Energy and Spectrum Efficient User Association in
Millimeter Wave Backhaul Small Cell Networks
Agapi Mesodiakaki, Member, IEEE, Ferran Adelantado, Member, IEEE, Luis Alonso, Senior Member, IEEE,
Marco Di Renzo Senior Member, IEEE, Christos Verikoukis, Senior Member, IEEE
Abstract—Macrocells are expected to be densely overlaid by
small cells (SCs) to meet the increasing capacity demands. Due to
their dense deployment, some SCs will not be connected directly
to the core network and thus they may forward their traffic to the
neighboring SCs until they reach it, thereby forming a multi-hop
backhaul (BH) network. This is a promising solution, since the
expected short length of BH links enables the use of millimeter
wave (mmWave) frequencies to provide high capacity BH. In this
context, user association becomes challenging due to the multi-
hop BH architecture and therefore new optimal solutions should
be developed. Thus, in this paper, we study the user association
problem aiming at the joint maximization of network energy and
spectrum efficiency, without compromising the user quality of
service. The problem is formulated as an ε-constraint problem,
which considers the transmit energy consumption both in the
access network, i.e., the links between the users and their serving
cells, and the BH links. The optimal Pareto front solutions of the
problem are analytically derived for different BH technologies
and insights are gained into the energy and spectrum efficiency
trade-off. The proposed optimal solutions, despite their high
complexity, can be used as a benchmark for the performance
evaluation of user association algorithms. We also propose a
heuristic algorithm, which is compared with reference solutions
under different traffic distribution scenarios and BH technologies.
Our results motivate the use of mmWave BH, while the proposed
algorithm is shown to achieve near-optimal performance.
Index Terms—Backhaul, cell selection, context-awareness,
green communications, LTE-Advanced, millimeter wave.
I. INTRODUCTION
T
HE mobile data traffic is expected to grow significantly
during the next few years, which results in an urgent
need for mobile operators to maintain capacity growth. Serving
more traffic leads to increased energy consumption, and there-
fore, how to minimize the energy consumption becomes also
important. In parallel, the spectrum scarcity problem stresses
the need for spectral efficient solutions. The aforementioned
goals can be summarized into the joint maximization of energy
and spectrum efficiency, which constitutes a fundamental
design objective for next generation cellular networks.
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
A. Mesodiakaki is with the Department of Computer Science, Karlstad
University, Karlstad, Sweden. E-mail: agapi.mesodiakaki@kau.se
F. Adelantado is with Open University of Catalonia, Barcelona, Spain. E-
mail: ferranadelantado@uoc.edu
L. Alonso is with the Signal Theory and Communications Department,
Technical University of Catalonia, Spain. E-mail: luisg@tsc.upc.edu
M. Di Renzo is with the Laboratoire des Signaux et Syst
`
emes, CNRS,
CentraleSup
´
elec, Univ Paris Sud, Universit
´
e Paris-Saclay, Gif-sur-Yvette,
France. E-mail: marco.direnzo@l2s.centralesupelec.fr
C. Verikoukis is with Telecommunications Technological Centre of Catalo-
nia, Castelldefels, Spain. E-mail: cveri@cttc.es
To that end, the dense deployment of small cells (SCs),
overlaying the existing macrocell networks, is a promising so-
lution. The SC deployment reduces the distance between user
equipments (UEs) and base stations (BSs)
1
and, consequently,
i) the area spectral efficiency (bps/Hz/m
2
) increases, and ii) the
energy consumption in the access network (AN), i.e., the links
between the UEs and their serving BSs, decreases. Hence,
dense deployment of SCs is expected during the next years,
with SC radius being eventually of the order of 50 meters [1].
