1
Achieving Maximum Energy-Efficiency in
Multi-Relay OFDMA Cellular Networks:
A Fractional Programming Approach
Kent Tsz Kan Cheung, Shaoshi Yang, and Lajos Hanzo, Fellow, IEEE
Abstract—In this paper, the joint power and subcarrier al-
location problem is solved in the context of maximizing the
energy-efficiency (EE) of a multi-user, multi-relay orthogonal
frequency division multiple access (OFDMA) cellular network,
where the objective function is formulated as the ratio of the
spectral-efficiency (SE) over the total power dissipation. It is
proven that the fractional programming problem considered is
quasi-concave so that Dinkelbach’s method may be employed for
finding the optimal solution at a low complexity. This method
solves the above-mentioned master problem by solving a series
of parameterized concave secondary problems. These secondary
problems are solved using a dual decomposition approach, where
each secondary problem is further decomposed into a number of
similar subproblems. The impact of various system parameters
on the attainable EE and SE of the system employing both EE
maximization (EEM) and SE maximization (SEM) algorithms is
characterized. In particular, it is observed that increasing the
number of relays for a range of cell sizes, although marginally
increases the attainable SE, reduces the EE significantly. It
is noted that the highest SE and EE are achieved, when the
relays are placed closer to the BS to take advantage of the
resultant line-of-sight link. Furthermore, increasing both the
number of available subcarriers and the number of active user
equipment (UE) increases both the EE and the total SE of
the system as a benefit of the increased frequency and multi-
user diversity, respectively. Finally, it is demonstrated that as
expected, increasing the available power tends to improve the SE,
when using the SEM algorithm. By contrast, given a sufficiently
high available power, the EEM algorithm attains the maximum
achievable EE and a suboptimal SE.
Index Terms—Subcarrier/power allocation, green communi-
cations, energy-efficiency, multiple relays, dual decomposition,
fractional programming.
I. INTRODUCTION
E
NERGY-efficiency (EE) is becoming of great concern in
the telecommunications community owing to the rapidly
increasing data rate requirements, increasing energy prices,
and societal as well as political pressures on mobile phone
This research has been funded by the Industrial Companies who are
Members of the Mobile VCE, with additional financial support from the UK
Government’s Engineering & Physical Sciences Research Council (EPSRC).
The financial support of the China Scholarship Council (CSC), of the Research
Councils UK (RCUK) under the India-UK Advanced Technology Center (IU-
ATC), of the EU under the auspices of the Concerto project, and of the
European Research Council’s Senior Research Fellow Grant is also gratefully
acknowledged.
K. T. K. Cheung is with the School of Electronics and Computer Sci-
ence, University of Southampton, Southampton, SO17 1BJ, UK (e-mail:
ktkc106@ecs.soton.ac.uk).
S. Yang is with the School of Electronics and Computer Science, Uni-
versity of Southampton, Southampton, SO17 1BJ, UK. He is also with
the School of Information and Communication Engineering, Beijing Uni-
versity of Posts and Telecommunications, Beijing, 100876, China (e-mail:
sy7g09@ecs.soton.ac.uk).
L. Hanzo is with the School of Electronics and Computer Sci-
ence, University of Southampton, Southampton, SO17 1BJ, UK (e-mail:
lh@ecs.soton.ac.uk).
operators to reduce their ’carbon footprint’ [1]. This has
led to several joint academic and industrial research ef-
forts dedicated to developing novel energy-saving techniques,
such as the ’green radio’ project [2], the GreenTouch al-
liance [3], and the energy aware radio and network technolo-
gies (EARTH) project [4]. Substantial research efforts have
also been dedicated to the next-generation wireless networks,
such as the third generation partnership project’s (3GPP) long
term evolution-advanced (LTE-A) and IEEE 802.16 world-
wide interoperability for microwave access (WiMAX) [5]
standards, which may rely on relaying between the central
base station (BS) and the user equipment (UE). As a benefit
of reduced transmission distances, either the quality of the
communication is maintained at reduced power requirements,
or the transmission integrity is improved at the same power
consumption. This allows the need for expensive deployment
and maintenance of additional BSs to be circumvented. The
two most popular relaying techniques are the amplify-and-
forward (AF) and the decode-and-forward (DF) schemes [6].
