$\displaystyle\alpha_{u,j}=\left\\{0,1\right\\},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$
$\displaystyle\mathrm{C_{2}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}\leq 1,\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},$
$\displaystyle\mathrm{C_{3}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{l\in\mathcal{L}}\left\|\mathbf{m}_{l}\right\|_{2}^{2}\leq
P_{\mathrm{tx}}^{\mathrm{MBS}},$
$\displaystyle\mathrm{\bar{C}_{4}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{u\in\mathcal{U}_{l}}\left\|\mathbf{w}_{b,u}\kappa_{b,u}\right\|_{2}^{2}\leq
P_{\mathrm{tx}}^{\mathrm{SBS}},\forall l\in\mathcal{L},b\in\mathcal{B}_{l},$
$\displaystyle\mathrm{\bar{C}_{5}}:\leavevmode\nobreak\ $
$\displaystyle\mathrm{SINR}^{\mathrm{UE}}_{u}\geq\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}\Gamma_{j}^{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},$
$\displaystyle\mathrm{C_{6}}:\leavevmode\nobreak\ $
$\displaystyle\kappa_{b,u}=\left\\{0,1\right\\},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l},$
$\displaystyle\mathrm{C_{7}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{u\in\mathcal{U}_{l}}\kappa_{b,u}\leq
N^{\mathrm{SBS}}_{\mathrm{streams}},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},$
$\displaystyle\mathrm{C_{8}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{u\in\mathcal{U}_{l}}\kappa_{b,u}\geq 1,\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},$
$\displaystyle\mathrm{C_{9}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{b\in\mathcal{B}_{l}}\kappa_{b,u}\leq
B_{\mathrm{max}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},$
$\displaystyle\mathrm{C_{10}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{b\in\mathcal{B}_{l}}\kappa_{b,u}\geq
B_{\mathrm{min}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},$
$\displaystyle\mathrm{C_{11}}:\leavevmode\nobreak\ $
$\displaystyle\beta_{l,j}=\left\\{0,1\right\\},\forall
l\in\mathcal{L},j\in\mathcal{J}^{\mathrm{SBS}},$
$\displaystyle\mathrm{C_{12}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}=1,\forall
l\in\mathcal{L},$ $\displaystyle\mathrm{C_{13}}:\leavevmode\nobreak\ $
$\displaystyle
W_{\mathrm{BW}}^{\mathrm{access}}\sum_{u\in\mathcal{U}_{l}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}R_{j}^{\mathrm{UE}}\leq
W_{\mathrm{BW}}^{\mathrm{backhaul}}\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}R_{j}^{\mathrm{SBS}},\forall
l\in\mathcal{L},$ $\displaystyle\mathrm{C_{14}}:\leavevmode\nobreak\ $
$\displaystyle\sum_{u\in\mathcal{U}_{l}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}=U_{\mathrm{served}},\forall
l\in\mathcal{L},$ $\displaystyle\mathrm{\bar{C}_{15}}:\leavevmode\nobreak\ $
$\displaystyle\widetilde{\mathrm{SINR}}_{l}^{\mathrm{SBS}}\geq\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}\Gamma_{j}^{\mathrm{SBS}},\forall
l\in\mathcal{L},$

Rate Selection and Admission Control: Similarly to Section II-A, the rate
assigned to a served UE can only be one within a set of predefined values. To
depict the rate selection for the UEs, we introduce the binary variables
$\alpha_{u,j}\in\left\\{0,1\right\\}$. These variables perform the dual task
of admission control and rate selection, which is ensured by
$\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}\leq 1,\forall
l\in\mathcal{L},u\in\mathcal{U}_{l}$, where $\mathcal{J}^{\mathrm{UE}}$
represents the set of possible rate values. A UE $u$ is served when
$\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}=1$, meaning that one rate
has been assigned. Otherwise, when
$\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}=0$, the UE is not served. We
denote the rates and target SINRs for UEs with $R_{j}^{\mathrm{UE}}$ and
$\Gamma_{j}^{\mathrm{UE}}$, respectively. To assign $R_{j}^{\mathrm{UE}}$ to
UE $u$, it is required that
$\mathrm{SINR}^{\mathrm{UE}}_{u}\geq\Gamma_{j}^{\mathrm{UE}}$,
$j\in\mathcal{J}^{\mathrm{UE}}$, for which we assume the same values shown in
Table III in Section III. Further, not all UEs shall be admitted since each
SBS can support up to $N^{\mathrm{SBS}}_{\mathrm{streams}}$ streams
simultaneously.

######  Proposition 1.

Due to existence of $\mathrm{C}_{1}-\mathrm{C}_{2}$, constraint
$\mathrm{\bar{C}_{5}}$ can be equivalently rewritten as

$\displaystyle\mathrm{C_{5}}:\mathrm{SINR}^{\mathrm{UE}}_{u}\geq\alpha_{u,j}\Gamma_{j}^{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

Proof: Because of $\mathrm{C}_{2}$, there is at most one variable at a time
that is $1$. As a result, the SINR constraints can be decomposed into multiple
constraints, each being related to only one binary variable.

######  Proposition 2.

Due to existence of $\mathrm{C_{6}}$, constraints
$\mathrm{\bar{C}_{4}}-\mathrm{C_{5}}$ can be equivalently rewritten as
$\mathrm{C_{17}}$, $\mathrm{C_{18}}$, $\mathrm{C_{19}}$, $\mathrm{C_{20}}$,
$\mathrm{\bar{C}_{21}}$, where

$\displaystyle\mathrm{\bar{C}_{4}}-\mathrm{C_{5}}=\begin{cases}\mathrm{C_{17}}:p_{b,u}\geq
0,\forall l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l},\\\
\mathrm{C_{18}}:\sum_{u\in\mathcal{U}_{l}}p_{b,u}\leq
P_{\mathrm{tx}}^{\mathrm{SBS}},\forall l\in\mathcal{L},b\in\mathcal{B}_{l},\\\
\mathrm{C_{19}}:p_{b,u}\leq\kappa_{b,u}P_{\mathrm{tx}}^{\mathrm{SBS}},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l},\\\
\mathrm{C_{20}}:\left\|\left[2\mathbf{w}_{b,u}^{H},\kappa_{b,u}-p_{b,u}\right]\right\|_{2}\leq\kappa_{b,u}+p_{b,u},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l},\\\
\mathrm{\bar{C}_{21}}:\frac{\big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\big{|}^{2}}{\sum_{\begin{subarray}{c}u^{\prime}\in\mathcal{U}_{l}\\\
u^{\prime}\neq
u\end{subarray}}\big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u^{\prime}}\big{|}^{2}+\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\\\
l^{\prime}\neq
l\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\big{|}^{2}+\sigma_{\mathrm{UE}}^{2}}\geq\alpha_{u,j}\Gamma_{j}^{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},\\\
\end{cases}$

Proof: See Appendix A.

######  Proposition 3.

Due to existence of $\mathrm{C_{1}}$, constraint $\mathrm{\bar{C}_{21}}$ can
be rewritten as $\mathrm{C_{21}}$, where

$\displaystyle\mathrm{C_{21}}:\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\Big{|}^{2}+\sigma_{\mathrm{UE}}^{2}\leq\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\Big{|}^{2}+\left(1-\alpha_{u,j}\right)^{2}Q_{u}^{2},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$

and
$Q_{u}^{2}=P_{\mathrm{tx}}^{\mathrm{SBS}}\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\left\|\mathbf{h}_{b^{\prime},u}\right\|_{2}^{2}+\sigma^{2}_{\mathrm{UE}}$
is an upper bound for the left-hand side (LHS) term of $\mathrm{C_{21}}$.

Proof: See Appendix B.

###  II-C Problem Formulation

We investigate the problem of joint optimization of _beamforming_ , _user
association_ , _rate selection_ , _admission control_ in the access network
and _beamforming_ , _rate selection_ in the backhaul network aiming to
maximize the weighted sum-rate at the access network (i,e., for the UEs),
which is formulated as $\mathcal{P}^{\prime}$ in the previous page.

In $\mathcal{P}^{\prime}$,
$R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)$ denotes
the weighted sum-rate achieved by all UEs in the access network. Besides,
$\omega_{u}$ represents the weight associated to UE $u$, which can be adjusted
by the network operator to assign different priorities, for instance, to
balance fairness among UEs. Formally, the objective function is expressed as
$R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\equiv
W_{\mathrm{BW}}^{\mathrm{access}}\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\omega_{u}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}R_{j}^{\mathrm{UE}}$.
However, since $W_{\mathrm{BW}}^{\mathrm{access}}$ is constant, we have
redefined it as
$R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\equiv\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\omega_{u}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}R_{j}^{\mathrm{UE}}$
without altering the nature of the problem.

Constraints $\mathrm{C_{1}}$, $\mathrm{C_{2}}$, $\mathrm{\bar{C}_{4}}$,
$\mathrm{\bar{C}_{5}}$, $\mathrm{C_{6}}$, $\mathrm{C_{7}}$, $\mathrm{C_{8}}$,
$\mathrm{C_{9}}$, $\mathrm{C_{10}}$, $\mathrm{C_{14}}$ are related to the
access network, $\mathrm{C_{3}}$, $\mathrm{C_{11}}$, $\mathrm{C_{12}}$,
$\mathrm{\bar{C}_{15}}$ are related to the backhaul network whereas
$\mathrm{C_{13}}$ is related to both networks. Constraints
$\mathrm{C_{1}}-\mathrm{C_{2}}$ depict the rate selection for all UEs,
constraint $\mathrm{C_{3}}$ restricts the transmit power of the MBS,
constraint $\mathrm{\bar{C}_{4}}$ restricts the transmit power of the SBSs,
constraint $\mathrm{\bar{C}_{5}}$ guarantees that the unicast SINR is larger
than the corresponding target SINR (specified in Table III), constraints
$\mathrm{C_{6}}-\mathrm{C_{8}}$ ensure that each SBS serves at least one UE
but cannot serve more UEs than the number of streams it can handle,
constraints $\mathrm{C_{9}}-\mathrm{C_{10}}$ ensure that each admitted UE is
served by at least $B_{\mathrm{min}}$ and by at most $B_{\mathrm{max}}$ SBSs,
constraints $\mathrm{C_{11}}-\mathrm{C_{12}}$ guarantee a rate selection for
every SBS cluster, constraint $\mathrm{C_{13}}$ guarantees that the total
access throughput in a cluster does not exceed the throughput of the
corresponding serving backhaul link, $\mathrm{C_{14}}$ ensures that there are
$U_{\mathrm{served}}$ served UEs per cluster, constraint
$\mathrm{\bar{C}_{15}}$ guarantees that the SINR per SBS cluster is larger
than the selected target SINR (specified in Table III).

Remark: In the strict sense, the integrality constraints (i.e.,
$\mathrm{C_{1}}$, $\mathrm{C_{6}}$, $\mathrm{C_{11}}$) make
$\mathcal{P}^{\prime}$ nonconvex. Nevertheless, in the MINLP literature, a
MINLP is referred to as nonconvex if it remains nonconvex even after excluding
the integral variables. Otherwise, it is called convex [41]. In general, both
convex and nonconvex MINLPs are NP-hard but the latter ones are more
challenging to solve. Specifically, $\mathcal{P}^{\prime}$ is a nonconvex
MINLP and the nonconvexity nature is conferred by the constraints
$\mathrm{\bar{C}_{4}}$, $\mathrm{\bar{C}_{5}}$, $\mathrm{\bar{C}_{15}}$.