However, the dense deployment of SCs also poses new
challenges. Due to the high number of deployed SCs, the
direct connection of all SCs to the core network becomes
complicated. Fiber connections, which have been traditionally
considered as the best backhaul (BH) solution, are prohibitive
in this case due to their high deployment cost [2]. A promising
solution lies in exploiting the existing connection between the
macrocell site and the core network (most of the times it is a
fiber connection), and to provide core network connectivity to
SCs through the macrocell site [3]. Still, in order to connect the
SCs to the macrocell site (thus providing them core network
connectivity), new cheap wireless BH solutions are required.
In addition, this wireless BH is expected to provide high-
capacity services from the SCs to the core network, in order
to meet the expected traffic demands of the order of Gbps
[1]. Therefore, a promising solution for high capacity wireless
BH connections between the SCs and the core network lies
in using millimeter wave (mmWave) frequencies, due to their
high bandwidth availability [2]. It has been shown, however,
that mmWave frequencies are capable of providing good
coverage only if the transmission distance is shorter than 200
meters [1]–[3]. Otherwise, links may not be established. In
parallel, small wavelengths enable highly directive antennas to
compensate the high path loss with the use of pencil beams [2].
Since the macrocell radius is even in dense deployments of the
order of 500 meters, this implies that a multi-hop architecture
of point-to-point line-of-sight (LOS) links is needed, in order
to allow each of the SCs to reach the macrocell site [3], [4].
In this context, user association becomes challenging due to
the multi-hop BH architecture [5] and therefore new optimal
solutions need to be developed aiming at the joint energy and
spectrum efficiency maximization of the network.
A. State-of-the-art and Contribution
The user association problem has received a lot of research
attention, since it impacts both the network and UE perfor-
1
In this paper, we will use the term BS to refer to a macrocell BS and/or a
SC BS (i.e., an eNodeB (eNB) and/or a Home eNB in LTE-A, respectively).
2
mance. In LTE-Advanced, the user association is based on
the reference signal received power (RSRP), which measures
the average received power over the resource elements that
carry cell-specific reference signals within certain bandwidth
[4]. Although RSRP maximizes the signal-to-interference-
plus-noise ratio (SINR) of UEs, it was shown that it does not
significantly increase the overall throughput, since few users
get connected to SCs [6]. Thus, range expansion (RE) (also
known as biasing) was introduced, whereby UEs are actively
pushed onto SCs [6]. In this case, although a UE may be
associated with a BS not providing the best SINR, better load
balancing is achieved between SCs and macrocell.
In [7], the authors propose a low-complexity distributed
algorithm that converges to a near-optimal solution and they
show that a per-tier biasing loses little, if the bias values
are chosen carefully. In [8], the joint user association and
resource allocation problem is studied. The authors aim to find
the optimal association so that the total resources required to
satisfy the given UE traffic demands are minimized. Focusing
also on the joint spectrum allocation and user association
problem, in [9], a proportionally fair utility function based
on the coverage rate is defined. The authors associate the UEs
with BSs based on the biased downlink received power, while
stochastic geometry is used to model the placement of BSs.
In [10], the authors formulate two different user association
problems. The first one is based on a sum utility of long-
term rate maximization with rate quality of service (QoS)
constraints, and the second on minimizing a global outage
probability with outage QoS constraints.
Taking into account the BH, in [11], the authors model a
BH-aware BS assignment problem as a multiple-choice multi-
dimensional Knapsack problem. In the considered framework,
they impose constraints on both AN and BH resources. The
main idea behind their algorithm is to distribute traffic among
BSs according to a load balancing strategy, considering both
AN and BH load status. Yet, the proposed algorithm, reduces
the BH congestion at the expense of lower spectral efficiency,
since some UEs may be assigned to non-optimal BSs in terms
of RSRP. In [12], a load-balancing based mobile association
framework is proposed under both full and partial frequency
reuse, and pseudo-optimal solutions are derived using gradient
descent method. In [13], a new theoretical framework is
introduced to model the downlink user association problem,
while upper bounds are derived for the achievable sum rate
and minimum rate using convex optimization. In [14], a joint
user association and resource allocation optimization problem
is proposed, which is shown to be NP-hard. Therefore, the
authors develop techniques to obtain upper bounds on the
system performance. In [15], the joint problem of downlink
user association and wireless BH bandwidth allocation is
studied in two-tier cellular heterogeneous networks (HetNets).