The AF regime is less complex than DF, since the relay
node (RN) needs only to receive and linearly amplify the
source’s transmissions, before forwarding it to the destination.
The effects of scheduling and frequency reuse in the context
of the above-mentioned networks was studied in [7].
Both LTE-A and WiMAX employ the orthogonal frequency
division multiple access (OFDMA) technique. In OFDMA,
the whole channel’s bandwidth is divided into multiple sub-
carriers, where subsets of subcarriers may be allocated for
transmission to different users [8]. In OFDMA, the system at-
tains two types of diversity, which may be jointly exploited for
improving the achievable sum-rate (SR) of the system. Firstly,
multi-user diversity is attained with the aid of appropriate user
mapping, because when the channel from the BS to a specific
UE is undergoing severe fading on a particular subcarrier, then
this subcarrier may be assigned for transmission to another
user, whose channel might be more friendly. On the other
hand, activating only those subcarriers that are suitable for
high-quality transmission to a particular UE leads to frequency
diversity. These philosophies underpin several contributions
in the literature, where the goal is to assign the available
resources, for example power and subcarriers, so that a system-
wide metric is maximized. These methods belong to the family
of resource allocation policies and typically aim for solving
one of two problems: either the spectral-efficiency (SE)
1
[9],
[10], [11] of the system is maximized while a maximum power
constraint is enforced, or the power consumption is minimized
under a minimum total system throughput or individual UE
1
For a given system bandwidth, the SR maximization and SE maximiza-
tion (SEM) solutions are identical. To avoid any additional abbreviations, they
are both henceforth referred to as SEM.
2
rate constraint
2
[12], [13], [14], [15], [16].
An example of the SE Maximization (SEM) problem was
considered in [11], where the authors formulate the optimiza-
tion problem for the downlink (DL) of an AF relaying-aided
OFDMA cellular network, and their goal was to optimize the
power and subcarrier allocation so that the SE of the system
was maximized under maximum outage probability and total
power constraints. In the class of power minimization prob-
lems, an example is the often-cited work by Wong et. al [12],
where a heuristic bit allocation algorithm was conceived for a
multi-user OFDMA system with the aim of minimizing the
power consumption under a minimum individual user rate
constraint. With a similar goal, Piazzo [13] developed a sub-
optimal bit allocation algorithm for an orthogonal frequency
division multiplexing (OFDM) system. This work was later
extended to provide the optimal bit allocation in [15].
However, the SEM and the power minimization problems
do not directly consider an EE objective function (OF), and in
general they do not deliver the EE maximization (EEM) so-
lution. In recent years, research into resource allocation using
an EE OF has become increasingly popular. In reality, EEM
may be viewed as an example of multi-objective optimization,
since typically the goal is to maximize the SE achieved, whilst
concurrently minimizing the power consumption required.
From this perspective, [17] derives an aggregate OF, which
consists of a weighted sum of the SR achieved and the power
dissipated. However, selecting appropriate weights for the two
OFs is not straightforward, and different combinations of
weights can lead to very different results. Another example
is given in [18], where the EEM problem is considered in
a multi-relay network employing the AF protocol. However,
both [17], [18] only optimize the user selection and power
allocation without considering the subcarrier allocation in the
network. Another formulation, demonstrated in [19], [20], con-
siders power and subcarrier allocation in an OFDMA cellular
network, but without a maximum total power constraint and
without relaying. The authors of [21] formulate the EEM
problem in a OFDMA cellular network under a maximum
total power constraint, however relaying is not considered.