##  III Proposed Problem Reformulation

In this section, we propose a series of transformations to simplify the
nonconvex constraints $\mathrm{\bar{C}_{4}}$, $\mathrm{\bar{C}_{5}}$,
$\mathrm{\bar{C}_{15}}$. The resulting reformulation $\mathcal{P}$ (shown in
Section III-D) is used in Section IV, Section VI, Section VII, where we
propose three algorithms: BnC-MISOCP, RnP-SOCP-1 and RnP-SOCP-2.

###  III-A Eliminating Additive Coupling between Binary Variables

To deal with the additive coupling of the binary variables at the right-hand
side (RHS) of $\mathrm{\bar{C}_{5}}$ (i.e. sum of variables), we separate
$\mathrm{\bar{C}_{5}}$ into multiple equivalent constraints, as described in
_Proposition 1_.

###  III-B Eliminating the Multiplicative Coupling between Continuous and
Binary Variables

To deal with the multiplicative coupling between the unicast beamforming
vectors and binary variables (in the form $\mathbf{w}_{b,u}\kappa_{b,u}$) in
$\mathrm{\bar{C}_{4}}-\mathrm{C_{5}}$, we reformulate such interdependencies
as equivalent additive couplings, which are simpler to handle, as described in
_Proposition 2_. In addition, note that $\mathrm{C_{17}-C_{20}}$ are convex,
whereas $\mathrm{\bar{C}_{21}}$ is a nonconvex mixed-integer nonlinear
constraint. To circumvent the involved structure $\mathrm{\bar{C}_{21}}$, we
remodel it (without loss of optimality) harnessing the big-M method [42],
which allows to remove the multiplicative tie between the beamformers and
binary variables, as described in _Proposition 3_. Finally, because constraint
$\mathrm{\bar{C}_{15}}$ has a similar structure as $\mathrm{\bar{C}_{5}}$, we
can reformulate it in an equivalent manner, as described in _Proposition 4_.

######  Proposition 4.

Due to existence of $\mathrm{C_{11}}$, constraint $\mathrm{\bar{C}_{15}}$ can
be equivalently recast as $\mathrm{C_{15}}$, where

$\displaystyle\mathrm{C_{15}}:\sum_{l^{\prime}\in\mathcal{L}}\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l^{\prime}}\right|^{2}+\sigma^{2}_{\mathrm{SBS}}\leq\left(1+{\Gamma_{j}^{\mathrm{SBS}}}^{-1}\right)\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right|^{2}+\left(1-\beta_{l,j}\right)^{2}Q_{b}^{2},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}},$

and
$Q_{b}^{2}=P_{\mathrm{tx}}^{\mathrm{MBS}}\left\|\mathbf{g}_{b}\right\|_{2}^{2}+\sigma^{2}_{\mathrm{SBS}}$
is an upper bound for the LHS term of $\mathrm{C_{15}}$.

Proof: The proof is along the same lines as the procedures adopted in
Proposition 1, Proposition 2 and Proposition 3. Therefore, it is omitted.

######  Proposition 5.

The nonconvex constraints $\mathrm{C_{21}}-\mathrm{C_{22}}$ can be
equivalently expressed as SOC constraints $\mathrm{C_{23}-C_{25}}$, i.e.,

$\displaystyle\mathrm{C_{21}-C_{22}}=\begin{cases}\mathrm{C_{23}}:\left\|\left[\bar{\mathbf{h}}_{u}^{H}\mathbf{W},\sigma_{\mathrm{UE}}\right]\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}+\left(1-\alpha_{u,j}\right)Q_{u},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},\\\
\mathrm{C_{24}}:\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}\geq\alpha_{u,j}\sqrt{\Gamma_{j}^{\mathrm{UE}}}\sigma_{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},\\\
\mathrm{C_{25}}:\mathsf{Im}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}=0,\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.\\\
\end{cases}$

Proof: See Appendix C.

######  Proposition 6.

The nonconvex constraints $\mathrm{C_{15}}-\mathrm{C_{16}}$ can be recast as
the more conservative SOC constraints $\mathrm{C_{26}-C_{27}}$, where

$\displaystyle\mathrm{C_{15}-C_{16}}=\begin{cases}\mathrm{C_{26}}:\left\|\left[\mathbf{g}_{b}^{H}\mathbf{M},\sigma_{\mathrm{SBS}}\right]\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{SBS}}}^{-1}}\mathsf{Re}\left\\{\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right\\}+\left(1-\beta_{l,j}\right)Q_{b},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}},\\\
\mathrm{C_{27}}:\mathsf{Re}\left\\{\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right\\}\geq\beta_{l,j}\sqrt{\Gamma_{j}^{\mathrm{SBS}}}\sigma_{\mathrm{SBS}},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}}.\end{cases}$

Proof: See Appendix D.

###  III-C Adding Cuts to Tighten the Feasible Set

To reduce the number of branches to be evaluated by MINLP solvers, we include
valid inequalities (cuts) for certain constraints involving integer variables.
Thus, we add the constraints $\mathrm{C_{16}}$ and $\mathrm{C_{22}}$, defined
as

$\displaystyle\mathrm{C_{16}}:\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right|^{2}\geq\beta_{l,j}\Gamma_{j}^{\mathrm{SBS}}\sigma^{2}_{\mathrm{SBS}},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}},$
$\displaystyle\mathrm{C_{22}}:\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\Big{|}^{2}\geq\alpha_{u,j}\Gamma_{j}^{\mathrm{UE}}\sigma^{2}_{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

Note that $\mathrm{C_{16}}$ is a lower bound for the multicast SINR numerator,
which becomes tight when the interference term is zero. This constraint is
always satisfied when $\beta_{l,j}$ are binary thus reducing the feasible set
and tightening the problem relaxation when the binary variables are recast as
real values (as in the proposed algorithms in Section VI and Section VII).
Adding $\mathrm{C_{16}}$ does not change the nature of the problem nor affects
its optimality. Similarly, $\mathrm{C_{22}}$ is a lower bound for the unicast
SINR numerator.

###  III-D Redefining the Problem

After applying the transformations in Section III-A, Section III-B and Section
III-C, the nonconvex constraints $\mathrm{\bar{C}_{4}}$,
$\mathrm{\bar{C}_{5}}$, $\mathrm{\bar{C}_{15}}$ have been replaced by the
convex constraints $\mathrm{C_{17}}$, $\mathrm{C_{18}}$, $\mathrm{C_{19}}$,
$\mathrm{C_{20}}$ and the nonconvex constraints $\mathrm{C_{15}}$,
$\mathrm{C_{21}}$. In addition, the nonconvex constraints $\mathrm{C_{16}}$,
$\mathrm{C_{22}}$ have been added to contract the feasible set. Collecting
these outcomes, we define $\mathcal{P}$ as

$\displaystyle\mathcal{P}:$
$\displaystyle\max_{\begin{subarray}{c}\mathbf{m}_{l},\mathbf{w}_{b,u},p_{b,u},\\\
\alpha_{u,j},\beta_{l,j},\kappa_{b,u}\end{subarray}}$ convex: $\displaystyle
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
convex:
$\displaystyle\mathrm{C_{2}}-\mathrm{C_{3}},\mathrm{C_{7}}-\mathrm{C_{10}},\mathrm{C_{12}}-\mathrm{C_{14}},$
$\displaystyle\mathrm{C_{17}}-\mathrm{C_{20}},$ nonconvex:
$\displaystyle\mathrm{C_{15}}-\mathrm{C_{16}},\mathrm{C_{21}}-\mathrm{C_{22}},$
binary: $\displaystyle\mathrm{C_{1}},\mathrm{C_{6}},\mathrm{C_{11}}.$

Remark: Notice that $\mathcal{P}$ is also a nonconvex MINLP and has the same
optimal solution as $\mathcal{P}^{\prime}$ since the introduced
transformations do not affect the original feasible set. However, the
structure of $\mathcal{P}$ is simpler, thus allowing us to tailor algorithms
for solving the problem more efficiently.

##  IV BnC-MISOCP: Proposed MISOCP Formulation

In this section, we recast $\mathcal{P}$ as a MISOCP by transforming the
nonconvex constraints into convex ones. We remodel
$\mathrm{C_{21}}-\mathrm{C_{22}}$ as convex constraints and replace
$\mathrm{C_{15}}-\mathrm{C_{16}}$ with convex inner surrogates.

###  IV-A Transforming Nonconvex Constraints into Convex Constraints

To deal with the nonconvex constraints $\mathrm{C_{21}}-\mathrm{C_{22}}$, we
recast them as convex conic constraints as they have hidden convexity. To
simplify notation, we first rewrite $\mathrm{C_{21}}-\mathrm{C_{22}}$ as

$\displaystyle\mathrm{C_{21}}:\left|\bar{\mathbf{h}}_{u}^{H}\mathbf{W}\right|^{2}+\sigma^{2}_{\mathrm{UE}}\leq\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\left(1-\alpha_{u,j}\right)^{2}Q_{u}^{2},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$
$\displaystyle\mathrm{C_{22}}:\alpha_{u,j}\Gamma_{j}^{\mathrm{UE}}\sigma^{2}_{\mathrm{UE}}\leq\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|^{2},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$

where
$\big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\big{|}^{2}=\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|^{2},u\in\mathcal{U}_{l}$
and
$\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\big{|}^{2}=\left|\bar{\mathbf{h}}_{u}^{H}\mathbf{W}\right|^{2}$.
In particular,
$\mathbf{h}_{u}=\left[\mathbf{h}_{b_{1},u}^{H},\cdots,\mathbf{h}_{b_{B},u}^{H}\right]^{H}$
and
$\mathbf{w}_{u}=\left[\mathbf{w}_{b_{1},u}^{H},\cdots,\mathbf{w}_{b_{B},u}^{H}\right]^{H}$,
denote respectively the channels and beamforming vectors from all SBS in the
same cluster that UE $u$ is located. Further, $\bar{\mathbf{h}}_{u}$ denotes
the channel between UE $u$ and all SBSs in the system whereas $\mathbf{W}$ is
a block diagonal matrix collecting all beamforming vectors between SBSs and
UEs. After applying these changes, we are in the position of expressing the
nonconvex constraints $\mathrm{C_{21}}-\mathrm{C_{22}}$ as exactly equivalent
SOC constraints, as described in _Proposition 5_.