According to the considered architecture, SCs are connected
through wireless BH with the macrocell BS. The problem
is formulated as a sum logarithmic user rate maximization
problem, and wireless BH constraints are also considered.
However, the aforementioned approaches either consider
only the AN [4], [6]–[10], thus totally overlooking the BH
capacity constraints and energy impact, or do not take into
account the energy consumption of the network and hence,
their energy efficiency cannot be guaranteed [11]–[15].
To that end, in this paper, we study the user association
problem aiming at the joint energy and spectrum efficiency
maximization, while taking into account both the AN and BH
and without compromising the UE throughput demands. Pre-
liminary results of this research have been published in [16].
However, in this paper, we provide the following contributions:
• The aforementioned problem is formulated as an ε-
constraint problem [17], where the total transmit power
consumption of AN and BH is the objective to be
minimized and the amount of spectrum resources needed
is set as constraint, with its upper bound denoted by ε.
• We study the trade-off between energy and spectrum
efficiency analytically for different BH technologies by
solving the ε-constraint problem for all different ε.
Thereby, we derive the Pareto front solutions of the
problem, i.e., the set of optimal solutions for all ε values,
which can be used as a benchmark for the performance
evaluation of user association algorithms.
• Due to the high complexity of the derived optimal solu-
tions, which increases for a higher number of UEs and
BSs, we also propose a low-complexity user association
algorithm, which aims at the maximization of the energy
efficiency given a specific spectral efficiency target. The
algorithm is able to select any point of the Pareto front,
by accordingly tuning a single parameter, i.e., the spectral
efficiency target. Moreover, for each UE, it considers the
total transmit power consumption needed (both AN and
BH) to serve its traffic. This association metric relaxes the
assumption of [16] that all BH links are homogeneous,
by considering the actual transmit power consumption of
each BH link and not just the number of hops.
• Finally, we compare the energy and spectrum efficiency
of the proposed algorithm with existing user association
solutions as well as with the derived optimal solutions
under different spectral efficiency targets, traffic distribu-
tion scenarios and BH technologies. Our results motivate
the use of mmWave frequencies to provide high capacity
BH, while the proposed algorithm is shown to achieve
notable performance gains.
The rest of the paper is organized as follows: In Section II,
the system model is presented. In Section III, the problem
formulation and the solution methodology are provided. In
Section IV and Section V, the proposed algorithm is described
and compared, respectively, with existing user association
algorithms as well as with the analytical solutions derived in
Section III. Finally, Section VI concludes the paper.
II. SYSTEM MODEL
Without loss of generality and in accordance with the
scenarios proposed by 3GPP [18], we focus our analysis on a
single eNB sector, overlaid with multiple SCs. In particular,
we consider a set of BSs, denoted by C, which includes one
eNB (j=0) and C −1 SCs (j=1...C-1), with C representing the
cardinality of the set C. The SCs are divided in N
cl
clusters
(k=1...N
cl
), as depicted in Fig. 1, with SC
k
denoting the
3
mmWave BH link
eNB/SC
aggregation GW
fiber link
eNB j=0
MME / S-GW
SC j=1...C-1
k=1
k=N
cl
UE
Fig. 1. System model.
number of SCs in cluster k [18]. We study the downlink and
make the following assumptions:
• Each SC is connected to the core network through the
eNB aggregation gateway either directly or through one
or more SC aggregation gateways [3]–[5].
• There is a fiber connection between the core network and
the eNB site, and a set of point-to-point LOS mmWave
BH links between the eNB site and the SCs, denoted
by L={L
1
,L
2
,...L
l
,... L
C−1
}. Each mmWave BH link l
is represented by a set L
l
that includes all cells j that
backhaul their traffic through it (i.e., ∀j ∈ L
l
).