In light of the above discussions, this work focuses on
a solution method for the EEM problem in a multi-relay,
multi-user OFDMA cellular network, which jointly considers
both power and subcarrier allocation as well as a maximum
total power constraint. The contributions of this paper is
summarized as follows:
• The EEM problem in the context of a multi-relay, multi-
user OFDMA cellular network, in which both direct and
relayed transmissions are employed, is formulated as a
fractional programming problem, which jointly considers
both the power and subcarrier allocation. In contrast to
previous contributions such as [7], the aim is for finding
the optimal power and subcarrier allocations within a
network context. Furthermore, in contrast to [12], [13],
[9], [10], [14], [11], [15], [18], [17], the focus is placed
on an EE OF. It is demonstrated that in its original form
the problem is a mixed-integer non-linear programming
problem (MINLP) [22], which is challenging to solve. In
order to make the problem more tractable, both a variable
transformation and a relaxation of the integer variables is
2
In the latter case, the minimum rate constraint may be viewed as ensuring
fairness among the users, since each user achieves at least a minimum rate.
introduced.
• It is proven that the relaxed problem is quasi-concave and
consequently Dinkelbach’s method [23] may be employed
for obtaining the optimal solution by solving a sequence
of parameterized secondary problems. Each of these are
solved using the dual decomposition approach of [24].
It is demonstrated that the EEM algorithm reaches the
optimal solution within a low number of iterations and
reaches the optimal solution obtained via an exhaustive
search. Thus the original problem is solved at a low
complexity.
• Comparisons are made between two multi-relay resource
allocation problems, namely one that solves the EEM
problem and another that considers SEM. As an example,
it is shown that when the maximum affordable power
is lower than a given threshold, the two problems have
the same solutions. However, as the maximum afford-
able power is increased, the SEM algorithm attempts to
achieve a higher SE at the cost of a lower EE, while
given the total power, the EEM algorithm reaches the
upper limit of the maximum achievable SE for the sake
of maintaining the maximum EE.
• Since the system model is generalized, the EEM and SEM
algorithms may be employed for gaining insights into
network design, when the aim is for maximizing either
the EE or SE. To that end, a comprehensive range of
results are presented, which demonstrate both the effect
of increasing the number of available subcarriers and
UEs in the system, as well as quantifying the impact
of increasing the number of RNs in the system and its
relation not only to the cell radius, but also to the relays’
positions. The algorithm may be used for characterizing
the effects of many other system design choices on the
maximum SE and EE.
The rest of this paper is organized as follows. In Section II,
the multi-user, multi-relay OFDMA cellular network model is
described, which is followed by a formulation of the optimiza-
tion problem in Section III. Upon invoking a transformation of
variables and a relaxation of the integer variables, it is proven
that the OF is quasi-concave. The combined solution algorithm
of Dinkelbach’s method [23] and dual decomposition [24]
is outlined in Section IV. The performance of the EEM
algorithm is demonstrated in Section V, which includes results
obtained when the EEM and SEM algorithms are employed
for characterizing the effect of different system design choices
on the achievable SE and EE. Lastly, conclusions are given in
Section VI, where future work ideas are also listed.
II. SYSTEM MODEL
Consider an OFDMA DL cellular system relying on a single
BS, M fixed RNs and K uniformly-distributed UEs, as shown
in Fig. 1
3
. The cell is divided into M sectors, where each
sector is served by one of the fixed RNs. Naturally, the path-
loss is a major factor in determining the receiver’s signal-to-
noise ratios (SNRs) at the UEs, and thus has a substantial
effect on the EE. Therefore, in order to minimize the RN-
to-UE pathloss, all the UEs in a specific sector are only
3
Although it is more realistic to consider a multi-cell system, which would
lead to inter-cell interference, our system model assumes that intelligent
interference coordination or mitigation techniques are employed such that
the level of inter-cell interference is negligible [25].