###  IV-B Recasting Nonconvex Constraints as Convex Inner Approximations

To circumvent the nonconvex constraints $\mathrm{C_{15}}$, $\mathrm{C_{16}}$,
we replace them by convex surrogates. Assuming that
$\mathbf{M}=\left[\mathbf{m}_{1},\cdots,\mathbf{m}_{L}\right]$, we express
$\mathrm{C_{15}}$ as

$\displaystyle\mathrm{C_{15}}:\left\|\mathbf{g}_{b}^{H}\mathbf{M}\right\|^{2}_{2}+\sigma^{2}_{\mathrm{SBS}}\leq\left(1+{\Gamma_{j}^{\mathrm{SBS}}}^{-1}\right)\left|\mathbf{g}_{b}^{H}\mathbf{m}_{l}\right|^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left(1-\beta_{l,j}\right)^{2}Q_{b}^{2},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}}.$

Using this expression, we reformulate $\mathrm{C_{15}}-\mathrm{C_{16}}$ as
convex inner SOC approximations, as stated in _Proposition 6_. If constraints
$\mathrm{C_{26}}-\mathrm{C_{27}}$ are satisfied, then
$\mathrm{C_{15}}-\mathrm{C_{16}}$ are automatically guaranteed because the
feasible set of $\mathrm{C_{26}}-\mathrm{C_{27}}$ is contained in that of
$\mathrm{C_{15}}-\mathrm{C_{16}}$. Therefore, they are called inner
approximations.

###  IV-C Summarizing the Changes

After applying the transformations above, we define the following problem,

$\displaystyle\mathcal{P}_{0}:$
$\displaystyle\max_{\begin{subarray}{c}\mathbf{m}_{l},\mathbf{w}_{b,u},p_{b,u},\\\
\alpha_{u,j},\beta_{l,j},\kappa_{b,u}\end{subarray}}$ convex: $\displaystyle
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
convex:
$\displaystyle\mathrm{C_{2}}-\mathrm{C_{3}},\mathrm{C_{7}}-\mathrm{C_{10}},\mathrm{C_{12}}-\mathrm{C_{14}},$
$\displaystyle\mathrm{C_{17}}-\mathrm{C_{20}},\mathrm{C_{23}}-\mathrm{C_{27}},$
binary: $\displaystyle\mathrm{C_{1}},\mathrm{C_{6}},\mathrm{C_{11}},$

which is an inner approximation of problem $\mathcal{P}$ due to
convexification of its original feasible set upon replacing
$\mathrm{C_{15}}-\mathrm{C_{16}}$ by $\mathrm{C_{26}}-\mathrm{C_{27}}$. Thus,
any feasible solution to $\mathcal{P}_{0}$ will also be feasible to
$\mathcal{P}^{\prime}$ and $\mathcal{P}$. Here, $\mathcal{P}_{0}$ has
$N_{v}=2LN_{\mathrm{tx}}^{\mathrm{MBS}}+2LBUN_{\mathrm{tx}}^{\mathrm{SBS}}+2LBU+LJ^{\mathrm{SBS}}+LUJ^{\mathrm{UE}}$
variables, $N_{l}=3L+2LU+3LB+2LBU+3LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$
linear constraints and $N_{c}=1+LBU+LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$
convex constraints. The complexity is
$\mathcal{O}\left(N_{s}(N_{v})^{3}(N_{l}+N_{c})\right)$, where $N_{s}$ is the
total number of evaluations needed by the mixed-integer (MIP) solver.

Remark: Note that $\mathcal{P}_{0}$ is a convex MINLP, and as such it can be
solved optimally by MIP solvers which exploit BnC techniques to prune
infeasible solutions thus reducing the search space of the problem. Although
BnC techniques can explore the binary space more efficiently and are faster
than exhaustive search, they may still require a considerable amount of time
to find the optimum, specially when the number of integral variables is large
as in $\mathcal{P}_{0}$. Thus, in order to expedite this process, we propose
suboptimal algorithms in Section VI and Section VII based on integrality
relaxation and penalization.

##  V Proposed Bounds

We derive an _upper bound_ and a _lower bound_ for $\mathcal{P}_{0}$. The
upper bound is defined as a MISOCP whereas the lower bound is a system- and
problem-specific rate value. When not possible to obtain a solution for
$\mathcal{P}_{0}$ (due to high time complexity), the upper and lower bounds
will be used as benchmarks for the developed algorithms in Section VI and
Section VII.

Upper Bound (UB): While the weighted sum-rate is a mechanism to balance rates,
i.e., to give higher priorities to the least favored UEs, the actual aggregate
rate in the network is given by the sum-rate
$R_{\mathrm{sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)=W_{\mathrm{BW}}^{\mathrm{access}}\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}R_{j}^{\mathrm{UE}}$
(without the weights). Thus, note that
$R_{\mathrm{sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)$ is
related to constraint $\mathrm{C_{13}}$, which ensures that the access sum-
rate per cluster does not exceed the rate of the serving backhaul link.
Therefore, the access sum-rate
$R_{\mathrm{sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)$ is
bounded from above by the backhaul sum-rate, defined as
$R_{\mathrm{sum}}^{\mathrm{backhaul}}\left(\boldsymbol{\beta}\right)\triangleq
W_{\mathrm{BW}}^{\mathrm{backhaul}}\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}R_{j}^{\mathrm{SBS}}$,
i.e., $R_{\mathrm{sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\leq
R_{\mathrm{sum}}^{\mathrm{backhaul}}\left(\boldsymbol{\beta}\right)$. Since
the backhaul sum-rate depends only on $\mathbf{m}_{l}$ and $\beta_{l,j}$, the
upper bound is given by

$\displaystyle\mathcal{P}_{\mathrm{UB}}:$
$\displaystyle\max_{\begin{subarray}{c}\mathbf{m}_{l},\beta_{l,j}\end{subarray}}\leavevmode\nobreak\
R_{\mathrm{sum}}^{\mathrm{backhaul}}\left(\boldsymbol{\beta}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{C_{3}},\mathrm{C_{11}},\mathrm{C_{12}},\mathrm{{C}_{26}},\mathrm{{C}_{27}},$

which is a MISOCP that can be solved optimally. _The upper bound essentially
maximizes the backhaul network throughput without considering the access
network requirements._ Note that $\mathcal{P}_{\mathrm{UB}}$ has
$N_{v}=LJ^{\mathrm{SBS}}+2LN_{\mathrm{tx}}^{\mathrm{MBS}}$ variables,
$N_{l}=L+LBJ^{\mathrm{SBS}}$ linear constraints and
$N_{c}=1+LBJ^{\mathrm{SBS}}$ convex constraints. Thus, its complexity is
$\mathcal{O}\left(N_{s}(N_{v})^{3}(N_{l}+N_{c})\right)$, where $N_{s}$
represents the total number of evaluations needed by the MIP solver.

Remark: $\mathcal{P}_{\mathrm{UB}}$ can be interpreted as joint multigroup
multicast beamforming and rate selection, which has not been investigated
before. A similar problem was studied in [43] but with continuous rates.
Although we do not investigate this new problem alone but in conjunction with
the additional access network constraints, we believe it is important to
highlight its novelty as it represents the discrete counterpart of the
aforementioned problem thus filling a gap in the existing literature and
opening new avenues of research.

Lower Bound (LB): The lower bound is based on the analysis of
$\mathcal{P}_{0}$. From constraint $\mathrm{{C}_{13}}$, a number of
$U_{\mathrm{served}}$ UEs per cluster needs to be served. In the worst case,
these UEs are allocated the lowest rate possible, which based on Table III,
corresponds to $R^{\mathrm{UE}}_{1}=0.2344$ bps/Hz. With $L$ clusters, the
minimum sum-rate at the access network is defined as $R_{\mathrm{sum-
min}}^{\mathrm{access}}=R^{\mathrm{UE}}_{1}\cdot
W_{\mathrm{BW}}^{\mathrm{access}}\cdot U_{\mathrm{served}}\cdot L$ bps. We
underline that this bound corresponds to the worst possible case in which the
UEs are minimally served while still satisfying the system constraints.

##  VI RnP-SOCP-1: Proposed SOCP Formulation

_This formulation is derived from problem $\mathcal{P}_{0}$_. We propose a
relax-and-penalize SOCP algorithm denoted by RnP-SOCP-1, which iteratively
optimizes a SOCP. To cope with the integrality constraints $\mathrm{C_{1}}$,
$\mathrm{C_{6}}$, $\mathrm{C_{11}}$, we replace them with the intersection of
two continuous sets [44], as described in _Proposition 7_.

######  Proposition 7.

The constraints $\mathrm{C_{1}}$, $\mathrm{C_{6}}$, $\mathrm{C_{11}}$ can be
equivalently expressed as,

$\displaystyle\mathrm{C_{1}}=\begin{cases}\mathrm{X_{1}}:0\leq\alpha_{u,j}\leq
1,\\\ \mathrm{Z_{1}}:\sum_{l,u,j}\alpha_{u,j}-\alpha_{u,j}^{2}\leq
0,\end{cases}$
$\displaystyle\mathrm{C_{6}}=\begin{cases}\mathrm{X_{2}}:0\leq\kappa_{b,u}\leq
1,\\\ \mathrm{Z_{2}}:\sum_{l,b,u}\kappa_{b,u}-\kappa_{b,u}^{2}\leq
0,\end{cases}$
$\displaystyle\mathrm{C_{11}}=\begin{cases}\mathrm{X_{3}}:0\leq\beta_{l,j}\leq
1,\\\ \mathrm{Z_{3}}:\sum_{l,j}\beta_{l,j}-\beta_{l,j}^{2}\leq 0.\end{cases}$

Proof: It is straightforward to see that $\mathrm{X_{1}}$ and $\mathrm{Z_{1}}$
intersect only at $\left\\{0,1\right\\}$. Thus, we omit further details.

Notice that constraints $\mathrm{X_{1}}-\mathrm{X_{3}}$ are convex whereas
$\mathrm{Z_{1}}-\mathrm{Z_{3}}$ are nonconvex. Considering _Proposition 7_ ,
we define

$\displaystyle{\mathcal{P}}_{1}:\max_{\begin{subarray}{c}\boldsymbol{\Theta}\end{subarray}}\leavevmode\nobreak\
\leavevmode\nobreak\
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\underbrace{\boldsymbol{\Theta}\in\mathscr{D}}_{\text{{convex}}},\underbrace{\mathrm{Z_{1}}-\mathrm{Z_{3}}}_{\text{{nonconvex}}}$

which is equivalent to $\mathcal{P}_{0}$. Here,
$\boldsymbol{\Theta}=\left(\mathbf{M},\mathbf{W},\mathbf{p},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)$
groups all the optimization variables and $\mathscr{D}$ denotes the feasible
set spanned by the convex constraints
$\mathrm{X_{1}}-\mathrm{X_{3}},\mathrm{C_{2}}-\mathrm{C_{3}},\mathrm{C_{7}}-\mathrm{C_{10}},\mathrm{C_{12}}-\mathrm{C_{14}},\mathrm{C_{17}}-\mathrm{C_{20}},\mathrm{C_{23}}-\mathrm{C_{27}}$.
Although ${\mathcal{P}}_{1}$ is a nonconvex MINLP, its nonconvexity is only
due to simple polynomial constraints $\mathrm{Z_{1}}-\mathrm{Z_{3}}$, which
belong to the class of difference of convex (DC) functions.