• Flat slow-fading channels are considered [14]. Therefore,
we assume that the total transmission power of each BS
is equally distributed among its subcarriers [4].
• We consider a set of N UEs (i=1,...,N) with strict
guaranteed bit rate (GBR) QoS requirements, denoted as
r
i,net
, based on their service/application [19].
• Each UE can be associated only with one BS at a time.
• There is a maximum number of spectrum resource units
available to each BS j, i.e., physical resource blocks
(PRBs)
2
, denoted by c
j
max
.
In the following, the most important parameters involved in
the total network energy efficiency calculation are derived. The
SINR calculation is given in Section II-A, while both AN and
BH power consumption models are provided in Section II-B.
A. SINR calculation
The signal-to-noise ratio (SNR) received by UE i from BS
j is given by [20]
SNR
ij
(dB)
= P
j
P RB
(dBm)
+ G
T
x
j
(dBi)
− L
cb
j
(dB)
−L
p
ij
(dB)
− L
f
ij
(dB)
− N
th
(dBm)
− NF
(dB)
(1)
with P
j
P RB
=10log
10
(P
j
max
/c
j
max
) being the power allocated
by BS j to a PRB, where P
j
max
is its maximum transmission
power (mW), and c
j
max
is the maximum number of PRBs
allocated to it. The parameter G
T
x
j
is the antenna gain of BS j
and L
cb
j
is the cable loss between the radio RF connector and
2
Please note that 1 PRB is equal to 12 subcarriers in the frequency domain
and 0.5 ms in the time domain [4].
the antenna. The path loss between UE i and BS j is denoted
by L
p
ij
, while L
f
ij
represents the losses due to shadowing.
Finally, N
th
stands for the thermal noise and N F is the noise
figure. The SINR of UE i from BS j is given by
SINR
ij
(dB)
= SNR
ij
(dB)
− 10log
10
I
ij(mW )
N
total
(mW )
+ 1
(2)
where I
ij
is the total interference experienced by UE i, when
associated with BS j, which depends on the applied frequency
allocation scheme. Due to the constant power allocation, the
SINR
ij
of UE i from BS j can be estimated a priori
3
, and
be given as an input to the problem. Hence, the proposed work
can be applied regardless of the employed channel allocation
scheme. Still, although it is out of the scope of this paper,
the combination of our proposal with a sophisticated channel
allocation could further improve the system performance.
Finally, N
total
= 10
(N
th
(dBm)
+NF
(dB)
)/10
denotes the total
noise power (mW) experienced by UE i.
B. Power consumption models
The total network power consumption can be divided into
the power consumed in the BSs (i.e, in the AN) and in the
BH links. The first is given by [16], [21]
P
AN
(W )
=
X
j∈C
P
AN
j
stat
(W )
+ P
AN
j
var
(W )
(3)
where P
AN
j
stat
is the fixed power consumption of BS j attributed
to e.g., power supply, cooling, and baseband unit operation
[21] and P
AN
j
var
is the load-dependent power consumption of
BS j. Without loss of generality, we assume ideal electronics
in terms of power efficiency and therefore the load dependent
power consumption part becomes equal to the radio frequency
(RF) transmit power consumption part, which is given by [16]
P
AN
j
var
(W )
= P
AN
j
RF
(W )
=
X
i∈N
(P
j
P RB
(W )
)dc
ij
ea
ij
=
X
i∈N
P
j
max
c
j
max
!
r
i,net
(1 − BLE R)(1 − k
ov
)
1
b log
2
(1 + SINR
ij
)
a
ij
(4)
where c
ij
represents the number of PRBs needed for the
association of UE i with BS j and r
i,net
is the rate demand
of UE i. The parameter BLER stands for the block error rate
(BLER), i.e, for the number of erroneous blocks divided by the
total number of received blocks [22] and k
ov
is the percentage
of overhead bits (e.g., cyclic prefixes, reference signals) [23].