3
Mobile
Mobile
MobileMobile
Mobile
Mobile
Mobile
Mobile
Mobile
Mobile
Relay
Mobile
Mobile
Relay
Mobile
Mobile
Mobile
Mobile
Mobile
Relay
Mobile
Base station
Figure 1: An example of a cellular network with M = 3 RNs
and K = 18 UEs.
supported by that sector’s RN, and therefore relay selection is
implicitly accomplished. Although the model may be readily
extended to include relay selection, for the sake of mathemat-
ical tractability, it is not included in this work. The model
accounts for both the AF relayed link as well as for the direct
link between the BS and UEs, while the variables related
to these two communication protocols are distinguishable
by the superscripts A and D, respectively. When defining
links, the subscript 0 is used for indicating the BS, whilst
M(k) ∈ {1, · · · , M } indicates the RN selected for assisting
the DL-transmissions to user k. The proportion of the BS-to-
RN distance to the cell radius is denoted by D
r
, while the total
available instantaneous transmission power of the network is
P
max
. Although it is more realistic to consider a system with
separate power constraints for each transmitting entity, for
simplicity, a certain total power constraint is considered
4
. The
results obtained provide insights into holistic system design
by granting a higher grade of freedom in terms of sharing the
power among the transmitting entities, and thus attaining a
higher performance.
Using the direct transmission protocol, the receiver’s SNR at
UE k on subcarrier n may be expressed as Γ
D,n
k
(P), whereas
when using the AF relaying protocol, the receiver’s SNR at
UE k on subcarrier n may be expressed as [6]
Γ
A,n
k
(P) =
γ
A,n
0,M(k)
γ
A,n
M(k),k
γ
A,n
0,M(k)
+ γ
A,n
M(k),k
+ 1
, (1)
where γ
X,n
a,b
= P
X,n
a,b
G
n
a,b
/∆γN
0
W is the SNR at receiver
b ∈ {1, · · · , M, 1, · · · , K} on subcarrier n ∈ {1, · · · , N}, and
P
X,n
a,b
is allocated to transmitter a ∈ {0, · · · , M} using proto-
col X ∈ {D, A} for transmission to receiver b. Furthermore,
G
n
a,b
represents the channel’s attenuation between transmitter
a and receiver b on subcarrier n, which is assumed to be
known at the BS for all links. The channel’s attenuation is
modeled by the path-loss and the Rayleigh fading between the
transmitter and receiver. Furthermore, N
0
is the additive white
4
Additionally, it was empirically shown the dual decomposition approach
only obtains a local optimum when separate BS and RN power constraints
are imposed.
Gaussian noise (AWGN) variance and W is the bandwidth of
a single subcarrier. Still referring to (1), ∆γ is the SNR gap
at the system’s bit error ratio (BER) target between the SNR
required at the discrete-input continuous-output memoryless
channel (DCMC) capacity and the actual SNR required the
modulation and coding schemes of the practical physical layer
transceivers employed. For example, making the simplifying
assumption that idealized transceivers operating exactly at the
DCMC capacity are employed, then ∆γ = 0 dB. Although
it is not possible to operate exactly at the DCMC channel
capacity, several physical layer transceiver designs exist that
operate arbitrarily close to it [26]. Additionally, the power
allocation policy of the system is denoted by P, which
determines the values of P
X,n
a,b
.
Assuming sufficiently high receiver’s SNR values, the fol-
lowing approximation can be made
Γ
A,n
k
(P) ≈
P
A,n
0,M(k)
G
n
0,M(k)
P
A,n
M(k),k
G
n
M(k),k
∆γN
0
W
P
A,n
0,M(k)
G
n
0,M(k)
+ P
A,n
M(k),k
G
n
M(k),k
,
(2)
which is valid
5
for P
A,n
0,M(k)
G
n
0,M(k)
+ P
A,n
M(k),k
G
n
M(k),k
∆γN
0
W . The SE of an AF link to UE k on subcarrier n is
then given by
R
A,n
k
(P) =
1
2
log
2
1 + Γ
A,n
k
[bits/s/Hz], (3)
where the factor of
1
2
accounts for the fact that two time slots
are required for the two-hop AF transmission. The SE of a
direct link to UE k on subcarrier n is similarly given by
R
D,n
k
(P) = log
2
1 + Γ
D,n
k
[bits/s/Hz]. (4)
The subcarrier indicator variable s
X,n
k
∈ {0, 1} is now
introduced, which denotes the allocation of subcarrier n for
transmission to user k using protocol X for s
X,n
k
= 1, and
s
X,n
k
= 0 otherwise. The weighted average SE of the system
is calculated as
R
T
(P, S) =
1
N
K
X
k=1
ω
k
N
X
n=1
s
D,n
k
log
2
1 + Γ
D,n
k
+
s
A,n
k
2
log
2
1 + Γ
A,n
k
[bits/s/Hz], (5)
where S denotes the subcarrier allocation policy of the system,
which determines the values of the subcarrier indicator vari-
able s
X,n
k
. The weighting factor ω
k
may be varied for ensuring
fairness amongst users. However, since ensuring fairness is not
the focus of this work, ω
k
= 1, ∀k is assumed then the effect
of ω
k
may be ignored.