Since ${\mathcal{P}}_{1}$ is challenging to solve optimally, we aim to obtain
a locally optimal solution. To find a solution for ${\mathcal{P}}_{1}$, we
devise an algorithm based on the minorization-maximization (MM) principle. To
cope with $\mathrm{Z_{1}}-\mathrm{Z_{3}}$, we include them as penalty terms in
the objective function [45]. Thus, we define $\widetilde{\mathcal{P}}_{1}$ in
(7)

$\widetilde{\mathcal{P}}_{1}:\max_{\begin{subarray}{c}\boldsymbol{\Theta}\in\mathscr{D}\end{subarray}}\leavevmode\nobreak\
\leavevmode\nobreak\
R\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)\triangleq
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\underbrace{-\lambda_{\alpha}f_{\alpha}\left(\boldsymbol{\alpha}\right)-\lambda_{\beta}f_{\beta}\left(\boldsymbol{\beta}\right)-\lambda_{\kappa}f_{\kappa}\left(\boldsymbol{\kappa}\right)}_{\text{nonconvex
DC functions}}$ (7)

$\displaystyle f_{\alpha}\left(\boldsymbol{\alpha}\right)\triangleq
p_{\alpha}\left(\boldsymbol{\alpha}\right)+q_{\alpha}\left(\boldsymbol{\alpha}\right),$
$\displaystyle
p_{\alpha}\left(\boldsymbol{\alpha}\right)\triangleq\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j},$
$\displaystyle
q_{\alpha}\left(\boldsymbol{\alpha}\right)\triangleq-\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\sum_{j\in\mathcal{J}^{\mathrm{UE}}}\alpha_{u,j}^{2},$
$\displaystyle f_{\beta}\left(\boldsymbol{\beta}\right)\triangleq
p_{\beta}\left(\boldsymbol{\beta}\right)+q_{\beta}\left(\boldsymbol{\beta}\right),$
$\displaystyle
p_{\beta}\left(\boldsymbol{\beta}\right)\triangleq\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j},$
$\displaystyle
q_{\beta}\left(\boldsymbol{\beta}\right)\triangleq-\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}^{\mathrm{SBS}}}\beta_{l,j}^{2},$
$\displaystyle f_{\kappa}\left(\boldsymbol{\kappa}\right)\triangleq
p_{\kappa}\left(\boldsymbol{\kappa}\right)+q_{\kappa}\left(\boldsymbol{\kappa}\right),$
$\displaystyle
p_{\kappa}\left(\boldsymbol{\kappa}\right)\triangleq\sum_{l\in\mathcal{L}}\sum_{b\in\mathcal{B}_{l}}\sum_{u\in\mathcal{U}_{l}}\kappa_{b,u},$
$\displaystyle
q_{\kappa}\left(\boldsymbol{\kappa}\right)\triangleq-\sum_{l\in\mathcal{L}}\sum_{b\in\mathcal{B}_{l}}\sum_{u\in\mathcal{U}_{l}}\kappa_{b,u}^{2}.$

$\widetilde{\mathcal{P}}_{1}^{(t)}:\max_{\begin{subarray}{c}\boldsymbol{\Theta}\in\mathscr{D}\end{subarray}}\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{R}^{(t)}\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)\triangleq
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)-\lambda_{\alpha}\tilde{f}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)-\lambda_{\beta}\tilde{f}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)-\lambda_{\kappa}\tilde{f}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)$
(8)

$\displaystyle\tilde{f}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)\triangleq
p_{\alpha}\left(\boldsymbol{\alpha}\right)+\tilde{q}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{f}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)\triangleq
p_{\beta}\left(\boldsymbol{\beta}\right)+\tilde{q}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{f}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)\triangleq
p_{\kappa}\left(\boldsymbol{\kappa}\right)+\tilde{q}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right).$

where $\lambda_{\alpha}\geq 0$, $\lambda_{\beta}\geq 0$, $\lambda_{\kappa}\geq
0$. Whenever $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$,
$\boldsymbol{\kappa}$ are not binary, the functions
$f_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$f_{\beta}\left(\boldsymbol{\beta}\right)$,
$f_{\kappa}\left(\boldsymbol{\kappa}\right)$ are positive. By including them
in the objective, they can be used as a measure of the degree of satisfaction
of the binary constraints, with $\lambda_{\alpha}$, $\lambda_{\beta}$,
$\lambda_{\kappa}$ representing penalty factors. Problems $\mathcal{P}_{1}$
and $\widetilde{\mathcal{P}}_{1}$ are related in the following sense. If
_Proposition 8_ is satisfied, $\mathcal{P}_{1}$ and
$\widetilde{\mathcal{P}}_{1}$ become equivalent [46, 45].

######  Proposition 8.

The optimization problems $\mathcal{P}_{1}$ and $\widetilde{\mathcal{P}}_{1}$
are equivalent for sufficiently large values of $\lambda_{\alpha}$,
$\lambda_{\beta}$, $\lambda_{\kappa}$, in which case both problems attain the
same optimal value and solution.

Proof: See Appendix E.

To solve $\widetilde{\mathcal{P}}_{1}$, the complication is in the objective
since $f_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$f_{\beta}\left(\boldsymbol{\beta}\right)$,
$f_{\kappa}\left(\boldsymbol{\kappa}\right)$ are nonconvex DC functions. Thus,
we apply first-order approximations to
$q_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$q_{\beta}\left(\boldsymbol{\beta}\right)$,
$q_{\kappa}\left(\boldsymbol{\kappa}\right)$, and define

$\displaystyle\tilde{q}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)\triangleq
q_{\alpha}\left(\boldsymbol{\alpha}^{(t-1)}\right)+\nabla_{\boldsymbol{\alpha}}q_{\alpha}^{T}\left(\boldsymbol{\alpha}^{(t-1)}\right)\left(\boldsymbol{\alpha}-\boldsymbol{\alpha}^{(t-1)}\right),$
$\displaystyle\tilde{q}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)\triangleq
q_{\beta}\left(\boldsymbol{\beta}^{(t-1)}\right)+\nabla_{\boldsymbol{\beta}}q_{\beta}^{T}\left(\boldsymbol{\beta}^{(t-1)}\right)\left(\boldsymbol{\beta}-\boldsymbol{\beta}^{(t-1)}\right),$
$\displaystyle\tilde{q}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)\triangleq
q_{\kappa}\left(\boldsymbol{\kappa}^{(t-1)}\right)+\nabla_{\boldsymbol{\kappa}}q_{\kappa}^{T}\left(\boldsymbol{\kappa}^{(t-1)}\right)\left(\boldsymbol{\kappa}-\boldsymbol{\kappa}^{(t-1)}\right),$

where $\tilde{q}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)\geq
q_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$\tilde{q}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)\geq
q_{\beta}\left(\boldsymbol{\beta}\right)$,
$\tilde{q}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)\geq
q_{\kappa}\left(\boldsymbol{\kappa}\right)$ are outer linear approximations
for $q_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$q_{\beta}\left(\boldsymbol{\beta}\right)$,
$q_{\kappa}\left(\boldsymbol{\kappa}\right)$, respectively.

Here, $\boldsymbol{\alpha}^{\left(t-1\right)}$,
$\boldsymbol{\beta}^{\left(t-1\right)}$,
$\boldsymbol{\kappa}^{\left(t-1\right)}$ denote a feasible solution (i.e.
reference point for linearization) whereas $\nabla_{\mathbf{x}}$ represents
the derivative with respect to variable $\mathbf{x}$. Using the MM principle
and constructing a sequence of surrogate functions
$\tilde{q}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)$,
$\tilde{q}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)$,
$\tilde{q}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)$ at every iteration
$t$, we solve problem $\widetilde{\mathcal{P}}_{1}^{(t)}$ defined in (8),
which is a SOCP where
$\tilde{f}_{\alpha}^{(t)}\left(\boldsymbol{\alpha}\right)\geq
f_{\alpha}\left(\boldsymbol{\alpha}\right)$,
$\tilde{f}_{\beta}^{(t)}\left(\boldsymbol{\beta}\right)\geq
f_{\beta}\left(\boldsymbol{\beta}\right)$,
$\tilde{f}_{\kappa}^{(t)}\left(\boldsymbol{\kappa}\right)\geq
f_{\kappa}\left(\boldsymbol{\kappa}\right)$. In particular, problem
$\widetilde{\mathcal{P}}_{1}^{(t)}$ is convex and can be solved using
interior-point methods. By solving $\widetilde{\mathcal{P}}_{1}^{(t)}$
iteratively, we show in _Proposition 9_ and _Proposition 10_ , that
$\widetilde{\mathcal{P}}_{1}^{(t)}$ is a global lower bound of
$\widetilde{\mathcal{P}}_{1}$ and the obtained solution is a KKT point.

######  Proposition 9.

Problem $\widetilde{\mathcal{P}}_{1}^{(t)}$ is a global lower bound for
$\widetilde{\mathcal{P}}_{1}$ since
$\tilde{R}^{(t)}\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)\leq
R\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)$.

Proof: See Appendix F.

######  Proposition 10.

Starting from a feasible point
$\boldsymbol{\Theta}^{(0)}=\left(\cdot,\cdot,\cdot,\boldsymbol{\alpha}^{(0)},\boldsymbol{\beta}^{(0)},\boldsymbol{\kappa}^{(0)}\right)$,
the sequence of solutions
$\boldsymbol{\Theta}^{(t)}=\left(\mathbf{M}^{(t)},\mathbf{W}^{(t)},\mathbf{p}^{(t)},\boldsymbol{\alpha}^{(t)},\boldsymbol{\beta}^{(t)},\boldsymbol{\kappa}^{(t)}\right)$,
for $t\geq 1$, obtained by iteratively solving
$\widetilde{\mathcal{P}}_{1}^{(t)}$ constitutes a sequence of enhanced points
for $\widetilde{\mathcal{P}}_{1}$, which converges to a KKT point.

Proof: See Appendix G.

To solve $\widetilde{\mathcal{P}}^{(t)}_{1}$, a feasible point
$\boldsymbol{\Theta}^{(0)}$ is needed to guarantee convergence as explained in
_Proposition 10_. We generate random initial feasible points and test them for
feasibility, as described in [47]. We use the best of these points as the
initial $\boldsymbol{\Theta}^{(0)}$, and iteratively solve
$\widetilde{\mathcal{P}}^{(t)}_{1}$ as shown in Algorithm 1.

_Step 1:_ |  Define $N_{\mathrm{iter}}$, $\delta$, $\lambda_{\alpha}$, $\lambda_{\beta}$, $\lambda_{\kappa}$.
---|---
_Step 2:_ |  Find an initial point $\boldsymbol{\Theta}^{(0)}=\left(\cdot,\cdot,\cdot,\boldsymbol{\alpha}^{(0)},\boldsymbol{\beta}^{(0)},\boldsymbol{\kappa}^{(0)}\right)$
|  using $\left\\{0,1\right\\}$ values.
_Step 3:_ |  Initialize $t=1$.
_Step 4:_ |  Solve $\mathcal{P}_{1}^{(t)}$ using $\boldsymbol{\Theta}^{(t-1)}$.
_Step 5:_ |  Assign $\boldsymbol{\Theta}^{(t)}\leftarrow\boldsymbol{\Theta}^{(t-1)}$.
_Step 6:_ |  Update the iteration index $t$ by one, i.e. $t=t+1$.
_Step 7:_ |  Verify if the stop criterion is attained. Otherwise,
|  return to _Step 4_.