From now on, we will denote as r
i
=
r
i,net
(1−BLER)(1−k
ov
)
, the
total rate needed for the satisfaction of r
i,net
. Parameter b
is the bandwidth of a PRB and d·e is the ceiling function
operator. The denominator of the third fraction is derived by
Shannon’s theorem and represents the maximum rate that can
be achieved with effective SINR
ij
[23] and bandwidth equal
to b. Finally, a
ij
is the association vector (equal to 1 when the
UE i is associated with BS j and 0 otherwise).
3
Please note that the SINR after the UE association may differ from the
estimated one, as the interference experienced by the UE depends on resource
allocation i.e., whether neighboring BSs allocate the same PRBs to other UEs,
and consequently on user association. In this work, to overcome this problem,
the worst-case scenario in terms of generated interference is considered.
4
Similar to the AN, the power consumption of the BH links
consists of a fixed and a variable part [21], and thus equals to
P
BH
(W )
=
X
L
l
∈L
P
BH
L
l
stat
(W )
+ P
BH
L
l
var
(W )
(5)
Under the assumption of ideal electronics, the load dependent
power consumption of a BH link L
l
, i.e., P
BH
L
l
var
, equals to the
RF transmit power consumption, which is given by [20], [24]
P
BH
L
l
RF
(dBm)
= SINR
trg
L
l
(dB)
+
α
L
l
z
}| {
L
p
o
(dB)
+ L
p
L
l
(dB)
+ IL
(dB)
z }| {
+N
th
(dBm)
+ N F
(dB)
− G
T
x
L
l
(dBi)
− G
R
x
L
l
(dBi)
(6)
where L
p
o
is the path loss at 1 m distance and
L
p
L
l
=20log
10
(4π
d
L
l
λ
) is the path loss at distance d
L
l
equal
to the length of the link. Moreover, λ is the signal wavelength
(e.g., for 60 GHz, λ = 0.005 m) and IL is the implementation
loss that may account for e.g., distortion, intermodulation
and/or phase noise. The over-braced equation, which is de-
rived by subtracting from the total losses, the transmitter and
receiver antenna gains of the BH link, will be denoted from
now on by α
L
l
. Finally, assuming that link adaptation is
employed [20], SINR
trg
L
l
corresponds to the (minimum) target
SINR that is needed so that the aggregated BH link traffic is
successfully transmitted and can be given by [20]
SINR
trg
L
l
(dB)
= 10log
10
2
P
i∈N
P
j∈L
l
r
i
a
ij
B
L
l
− 1
(7)
where B
L
l
is the bandwidth of the BH link L
l
and
P
i∈N
P
j∈L
l
r
i
a
ij
is the aggregated traffic that passes
through it. For mmWave, the generated interference is neg-
ligible due to high path loss, and thus SINR
trg
L
l
=SNR
trg
L
l
.
III. PROBLEM FORMULATION
The problem under study aims at the joint maximization of
the network energy and spectrum efficiency, without compro-
mising the UE QoS (i.e., the UE throughput demands). The
energy efficiency (bits/Joule) is expressed as the total number
of successfully transmitted useful bits divided by the total
energy consumption or equivalently as the total goodput of the
network divided by the total power consumption (i.e., the sum
of the power consumed in the AN and in the BH links). Under
the condition that the specific UE throughput demands are
satisfied, the network energy efficiency maximization is equiv-
alent to power consumption minimization, while the spectral
efficiency maximization is equivalent to PRBs minimization.
The aforementioned problem is a non-convex multi-
objective problem. Therefore, for its formulation, we employ
the ε-constraint method, which is able to find any Pareto
optimal solution even for non-convex problems [17]. Accord-
ing to it, one of the objectives is included in the utility
function to be optimized (i.e., minimization of the total power
consumption), while the others (i.e., minimization of the total
number of required PRBs) are converted into constraints by
setting an upper bound to them. Given that the fixed power
consumption
4
is independent of the user association decision,
4
Still, the inclusion of the fixed power would impact all the algorithms by
equally increasing their power consumption.
the minimization of the total power consumption is equivalent
to the minimization of the traffic-dependent part (i.e., the RF
transmit power consumption in our case). Therefore, our study
from now on focuses on this part, as depicted in (8).
Hence, the first term of the objective function in (8) repre-
sents the total RF transmit power consumption of the AN and
the second of the BH links. We remind that a
ij
5
denotes the
association vector that is equal to 1 when the UE i is associated
with BS j and 0 otherwise (8a). Each UE can be associated
only with one BS at a time (8b). The total number of PRBs
used by BS j, denoted by c
ij
, cannot exceed the maximum
number that is allocated to it (8c). The RF transmit power
consumption of the BH link L
l
cannot exceed a maximum
value, denoted by P
BH
max
(8d). The parameter s
L
l
j
is 1 if
the traffic of the BS j passes through the BH link L
l
and
0 otherwise (8e). Finally, constraint (8f) refers to the total
number of PRBs and thus to the network spectrum efficiency.
argmin
a
ij
f
1
(a
ij
) =
X
j∈C
P
AN
j
RF
(W )
z }| {
X
i∈N
P
j
P RB
c
ij
a
ij
+
+
X
L
l
∈L
P
BH
L
l
RF
(W )
z }| {
2
P
i∈N
P
j∈C
a
ij
s
L
l
j
r
i
B
L
l
− 1
10
α
L
l
−30
10
s.t. a) a
ij
∈ {0, 1}, ∀i ∈ N , ∀j ∈ C
b)
X
j∈C
a
ij
= 1, ∀i ∈ N
c)
X
i∈N
a
ij
c
ij
≤ c
j
max
, ∀j ∈ C
d)P
BH
L
l
RF
≤ P
BH
max
∀L
l
∈L
e) s
L
l
j
∈ {0, 1}, ∀L
l
∈ L, ∀j ∈ C
f)f
2
(a
ij
) =
X
i∈N
X
j∈C
a
ij
c
ij
≤ ε
(8)
Theorem 1. The solution of the ε-constraint problem in (8)
is weakly Pareto optimal.
Proof. Let a
?
ij
be a solution of the ε-constraint problem. Let
us assume that a
?
ij
is not weakly Pareto optimal. In this case
there exists some other a
ij
such that f
k
(a
ij
) < f
k
(a
?
ij
) for
k=1,2. This means that f
2
(a
ij
) < f
2
(a
?
ij
) ≤ ε. Hence, a
ij
is feasible with respect to the ε-constraint problem. While in
addition f
1
(a
ij
) < f
1
(a
?
ij
), we have a contradiction to the
assumption that a
?
ij
is a solution of the ε-constraint problem.
Thus, a
?
ij
6
has to be weakly Pareto optimal.
Although, according to Theorem 1, every solution of the ε-
constraint problem is weakly Pareto optimal, there is no Pareto
optimal solution, since there is no solution that optimizes
both objectives simultaneously. Therefore, it is reasonable
to search for a good trade-off between the two objectives
5
Due to the binary association vector and the non-linear objective function
and contraints, the problem is a 0-1 non-linear integer programming problem.
6
Please note that, in the rest of the paper, a
ij
is omitted.
5
instead. To that end, the increase of ε leads to a relaxation
of the spectral efficiency constraint (i.e., f
2
) and consequently
to a more energy efficient solution. On the contrary, the
decrease of ε improves the spectral efficiency of the solution
by degrading its energy efficiency. The set of solutions for
the subproblems resulting from the variation of ε define the
Pareto front, hereafter denoted by F. In practice, due to the
high number of subproblems and the difficulty to establish
an efficient variation scheme for the ε-vector, this approach
has mostly been integrated within heuristic and interactive
schemes. However, due to the nature of (8), it is possible
to derive the exact Pareto front with the use of an iterative
algorithm [25]. The idea is to construct a sequence of ε-
constraint problems based on a progressive reduction of ε.