Algorithm 1 Optimization of $\mathcal{P}_{1}$

We stop the iterative process when a criterion has been met, i.e.,
$t=N_{\mathrm{iter}}$ or
$\tilde{R}^{(t)}\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)-\tilde{R}^{(t-1)}\left(\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)\leq\delta$.
The computational complexity of $\widetilde{\mathcal{P}}^{(t)}_{1}$ is similar
to that of one evaluation of $\mathcal{P}_{0}$. In particular,
$N_{v}=2LN_{\mathrm{tx}}^{\mathrm{MBS}}+2LBUN_{\mathrm{tx}}^{\mathrm{SBS}}+2LBU+LJ^{\mathrm{SBS}}+LUJ^{\mathrm{UE}}$
variables,
$N_{l}=3L+2LU+3LB+2LJ^{\mathrm{SBS}}+4LBU+5LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$
linear constraints and $N_{c}=1+LBU+LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$
convex constraints. Therefore, the complexity is
$\mathcal{O}\left(N_{\mathrm{iter}}(N_{v})^{3}(N_{l}+N_{c})\right)$, where
$N_{\mathrm{iter}}$ is the number of iterations.

##  VII RnP-SOCP-2: Proposed SOCP Formulation

_This formulation is derived from problem $\mathcal{P}_{1}$_. We propose an
alternative relax-and-penalize SOCP algorithm, denoted by RnP-SOCP-2, whose
main characteristic is the reduced number of optimization variables compared
to RnP-SOCP-1, thus allowing to obtain solutions faster. To decrease the large
number of optimization variables in $\mathcal{P}_{1}$, (essentially dominated
by the number of antennas at the MBS and SBSs) we adopt a simpler approach in
which instead of optimizing high-dimensional beamforming vectors, we only
optimize their gains.

In particular, we define the variables $v_{b,u}$ and $t_{l}$ as the gains
(i.e., amplitude and phase) of predefined unicast (i.e., access) and multicast
(i.e., backhaul) beamforming vectors $\widehat{\mathbf{w}}_{b,u}$ and
$\widehat{\mathbf{m}}_{l}$, respectively, such that
$\mathbf{m}_{l}={t_{l}}\widehat{\mathbf{m}}_{l}$,
$\mathbf{w}_{b,u}={v_{b,u}}\widehat{\mathbf{w}}_{b,u}$,
$\left\|\widehat{\mathbf{m}}_{l}\right\|^{2}_{2}=1$,
$\left\|\widehat{\mathbf{w}}_{b,u}\right\|^{2}_{2}=1$. We design the unit-norm
unicast beamforming vectors $\widehat{\mathbf{w}}_{b,u}$ using the zero-
forcing (ZF) criterion. On the other hand, the unit-norm multicast beamforming
vectors $\widehat{\mathbf{m}}_{l}$ are obtained experimentally upon evaluating
the upper bound $\mathcal{P}_{\mathrm{UB}}$ for multiple realizations with
varying degrees of shadowing and small-scale fading, and then taking the
average of all these beamforming vectors. This procedure allows us to obtain a
fair estimation of the multicast beamforming vectors because the SBSs are
stationary and therefore the MBS-SBS channels geometry do not change
substantially. Thus, the constraints that are affected by
$\mathbf{m}_{l}={t_{l}}\widehat{\mathbf{m}}_{l}$,
$\mathbf{w}_{b,u}={v_{b,u}}\widehat{\mathbf{w}}_{b,u}$ are $\mathrm{C_{3}}$,
$\mathrm{C_{20}}$,$\mathrm{C_{23}}-\mathrm{C_{27}}$ which are redefined at the
top of this page, where $\mathbf{S}_{b}$ is a block diagonal matrix containing
the combinations of beamformers $\widehat{\mathbf{w}}_{b,u}$ and channels for
UE $u$, $c_{b,u}=\mathbf{h}_{b,u}^{H}\widehat{\mathbf{w}}_{b,u}$,
$\mathbf{R}_{b}=\mathrm{diag}\left(\mathbf{g}_{b}^{H}\widehat{\mathbf{M}}\right)$,
$r_{b,l}=\mathsf{Re}\left\\{\mathbf{g}_{b}^{H}\widehat{\mathbf{m}}_{l}\right\\}$.

$\displaystyle(\mathrm{C_{3}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{1}}:\sum_{l\in\mathcal{L}}t_{l}^{2}\leq P_{\mathrm{tx}}^{\mathrm{MBS}},$ $\displaystyle(\mathrm{C_{20}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{2}}:\left\|\left[2\widehat{\mathbf{w}}_{b,u}^{H}v_{b,u},\kappa_{b,u}-p_{b,u}\right]\right\|_{2}\leq\kappa_{b,u}+p_{b,u},\forall l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l},$ $\displaystyle(\mathrm{C_{23}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{3}}:\left\|\left[\mathbf{S}_{b}\mathbf{v},\sigma_{\mathrm{UE}}\right]\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\mathsf{Re}\leavevmode\nobreak\ \Big{\\{}\sum_{b\in\mathcal{B}_{l}}c_{b,u}v_{b,u}\Big{\\}}+\left(1-\alpha_{u,j}\right)Q_{u},\forall l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$ $\displaystyle(\mathrm{C_{24}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{4}}:\mathsf{Re}\leavevmode\nobreak\ \Big{\\{}\sum_{b\in\mathcal{B}_{l}}c_{b,u}v_{b,u}\Big{\\}}\geq\alpha_{u,j}\sqrt{\Gamma_{j}^{\mathrm{UE}}}\sigma_{\mathrm{UE}},\forall l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$ $\displaystyle(\mathrm{C_{25}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{5}}:\mathsf{Im}\leavevmode\nobreak\ \Big{\\{}\sum_{b\in\mathcal{B}_{l}}c_{b,u}v_{b,u}\Big{\\}}=0,\forall l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$ $\displaystyle(\mathrm{C_{26}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{6}}:\left\|\left[\mathbf{R}_{b}\mathbf{t},\sigma_{\mathrm{SBS}}\right]\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{SBS}}}^{-1}}r_{b,l}t_{l}+\left(1-\beta_{l,j}\right)Q_{b},\forall l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}},$ $\displaystyle(\mathrm{C_{27}})$ $\displaystyle\leavevmode\nobreak\ \mathrm{L_{7}}:r_{b,l}t_{l}\geq\beta_{l,j}\sqrt{\Gamma_{j}^{\mathrm{SBS}}}\sigma_{\mathrm{SBS}},\forall l\in\mathcal{L},b\in\mathcal{B}_{l},j\in\mathcal{J}^{\mathrm{SBS}},$ Table IV: Simulation settings Scenario | Backhaul network | Access network
---|---|---
$N^{\mathrm{MBS}}_{\mathrm{tx}}$ |  $P^{\mathrm{MBS}}_{\mathrm{tx}}$ [dBm] | $L$ | $B$ | $B_{\mathrm{total}}$ | $\chi_{\mathrm{backhaul}}$ |  $P^{\mathrm{SBS}}_{\mathrm{tx}}$ [dBm] | $U$ | $U_{\mathrm{total}}$ | $U_{\mathrm{served}}$ | $\chi_{\mathrm{access}}$
S1 | $64$ | $9,12,\dots,27$ | $2$ | $3$ | $6$ | $0$ | $6,10,14$ | $6$ | $12$ | $3$ | $0$
S2 | $16,32,48,64$ | $15,18,\dots,36$ | $1,2,\dots,6$ | $3$ | $3,6,\dots,18$ | $0$ | $-$ | $-$ | $-$ | $-$ | $-$
$64$ | $15,18,\dots,36$ | $5$ | $1,2,\dots,6$ | $5,10,\dots,30$ | $0$ | $-$ | $-$ | $-$ | $-$ | $-$
S3 | $64$ | $15,18,\dots,36$ | $5$ | $3$ | $15$ | $0$ | $0,2,\dots,14$ | $20$ | $100$ | $4$ | $0$
S4 | $64$ | $18,27,36$ | $2,3,\dots,6$ | $3$ | $6,9,\dots,18$ | $0$ | $14$ | $20$ | $100$ | $4$ | $0$
S5 | $64$ | $36$ | $5$ | $3$ | $15$ | $[0,1]$ | $14$ | $20$ | $100$ | $4$ | $[0,1]$
S6 | $64$ | $36$ | $5$ | $3$ | $15$ | $0$ | $14$ | $20$ | $100$ |  $4$ (slotted) | $0$

After applying these changes, we define,

$\displaystyle\mathcal{P}_{2}:\max_{\boldsymbol{\Theta}}\leavevmode\nobreak\
\leavevmode\nobreak\
R_{\mathrm{w-sum}}^{\mathrm{access}}\left(\boldsymbol{\alpha}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\underbrace{\boldsymbol{\Theta}\in\mathscr{D}}_{\text{{convex}}},\underbrace{\mathrm{Z_{1}}-\mathrm{Z_{3}}}_{\text{{nonconvex}}},$

where
$\boldsymbol{\Theta}=\left(\mathbf{t},\mathbf{v},\mathbf{p},\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\kappa}\right)$
with $\mathscr{D}$ denoting the feasible set spanned by the constraints
$\mathrm{L_{1}}-\mathrm{L_{7}},\mathrm{X_{1}}-\mathrm{X_{3}},\mathrm{C_{2}},\mathrm{C_{7}}-\mathrm{C_{10}},\mathrm{C_{12}}-\mathrm{C_{14}},\mathrm{C_{17}}-\mathrm{C_{19}}$.
In a similar manner as with $\widetilde{\mathcal{P}}_{1}$, we define
$\widetilde{\mathcal{P}}_{2}$, and thereupon its linearized version as
$\widetilde{\mathcal{P}}_{2}^{(t)}$, which can be solved via Algorithm 1.
$\widetilde{\mathcal{P}}_{2}^{(t)}$ is a SOCP program with
$N_{v}=2L+4LBU+LJ^{\mathrm{SBS}}+LUJ^{\mathrm{UE}}$ decision variables, which
roughly represent half of that used in BnC-MISOCP and RnP-SOCP-1 (for the
evaluated settings). In addition, $\widetilde{\mathcal{P}}_{2}^{(t)}$ has
$N_{l}=3L+2LU+3LB+2LBU+3LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$ linear
constraints and $N_{c}=1+LBU+LUJ^{\mathrm{UE}}+LBJ^{\mathrm{SBS}}$ convex
constraints. Thus, the complexity of RnP-SOCP-2 is
$\mathcal{O}\left(N_{\mathrm{iter}}(N_{v})^{3}(N_{l}+N_{c})\right)$, with
$N_{\mathrm{iter}}$ denoting the number of iterations. Further, we note that
RnP-SOCP-2 exhibits reduced complexity compared to RnP-SOCP-1.