Let
~
φ
I
= (φ
I
1
, φ
I
2
) be the ideal point, where φ
I
1
= min(f
1
)
and φ
I
2
= min(f
2
) stand for the minimum value of f
1
and
f
2
, respectively. Equivalently, let
~
φ
N
= (φ
N
1
, φ
N
2
) be the
nadir point, with φ
N
1
and φ
N
2
being the minimum values
of f
1
and f
2
, when f
2
= φ
I
2
and f
1
= φ
I
1
, respectively,
i.e., φ
N
1
= min{f
1
:f
2
=φ
I
2
} and φ
N
2
= min{f
2
:f
1
=φ
I
1
}. Thus,
(φ
I
1
, φ
N
2
) is the solution of the Pareto front that minimizes
the RF transmit power consumption (i.e., f
1
) without spectral
efficiency constraints, whereas (φ
N
1
, φ
I
2
) is the solution in F
that minimizes the total number of PRBs used (i.e., f
2
).
Lemma 1. Both (φ
I
1
, φ
N
2
) and (φ
N
1
, φ
I
2
) belong to F, i.e., (φ
I
1
,
φ
N
2
) ∈ F and (φ
N
1
, φ
I
2
) ∈ F.
Proof. Let us assume that (φ
I
1
, φ
N
2
) /∈ F. Then, ∃
~
f=(f
1
, f
2
)
∈ Φ: (f
1
, f
2
) (φ
I
1
, φ
N
2
), where Φ denotes the objective
space and the expression
~
f=(f
1
, f
2
) (φ
I
1
, φ
N
2
) denotes that
(f
1
, f
2
) dominates (φ
I
1
, φ
N
2
). In general, we say that
~
f=(f
1
,
f
2
) dominates
~
f
0
=(f
0
1
, f
0
2
), with
~
f,
~
f
0
∈ Φ if and only if (iff)
f
1
≤ f
0
1
and f
2
≤ f
0
2
, where at least one inequality is strict.
Thus,
~
f=(f
1
, f
2
) (φ
I
1
, φ
N
2
) is true when a) f
1
< φ
I
1
and f
2
< φ
N
2
or b) f
1
< φ
I
1
and f
2
= φ
N
2
or c) f
1
= φ
I
1
and f
2
<
φ
N
2
. Since a) and b) contradict the definition of an ideal point
and since c) contradicts the definition of a nadir point, then
(φ
I
1
, φ
N
2
) ∈ F. The proof of (φ
N
1
, φ
I
2
) ∈ F is analogous.
Lemma 2. For each (f
1
, f
2
) ∈ Φ, if (f
1
, f
2
) ∈ F, then φ
I
1
≤ f
1
≤ φ
N
1
and φ
I
2
≤ f
2
≤ φ
N
2
.
Proof. As proved in Lemma 1, (φ
I
1
, φ
N
2
) ∈ F, and thus it is
non-dominated. Since φ
I
1
= min(f
1
), f
1
≥ φ
I
1
, ∀ (f
1
, f
2
) ∈ F.
Moreover, if f
2
> φ
N
2
, (φ
I
1
, φ
N
2
) (f
1
,f
2
) and (f
1
,f
2
) /∈ F.
Hence, f
1
≥ φ
I
1
and f
2
≤ φ
N
2
∀ (f
1
, f
2
) ∈ F. The proof for
φ
I
2
≤ f
2
≤ φ
N
2
is analogous.
According to Lemma 1 and Lemma 2, Algorithm 1 gener-
ates the exact Pareto front of the problem described in (8).
Theorem 2. Algorithm 1 generates one feasible solution for
each point of the Pareto front.