##  VIII Simulation Results

We evaluate the performance of RadiOrchestra in different scenarios with
varying conditions. Throughout all simulations, we consider the following
default parameters, unless specified otherwise. The carrier frequency is
$f_{c}=41$ GHz (V-band in FR2) with
$W_{\mathrm{BW}}^{\mathrm{access}}=W_{\mathrm{BW}}^{\mathrm{backhaul}}=100$
MHz bandwidth [36]. The channel models are UMa LOS for the backhaul and UMi
LOS/NLOS for the access [36], which include path-loss, shadowing and small-
scale fading. In the system, there are $L=5$ clusters each having $B=3$ SBSs
and $U=20$ UEs, thus making a total of $B_{\mathrm{total}}=15$ SBSs and
$U_{\mathrm{total}}=100$ UEs. The MBS has a maximum transmit power of
$P^{\mathrm{MBS}}_{\mathrm{tx}}=36$ dBm and is equipped with a $16\times 4$
antenna array ($N_{\mathrm{tx}}^{\mathrm{MBS}}=64$) whereas the SBSs can
transmit at a maximum power of $P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm and
have smaller $4\times 4$ arrays ($N_{\mathrm{tx}}^{\mathrm{SBS}}=16$). We
assume that SBSs can support up to four UEs
($N^{\mathrm{SBS}}_{\mathrm{streams}}=4$) simultaneously, and there are
$U_{\mathrm{served}}=4$ UEs served concurrently (i.e., in one slot) in each
cluster. Further, all UEs have the same priority, i.e.,
$\omega_{u}=\frac{1}{L*U}$ and
$\sum_{l\in\mathcal{L}}\sum_{u\in\mathcal{U}_{l}}\omega_{u}=1$. In Table IV,
we show the parameters for each scenario. The algorithms have been implemented
using CVX and MOSEK on a computer with 16GB RAM and a Intel Core i7-6700
processor.

$\mathtt{UB}$$\mathtt{LB}$$\mathtt{BnC-MISOCP}$$\mathtt{RnP-
SOCP-1}$$\mathtt{RnP-SOCP-2}$

(a) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}$ when
$P^{\mathrm{SBS}}_{\mathrm{tx}}=6$ dBm

(b) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}$ when
$P^{\mathrm{SBS}}_{\mathrm{tx}}=10$ dBm

(c) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}$ when
$P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm

(d) Time complexity when $P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm

(e) Time complexity when $P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm

(f) Convergence

Figure 5: Evaluation of Scenario S1. _We notice the small performance gap of
RnP-SOCP-1 and RnP-SOCP-2 with respect to BnC-MISOCP, which is reasonable
considering that their time complexities are smaller by 3 orders of magnitude.
Because CVX needs to parse the mathematical model into a suitable structure
for MOSEK, the results showing time complexity consider the raw solving time
while neglecting the parsing time. Besides, we note that UB can be used for
quick benchmarking when the access throughput bottleneck is originated by the
backhaul network. In addition, we note that LB is loose as it is agnostic to
the network conditions but provides an idea of the worst-case scenario without
solving any problem. It becomes valuable when evaluating cases wherein the
transmit power at the MBS or SBSs are limited as in Fig. 5(a) because under
such conditions the lowest rates will very likely be allocated._

Scenario S1: Optimality gap and computational complexity. We benchmark the
algorithms considering a small setting, with the purpose of obtaining an
optimal solution for BnC-MISOCP within a reasonable amount of time and compare
its performance against that of RnP-SOCP-1 and RnP-SOCP-2. Fig. 5(a), Fig.
5(b), Fig. 5(c) show the access throughput with various MBS and SBSs transmit
powers. In particular, RnP-SOCP-1 and RnP-SOCP-2 are $5.1\%$ and $9.7\%$ below
BnC-MISOCP when $P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm (see Fig. 5(c)). Also,
UB becomes tighter with increasing $P^{\mathrm{SBS}}_{\mathrm{tx}}$, e.g.,
within only $9.6\%$ with respect to BnC-MISOCP in Fig. 5(c). This occurs
because UB only considers the backhaul throughput optimization, which depends
on $P^{\mathrm{MBS}}_{\mathrm{tx}}$. Thus, as long as the bottleneck is
originated in the access network (due to low transmit power at the SBSs), UB
will not capture such limitations. With higher
$P^{\mathrm{SBS}}_{\mathrm{tx}}$, as shown in Fig. 5(c), the access throughput
limitation is removed and is shifted to the backhaul network, where
$P^{\mathrm{MBS}}_{\mathrm{tx}}$ is varied from a low to a high transmit
power. As a result, in Fig. 5(c) the access throughput limitation is dominated
by $P^{\mathrm{MBS}}_{\mathrm{tx}}$, where we recognize a high degree of
similarity between UB and BnC-MISOCP. Therefore, UB can be used as a tight
bound to evaluate the performance of the system whenever the SBSs can transmit
at sufficiently high power.

On the other hand, Fig. 5(d) and Fig. 5(e) provide the time complexities when
$P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ (as in Fig. 5(c)) showing that RnP-SOCP-1
and RnP-SOCP-2 are roughly $1000$ and $2000$ times computationally faster than
BnC-MISOCP, respectively. Similarly, the time complexity of UB is
approximately $100$ times lighter than that of BnC-MISOCP. This huge
difference is because the complexity of BnC-MISOCP is combinatorial, i.e.,
collapsing to exhaustive search in the worst case. Although this case may not
be reached in practice, BnC-MISOCP requires to solve multiple convex problems
to prune the infeasible branches and thus abridge the search process. However,
RnP-SOCP-1 and RnP-SOCP-2 circumvent this issue by relaxing the binary
variables, penalizing them and solving the problem in the continuous domain,
which explains their reduced complexity. Besides, UB has a small number of
optimization variables compared to BnC-MISOCP, explaining its faster solving
time. Note that the time complexities grow with increasing
$P^{\mathrm{MBS}}_{\mathrm{tx}}$ because a higher
$P^{\mathrm{MBS}}_{\mathrm{tx}}$ enables the allocation of a wider range of
rates thus needing more evaluations, specially by BnC-MISOCP and UB. Further,
Fig. 5(f) shows the convergence of RnP-SOCP-1 and RnP-SOCP-2 for $5$ different
realizations. Here, we measured the error of the binary variables with respect
to their rounded versions and computed the mean squared error (MSE), showing
that after $6$ or $7$ iterations the error converges to zero, i.e., the
relaxed binary variables values become integer.

(a) Backhaul throughput for varying $P_{\mathrm{tx}}^{\mathrm{MBS}}$,
$N_{\mathrm{tx}}^{\mathrm{MBS}}$ and $L$ when $B=3$.

(b) Backhaul throughput for varying $B$ and $N_{\mathrm{tx}}^{\mathrm{MBS}}$
when $L=5$.

Figure 6: Evaluation of Scenario S2. _We note that UB can be used to evaluate
multiple network configurations, thus providing insights of potentially
optimal operations points that can be adopted in the planning phase of the
network._

(a) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}|P^{\mathrm{SBS}}_{\mathrm{tx}}=14$
dBm

(b) Varying $P^{\mathrm{SBS}}_{\mathrm{tx}}|P^{\mathrm{MBS}}_{\mathrm{tx}}=36$
dBm

(c) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}$ and
$P^{\mathrm{SBS}}_{\mathrm{tx}}$

(d) Varying $P^{\mathrm{MBS}}_{\mathrm{tx}}$ and
$P^{\mathrm{SBS}}_{\mathrm{tx}}$

(e) ${P_{\mathrm{tx}}^{\mathrm{MBS}}=36}$ dBm $\leavevmode\nobreak\
|\leavevmode\nobreak\ {P_{\mathrm{tx}}^{\mathrm{SBS}}=2}$ dBm

(f) ${P_{\mathrm{tx}}^{\mathrm{MBS}}=36}$ dBm $\leavevmode\nobreak\
|\leavevmode\nobreak\ {P_{\mathrm{tx}}^{\mathrm{SBS}}=6}$ dBm

(g) ${P_{\mathrm{tx}}^{\mathrm{MBS}}=36}$ dBm $\leavevmode\nobreak\
|\leavevmode\nobreak\ {P_{\mathrm{tx}}^{\mathrm{SBS}}=10}$ dBm

(h) ${P_{\mathrm{tx}}^{\mathrm{MBS}}=36}$ dBm $\leavevmode\nobreak\
|\leavevmode\nobreak\ {P_{\mathrm{tx}}^{\mathrm{SBS}}=14}$ dBm

Figure 7: Evaluation of Scenario S3. _We note that maximizing the access
throughput is highly dependent on both backhaul and access network parameters,
which highlights the importance of jointly optimizing them._

Scenario S2: Upper bound as a means of network planning. Since UB is much
simpler to solve than BnC-MISOCP (as shown in Fig. 5(d) and Fig. 5(e)), we can
use UB in larger settings to examine multiple configurations of number of
antennas, transmit power, number of clusters and cluster size. From the
planning perspective, these results are valuable as they allow us to choose
suitable operation points for the network. In Fig. 6(a), we show the backhaul
throughput (i.e., the objective of UB) for various combinations of
$P^{\mathrm{MBS}}_{\mathrm{tx}}$, $N^{\mathrm{MBS}}_{\mathrm{tx}}$, $L$, where
the bottommost and uppermost layers represent $L=1$ (one cluster) and $L=6$
(six clusters), respectively. We observe that the backhaul throughput improves
with increasing number of antennas and transmit power because more antennas
enhance the multiplexing capability while a higher power allows transmitting
at higher rates. However, when the number of clusters grows from $L=5$ to
$L=6$, the throughput saturates showing marginal improvement because the
scenario becomes more interference limited (due to more SBSs deployed). _We
realize that with $N^{\mathrm{MBS}}_{\mathrm{tx}}=64$ antennas,
$P^{\mathrm{MBS}}_{\mathrm{tx}}=36$ dBm transmit power and $L=5$ clusters, the
backhaul network can be operated at its full capacity_. In Fig. 6(a), we
considered $B=3$, but we validate such decision in Fig. 6(b), where we
illustrate the backhaul throughput for various combinations of
$P^{\mathrm{MBS}}_{\mathrm{tx}}$ and $B$ when $L=5$. We note that the
throughput decreases when the cluster size increases from $B=1$ to $B=6$
because, to reach more SBSs, higher MBS power is consumed but also more
interference is generated due to more SBSs scattered. However, a larger SBS
cluster is preferred because _(i)_ more UEs can be served (each SBS can serve
a limited number of UEs) and _(ii)_ UEs can be allocated higher rates by being
connected to more SBSs. With $B=3$, still the maximum backhaul throughput can
be achieved.

(a) $P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm and
$P_{\mathrm{tx}}^{\mathrm{SBS}}=14$ dBm

(b) $P_{\mathrm{tx}}^{\mathrm{MBS}}=27$ dBm and
$P_{\mathrm{tx}}^{\mathrm{SBS}}=14$ dBm

(c) $P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm and
$P_{\mathrm{tx}}^{\mathrm{SBS}}=14$ dBm

(d) $L=2\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(e) $L=3\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(f) $L=4\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(g) $L=5\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(h) $L=2\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(i) $L=3\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(j) $L=4\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

(k) $L=5\leavevmode\nobreak\ |\leavevmode\nobreak\
P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm and $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm

Figure 8: Evaluation of Scenario S4. _We note that the overall access
throughput can be expanded with more clusters (i.e., more SBSs and UEs).
However, this improvement may saturate beyond a number of clusters due to more
interference or insufficient transmit power._

Scenario S3: Impact of the transmit power. Fig. 7(a), Fig. 7(b), Fig. 7(c) and
Fig. 7(d) illustrate how the variation of transmit power at the MBS and SBSs
impacts the access network throughput. Fig. 7(a) shows the case when
$P^{\mathrm{SBS}}_{\mathrm{tx}}=14$ dBm and $P^{\mathrm{MBS}}_{\mathrm{tx}}$
is varied. As observed, the access throughput improves as the MBS increases
its transmit power, which is logical since the backhaul capacity is naturally
expanded with higher power. Similarly, Fig. 7(b) shows the case when
$P^{\mathrm{MBS}}_{\mathrm{tx}}=36$ dBm and $P^{\mathrm{SBS}}_{\mathrm{tx}}$
is varied. We note that the access throughput improves as the SBSs increase
their transmit power. This occurs because higher SBSs power enables UEs to be
assigned higher rates. We observe in Fig. 7(a) and Fig. 7(b) that when
$P^{\mathrm{MBS}}_{\mathrm{tx}}=36$ dBm and
$P^{\mathrm{SBS}}_{\mathrm{tx}}=14$, both RnP-SOCP-1 and RnP-SOCP-2 achieve
nearly the same performance although RnP-SOCP-2 grows at a slower rate. This
slower improvement stems from the fact that the beamforming vectors for RnP-
SOCP-2 are predesigned and only their gains can be optimized, thus allowing
for less flexibility compared to RnP-SOCP-1. Thus, their performance meet only
in the presence of high MBS/SBSs transmit power. At this point, the gap
compared to UB is $14.8\%$ and $16.5\%$ for RnP-SOCP-1 and RnP-SOCP-2,
respectively. Fig. 7(c) and Fig. 7(d) show the effect of varying both
$P^{\mathrm{SBS}}_{\mathrm{tx}}$ and $P^{\mathrm{MBS}}_{\mathrm{tx}}$. In Fig.
7(e), Fig. 7(f), Fig. 7(g), Fig. 7(h), we show the allocation of UE rates when
$P^{\mathrm{MBS}}_{\mathrm{tx}}=36$ and $P^{\mathrm{SBS}}_{\mathrm{tx}}$ is
varied gradually from a low to a high power. At lower
$P^{\mathrm{SBS}}_{\mathrm{tx}}$ as in Fig. 7(e), the UEs are mainly assigned
the lowest rates. As $P^{\mathrm{SBS}}_{\mathrm{tx}}$ becomes higher, it
becomes possible to allocate higher rates to the UEs, as observed in Fig.
7(h).

Scenario S4: Impact of the number of clusters. Fig. 8(a), Fig. 8(b), Fig. 8(c)
show the access throughput when $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$ dBm and
the number of clusters is varied from $L=2$ to $L=6$ for different
$P_{\mathrm{tx}}^{\mathrm{MBS}}$ values. The access throughput improves with
increasing $L$ because more clusters translates to more served UEs (there are
$U_{\mathrm{served}}$ UEs per cluster), and hence the higher aggregate rate.
Besides, higher $P_{\mathrm{tx}}^{\mathrm{MBS}}$ also improves the access
throughput because it boosts the backhaul network capacity. In particular, we
observe throughput saturation when increasing from $L=5$ to $L=6$, which is
consistent with the behavior observed in Fig. 6(a) where the backhaul network
throughput was evaluated. Further, we note that RnP-SOCP-1 outperforms RnP-
SOCP-2 when $P_{\mathrm{tx}}^{\mathrm{MBS}}=\left\\{18,27\right\\}$ dBm.
However, for sufficiently high $P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm, the
performance of both are comparable. Besides, we examine the UE rate allocation
in Fig. 8(d), Fig. 8(e), Fig. 8(f) and Fig. 8(g) assuming
$P_{\mathrm{tx}}^{\mathrm{MBS}}=18$ dBm, $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm. We observe that when the number of clusters is small, e.g. $L=2$ (see
Fig. 8(d)), the rates assigned to the UEs span a wider range compared to the
case when $L=5$ (see Fig. 8(g)). The reason for this behavior is that more
interference is generated in the backhaul network with $L=5$ than with $L=2$.
In particular, with $L=2$, only two signals are transmitted whereas with
$L=5$, five different signals are sent from the MBS, thus generating more
interference at the receiving SBSs. In Fig. 8(h), Fig. 8(i), Fig. 8(j) and
Fig. 8(k) we also examine the UE rates assuming
$P_{\mathrm{tx}}^{\mathrm{MBS}}=36$ dBm, $P_{\mathrm{tx}}^{\mathrm{SBS}}=14$
dBm. In this case, the backhaul network has sufficiently high power. As a
result, throughout Fig. 8(h), Fig. 8(i), Fig. 8(j) and Fig. 8(k), the
distribution of rates remains more or less similar.

(a) Imprecise backhaul channel

(b) Imprecise access channel

(c) Imprecise channels

(d) Imprecise channels

Figure 9: Evaluation of Scenario S5. _We have used the model
$\mathbf{c}=\sqrt{1-\chi^{2}}\hat{\mathbf{c}}+\chi\mathbf{p}$ to emulate
imprecise channel conditions, where $\mathbf{c}$ is the estimated channel,
$\hat{\mathbf{c}}$ is the exact access/backhaul channel (but unknown),
$\chi\in\left[0,1\right]$ is the degree at which the perturbation contaminates
the channel, and
$\mathbf{p}\sim\left(\mathbf{0},\left\|\hat{\mathbf{c}}\right\|^{2}_{2}\mathbf{I}/K\right)$
is a random perturbation, where $K$ is the length of $\hat{\mathbf{c}}$. We
note the importance of careful provision of the backhaul network because it is
the link with highest importance delivering data to the UEs. A potential
disruption affecting this link causes a degradation of the whole network
whereas impairments in the individual access links do not have a significant
impact on the overall network performance. We underline a fundamental
difference regarding the impact of imperfect CSI in system models assuming
discrete or continuous rates. While CSI variations affect both systems, it has
more detrimental consequences in the discrete-rate case. For instance, in
continuous-rate models, a CSI variation will produce a SINR different from the
expected thus also affecting the rate. However, the resulting rate will still
be feasible for the model due to being continuous. On the contrary, in
discrete-rate models, if the SINR is below the required target, the data will
not be decoded by the SBS/UE thus causing the resulting rate to drop to zero._

(a) Serving $U_{\mathrm{served}}=4$ UEs per cluster per slot.

(b) Displaying the individual rates of all UEs in cluster $\mathcal{U}_{1}$.

Figure 10: Evaluation of Scenario S6. _We observe that it is possible to serve
all UEs in a system by allocating them in multiple slots, showing that
RadiOrchestra is scalable. In addition, the UE rates can be adapted to enforce
different priorities based on any network policy of the operator. In this
example, we aimed at improving fairness among UEs._

Scenario S5: Impact of imprecise channel estimation. Fig. 9(a) shows the
access throughput when the access channels are estimated perfectly but the
backhaul channels inaccurately. Here, the channel energy variation is
represented by $\xi_{\mathrm{backhaul}}$. Although backhaul channels are
generally static due to fixed positions of MBS and SBS, it is important to
test the network against estimation errors that may arise due to hardware
miscalibration or impairments. We observe that as the degree of error in the
backhaul channels increases, the access throughput is affected more severely
due to information that cannot be decoded by the SBSs and therefore not
relayed to the UEs. Further, RnP-SOCP-1 is more robust than RnP-SOCP-2 to
dealing with such imprecisions because RnP-SOCP-2 only optimizes the
beamformers gains, making it less robust to perturbations. With RnP-SOCP-1 and
RnP-SOCP-2, the throughput decreases $4.2\%$ and $18.4\%$, respectively when
the channel energy varies within $\xi_{\mathrm{backhaul}}=5\%$, and $10.1\%$
and $58.5\%$, respectively when the channel energy varies within
$\xi_{\mathrm{backhaul}}=10\%$. Fig. 9(b) shows the access throughput when the
access channels are estimated inaccurately but the backhaul channels
perfectly, and the error energy is represented by $\xi_{\mathrm{access}}$. The
access channel may be inaccurately estimated due to UE mobility, feedback
quantization or unmanaged interference from other networks.

We note that the access throughput with RnP-SOCP-1 and RnP-SOCP-2 only suffers
a decay of $9.9\%$ and $31.6\%$, respectively, even when the access channels
change within $\xi_{\mathrm{access}}=40\%$, which is much less compared to the
case in Fig. 9(a). The reason for this outcome is that a disruption in an
access link may cause only a single UE not being able to decode its
information (since its SINR may decrease). In contrast, a disruption in a
backhaul link may cause many SBSs in a cluster to be automatically unsupplied,
thus making them unable to deliver data to the UEs. In addition, the multicast
topology of the backhaul network is more susceptible to channel variations,
since the link with the weakest condition limits the data rate for the whole
SBS cluster. On the other hand, Fig. 9(c) and Fig. 9(d) show the access
throughput performance when both the access and backhaul channels contain
estimation errors.

Scenario S6: Time-slotted evaluation. We have evaluated the access throughput
considering that all UEs have the same priorities. However, the UE priorities
(weights) can be adjusted, for instance, to balance the cumulative throughput
so that all UEs experience a similar degree of fairness over time. To realize
this, we evaluate the algorithms in a slotted manner. Assuming $L=5$, $U=20$,
$U_{\mathrm{served}}=4$, the network needs $5$ slots to allocate the $100$
UEs, i.e., in each slot, $20$ UEs are simultaneously served with $4$ UEs per
cluster. In Fig. 10(a), we show the access throughput for RnP-SOCP-1 and RnP-
SOCP-2 during $50$ slots of equal duration $T=T_{n}-T_{n-1}$ and assuming that
the channel is estimated every $5$ slots, i.e., once all the UEs have been
served, a new UE scheduling with a different channel is considered. In
particular, in every cluster, in time slot $T_{1}$, $4$ UEs out of $20$ are
chosen; in slot $T_{2}$, 4 out of $16$ are chosen; in slot $T_{3}$, $4$ out of
$12$ are chosen, in slot $T_{4}$, $4$ out of $8$, and in slot $T_{5}$ the
remaining $4$ UEs are served. In slot $T_{6}$, the weights are updated based
on the cumulative rate the UEs have experienced according to
$w_{u}^{(n)}=\frac{1}{T\sum_{i=1}^{n}r_{u}^{(i-1)}}$ (up to normalization),
where $r_{u}^{(i)}$ is the rate of UE $u$ in slot $T_{i}$. In slot $T_{6}$,
another $4$ UEs out of $20$ are chosen (possibly a different UE batch than in
slot $T_{1}$). The process continues in this manner, updating the weights
every $5$ slots. In Fig. 10(b), we show the individual cumulative throughput
for all $20$ UEs in cluster $\mathcal{U}_{1}$. We realize that the throughput
experienced by the UEs tend to be similar as the deviation from each other is
small, which is achieved due to the adaption of weights.