Proof. Let us denote the sequence of solutions of Algorithm 1
by {
~
f
∗
1
, . . . ,
~
f
∗
m
, . . . ,
~
f
∗
M
}, where, e.g.,
~
f
∗
m
=
(f
∗
m
)
1
, (f
∗
m
)
2
,
Algorithm 1 Exact Pareto front calculation of problem (8)
1: Calculate the ideal and nadir points,
~
φ
I
and
~
φ
N
.
2: Add
~
f
∗
1
= (φ
I
1
, φ
N
2
) to F.
3: Set m = 2.
4: Set ε
m
= φ
N
2
− ∆, with ∆ = 1.
5: while ε
m
≥ φ
I
2
do
6: Solve problem (8) and add the optimal solution value
~
f
∗
m
=
(f
∗
m
)
1
, (f
∗
m
)
2
to F.
7: Set ε
m+1
= (f
∗
m
)
2
− ∆.
8: Set m = m + 1.
9: end while
10: Remove dominated points if required.
with 1, 2 denoting the first and the second objective, re-
spectively. We have to prove that if
~
f ∈ Φ \ {
~
f
∗
1
, . . . ,
~
f
∗
m
, . . . ,
~
f
∗
M
}, then
~
f /∈ F. Let us assume that there is a
solution
~
f
0
= (f
0
1
, f
0
2
) ∈ Φ \ {
~
f
∗
1
, . . . ,
~
f
∗
m
, . . . ,
~
f
∗
M
} such
that
~
f
0
∈ F. By Lemma 2, for the first objective we have
φ
I
1
≤ f
0
1
≤ φ
N
1
. Thus, either a) f
0
1
= (f
∗
m
)
1
for a given
m = 1 . . . M or b) (f
∗
m−1
)
1
< f
0
1
< (f
∗
m
)
1
and (f
∗
m−1
)
2
< f
0
2
≤ (f
∗
m
)
2
for a given m = 1 . . . M. In the first case (i.e.,
case a), f
0
2
must be lower than (f
∗
m
)
2
for
~
f
0
to be efficient.
However, since ∆ = 1 and the second objective is integer by
definition, ∃ ε
m
0
that will eventually reach a value for which
the optimum of the corresponding ε-constraint problem is
~
f
0
for m + 1 ≤ m
0
≤ M, that is
~
f
0
∈ {
~
f
∗
m+1
, . . . ,
~
f
∗
M
}, which
contradicts the hypothesis. Regarding the second case (i.e.,
case b), f
0
2
must be such that (f
∗
m−1
)
2
< f
0
2
≤ (f
∗
m
)
2
, which is
impossible since
~
f
∗
m
is the optimal value of problem (8), with
ε
m
=ε
m−1
-∆, ∆=1, and the second objective is integer.
Some dominated solutions may be generated by the se-
quence of subproblems derived according to Theorem 2.
However, since all dominated points can be identified, one
can simply exclude the non-efficient solutions to obtain the
exact Pareto front. Furthermore, although Algorithm 1 limits
the number of subproblems, a subproblem may be very hard to
solve. This stems from the fact that an exhaustive search would
require the examination of C
N
possible solutions, which re-
sults in prohibitive complexity (O(n
n
)), as the number of BSs,
C, and the number of UEs, N , increase. Therefore, alternative
algorithms, available in the literature, should be used, able
to come up with very close to the optimal solutions with
acceptable computational complexity [17]. In this work, we
applied a meta-heuristic method [26], which has been shown
to lead to high-quality solutions (the average gap is less than
1% with respect to best-known solutions) in almost real time.
The applied method uses biased randomization together with
an iterated local search meta-heuristic algorithm. Although
the meta-heuristic algorithm involves lower complexity than
O(n
n
)
7
, it still requires a high number of iterations (50000
in our case). Therefore, there is need for low-complexity
algorithms, able to achieve solutions close to the Pareto front.
7
Meta-heuristics have no predefined end, and thus big O notation cannot
be used to describe their complexity. Yet, they can be compared empirically
(through number of objective function evaluations/iterations).