##  IX Conclusions

Self-backhauling millimeter-wave networks are a key enabler for dense
deployments by virtue of reducing costs (not needing fiber links) and
facilitating higher flexibility through usage of wireless links. However,
designing efficient and practical solutions for such systems are extremely
complex due to the intertwined nature of backhaul and radio access networks
that are not straightforward to model, and intrinsically result in complex
problems with coupled optimization variables that are challenging to solve. In
this paper, RadiOrchestra demonstrated how to tame this complexity with a
series of design choices in the system, and providing mathematical formulation
and optimization of radio resources. We proposed three formulations and their
respective algorithms, BnC-MISOCP, RnP-SOCP-1 and RnP-SOCP-2, to jointly
optimize beamforming, user association, rate selection and admission control
with the aim of maximizing the access network throughput. Our complexity
analysis showed that RnP-SOCP-1 and RnP-SOCP-2 are less complex than BnC-
MISOCP while the simulation results illustrated that their performance
remained within $16.5\%$ of the upper bound. We believe this attractive
complexity-performance trade-off is key to potential adaptation of
RadiOrchestra in future systems. RadiOrchestra can be extended in several
directions. In RadiOrchestra we considered that both the access and backhaul
networks operate over a fixed bandwidth. However, to make the approach more
flexible and therefore capable of dealing with unbalanced channel conditions,
bandwidth optimization could be incorporated as an additional degree of
freedom. Another direction is expanding RadiOrchestra to be robust against
channel imprecisions at both the access and backhaul networks to ultimately
preserve the integrity of data. While current networks are centralized,
enabling distributed optimization algorithms is desirable due to lower
latency. Thus, a possible direction of expanding RadiOrchestra is to
parallelize the optimization to let each SBS cluster optimize the resources
without a central coordinator. In RadiOrchestra, we assumed that the UEs are
pre-associated to a given SBS cluster. In dynamic networks, however, this
association can change. Therefore, it is interesting to investigate these
changes in contexts of transitions between different clusters.

## Acknowledgment

The research is in part funded by the Deutsche Forschungsgemeinschaft (DFG)
within the B5G-Cell project in SFB 1053 MAKI, by the LOEWE initiative (Hesse,
Germany) within the emergenCITY center, and by the European Commission through
Grant No. 101017109 (DAEMON project).

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##  Appendix A Proof of Proposition 2

In constraints $\mathrm{\bar{C}_{4}}$ and $\mathrm{C_{5}}$, the beamformer
$\mathbf{w}_{b,u}$ and binary variable $\kappa_{b,u}$ are tied. This leads to
obtain zero-beamformers for unserved UEs. To ensure the same effect after
removing the multiplicative coupling between $\mathbf{w}_{b,u}$ and
$\kappa_{b,u}$, additional constraints are required. First, we define the
auxiliary variable $p_{b,u}$ representing the power of the beamformer from SBS
$b$ to UE $u$, which leads us to declare the following constraint,
$\mathrm{C_{17}}:p_{b,u}\geq 0,\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l}$. Considering, the
newly introduced variable, constraint $\mathrm{\bar{C}_{4}}$ is redefined as
$\mathrm{C_{18}}:\sum_{u\in\mathcal{U}_{l}}p_{b,u}\leq
P_{\mathrm{tx}}^{\mathrm{SBS}},\forall l\in\mathcal{L},b\in\mathcal{B}_{l}$.
In addition, the power $p_{b,u}$ of a beamformer needs to be zero for unserved
UEs and positive for served UEs, which is enforced via
$\mathrm{C_{19}}:p_{b,u}\leq\kappa_{b,u}P_{\mathrm{tx}}^{\mathrm{SBS}},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l}$. To connect the
beamformer $\mathbf{w}_{b,u}$ and its power $p_{b,u}$, we define
$\left\|\mathbf{w}_{b,u}\right\|_{2}^{2}\leq\kappa_{b,u}p_{b,u}$, which
ensures that the beamformer is a zero-vector when $\kappa_{b,u}=0$. Note that
$\left\|\mathbf{w}_{b,u}\right\|_{2}^{2}\leq\kappa_{b,u}p_{b,u}$ is nonconvex
but it can be recast as a SOC constraint as shown in the following. Using the
difference of squares, the product $\kappa_{b,u}p_{b,u}$ is equivalent to
$\kappa_{b,u}p_{b,u}=\frac{\left(\kappa_{b,u}+p_{b,u}\right)^{2}-\left(\kappa_{b,u}-p_{b,u}\right)^{2}}{4}$,
which allows us to rearrange
$\left\|\mathbf{w}_{b,u}\right\|_{2}^{2}\leq\kappa_{b,u}p_{b,u}$ as a new
constraint
$\mathrm{C_{20}}:\left\|\left[2\mathbf{w}_{b,u}^{H},\kappa_{b,u}-p_{b,u}\right]\right\|_{2}\leq\kappa_{b,u}+p_{b,u},\forall
l\in\mathcal{L},b\in\mathcal{B}_{l},u\in\mathcal{U}_{l}$. After these changes,
$\mathbf{w}_{b,u}$ and $\kappa_{b,u}$ have been decoupled while still
guaranteeing the same effect as if coupled. Thus, the product
$\mathbf{w}_{b,u}\kappa_{b,u}$ can be replaced by $\mathbf{w}_{b,u}$ upon
including $\mathrm{C_{17}}-\mathrm{C_{20}}$. The, constraint
$\mathrm{\bar{C}_{21}}$ is obtained after replacing
$\mathbf{w}_{b,u}\kappa_{b,u}$ by $\mathbf{w}_{b,u}$ in $\mathrm{C_{5}}$.

##  Appendix B Proof of Proposition 3

We follow a similar procedure as in [20].We exchange positions between the
SINR denominator and the right-hand side (RHS) of $\mathrm{\bar{C}_{21}}$.
Then, we add
$\big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\big{|}^{2}$
to both sides, thus yielding
$\displaystyle\mathrm{\bar{C}_{21}}:\left(1+{\alpha_{u,j}}^{-1}{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\Big{|}^{2}\geq$
$\displaystyle\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\Big{|}^{2}+\sigma^{2}_{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

To deal with this nonconvex constraint, we first derive expressions for its
two cases.

1 $\displaystyle\alpha_{u,j}=0\Rightarrow$
$\displaystyle\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\Big{|}^{2}$
$\displaystyle+\sigma^{2}_{\mathrm{UE}}\leq\infty,\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},$ 2
$\displaystyle\alpha_{u,j}=1\Rightarrow$
$\displaystyle\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\Big{|}^{2}+\sigma^{2}_{\mathrm{UE}}\leq$
$\displaystyle\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\Big{|}^{2},\forall
l\in\mathcal{L},$ $\displaystyle
u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

In case 1, the inequality is satisfied by default. Besides, it is possible to
find an upper bound $Q_{u}^{2}$ for
$\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\big{|}^{2}+\sigma^{2}_{\mathrm{UE}}$
to prevent using $\infty$. By harnessing the big-M method, we can equivalently
combine the two cases into $\mathrm{C_{21}}$, shown at the top of this page.
The upper bound
$Q_{u}^{2}=P_{\mathrm{tx}}^{\mathrm{SBS}}\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\left\|\mathbf{h}_{b^{\prime},u}\right\|_{2}^{2}+\sigma^{2}_{\mathrm{UE}}$
is obtained by maximizing the LHS of $\mathrm{C_{21}}$.

$\displaystyle\mathrm{C_{21}}:\sum_{\begin{subarray}{c}l^{\prime}\in\mathcal{L}\end{subarray}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\Big{|}\sum_{b^{\prime}\in\mathcal{B}_{l^{\prime}}}\mathbf{h}_{b^{\prime},u}^{H}\mathbf{w}_{b^{\prime},u^{\prime}}\Big{|}^{2}+\sigma^{2}_{\mathrm{UE}}\leq\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\Big{|}\sum_{b\in\mathcal{B}_{l}}\mathbf{h}_{b,u}^{H}\mathbf{w}_{b,u}\Big{|}^{2}+\left(1-\alpha_{u,j}\right)^{2}Q_{u}^{2},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

$\displaystyle\mathrm{C_{21}-C_{22}}=\begin{cases}\mathrm{C_{23}}:\left\|\left[\bar{\mathbf{h}}_{u}^{H}\mathbf{W},\sigma_{\mathrm{UE}}\right]\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}+\left(1-\alpha_{u,j}\right)Q_{u},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},\\\
\mathrm{C_{24}}:\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}\geq\alpha_{u,j}\sqrt{\Gamma_{j}^{\mathrm{UE}}}\sigma_{\mathrm{UE}},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}},\\\
\mathrm{C_{25}}:\mathsf{Im}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}=0,\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.\\\
\end{cases}$

##  Appendix C Proof of Proposition 5

Assuming that
$\mathbf{x}=\left[\bar{\mathbf{h}}_{u}^{H}\mathbf{W},\sigma_{\mathrm{UE}}\right]$,
constraint $\mathrm{C_{21}}$ can be expressed as

$\displaystyle\left\|\mathbf{x}\right\|^{2}_{2}\leq\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\left(1-\alpha_{u,j}\right)^{2}Q_{u}^{2},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

Taking the square root at both sides and applying the Jensen’s inequality to
the RHS expression, we obtain

$\displaystyle\sqrt{\left(1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}\right)\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|^{2}+\left(1-\alpha_{u,j}\right)^{2}Q_{u}^{2}}\leq$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|+\left(1-\alpha_{u,j}\right)Q_{u}.$

When $\alpha_{u,j}=1$, the inequality is tight, because the RHS and LHS of the
expression above become equivalent, i.e.,
$\left\|\mathbf{x}\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|$.
When $\alpha_{u,j}=0$, the inequality still remains valid, i.e.
$\left\|\mathbf{x}\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|+Q_{u}$,
because $Q_{u}$ is an upper bound for $\left\|\mathbf{x}\right\|_{2}$. As a
result, the following expression is equivalent to $\mathrm{C_{21}}$

$\displaystyle\left\|\mathbf{x}\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left(1-\alpha_{u,j}\right)Q_{u},\forall
l\in\mathcal{L},u\in\mathcal{U}_{l},j\in\mathcal{J}^{\mathrm{UE}}.$

Notice that the beamforming vectors are invariant to phase shift. In
particular, $\mathbf{w}_{u}$ and $\mathbf{w}_{u}e^{j\theta_{u}}$ yield the
same received SINR at the UE $u$. Thus, it is possible to choose a phase
$e^{j\theta_{u}}$ such that $\mathbf{h}_{u}^{H}\mathbf{w}_{u}$ becomes purely
real and nonnnegative [48, ch. 18]. Therefore, the following holds

$\displaystyle\left\|\mathbf{x}\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|+\left(1-\alpha_{u,j}\right)Q_{u}\triangleq$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\begin{cases}\left\|\mathbf{x}\right\|_{2}\leq\sqrt{1+{\Gamma_{j}^{\mathrm{UE}}}^{-1}}\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}+\left(1-\alpha_{u,j}\right)Q_{u},\\\
\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}\geq 0,\\\
\mathsf{Im}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}=0.\end{cases}$

Similarly, constraint $\mathrm{C_{22}}$ can be expressed as

$\alpha_{u,j}\sqrt{\Gamma_{j}^{\mathrm{UE}}}\sigma_{\mathrm{UE}}\leq\left|\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right|\triangleq\begin{cases}\alpha_{u,j}\sqrt{\Gamma_{j}^{\mathrm{UE}}}\sigma_{\mathrm{UE}}\leq\mathsf{Re}\left\\{\mathbf{h}_{u}^{H}\mathbf{w}_{u}\right\\}\\\