In order to compute the energy used in these transmissions,
a model similar to [27] is adopted and the total power
consumption of the system is assumed be governed by a
constant term and a term that varies with the transmission
powers, which may be written as (6).
Here, P
(B)
C
and P
(R)
C
represent the fixed power consumption
of each BS and each RN, respectively, while ξ
(B)
> 1
and ξ
(R)
> 1 denote the reciprocal of the drain efficiencies
of the power amplifiers employed at the BS and the RNs,
5
It is plausible that in next-generation systems, through the combination
of multi-user and frequency diversity, this assumption holds true when an
intelligent scheduler is employed [7].
4
P
T
(P, S) =
P
(B)
C
+ M · P
(R)
C
+
K
X
k=1
N
X
n=1
s
D,n
k
ξ
(B)
P
D,n
0,k
+
1
2
s
A,n
k
·
ξ
(B)
P
A,n
0,M(k)
+ ξ
(R)
P
A,n
M(k),k
[Watts] (6)
respectively. For example, an amplifier having a 25% drain
efficiency would have ξ =
1
0.25
= 4.
Finally, the average EE metric of the system is expressed
as
η
E
(P, S) =
R
T
(P, S)
P
T
(P, S)
[bits/Joule/Hz]. (7)
III. PROBLEM FORMULATION
The aim of this work is to maximize the energy efficiency
metric of (7) subject to a maximum total instantaneous trans-
mit power constraint. In its current form, (7) is dependent
on 3KN continuous power variables P
D,n
0,k
, P
A,n
0,M(k)
and
P
A,n
M(k),k
, ∀k, n, and 2KN binary subcarrier indicator variables
s
D,n
k
and s
A,n
k
, ∀k, n. Thus, it may be regarded as a MINLP
problem [22], and can be solved using the branch-and-bound
method of [28]. However, the computational effort required for
branch-and-bound techniques typically increases exponentially
with the problem size. Therefore, a simpler solution is derived
by relaxing the binary constraint imposed on the subcarrier
indicator variables, s
D,n
k
and s
A,n
k
, so that they may assume
continuous values from the interval [0, 1], as demonstrated
in [12], [29]. Furthermore, the variables
e
P
D,n
0,k
= P
D,n
0,k
s
D,n
k
,
e
P
A,n
0,M(k)
= P
A,n
0,M(k)
s
A,n
k
and
e
P
A,n
0,M(k)
= P
A,n
0,M(k)
s
A,n
k
are
introduced.
The relaxation of the binary constraints imposed on the
variables s
D,n
k
and s
A,n
k
allows them to assume continuous
values, which leads to a time-sharing subcarrier allocation
between the UEs. Naturally, the original problem is not actu-
ally solved. However, it has been shown that solving the dual
of the relaxed problem provides solutions that are arbitrarily
close to the original, non-relaxed problem, provided that the
number of available subcarriers tends to infinity [29]. It has
empirically been shown that in some cases only 8 subcarriers
are required for obtaining close-to-optimal results [30]. It
shall be demonstrated in Section V that even for as few as
two subcarriers, the solution algorithm employed in this work
approaches the optimal EE achieved by an exhaustive search.
The optimization problem is formulated as shown as fol-
lows:
Relaxed Problem (P):
maximize
e
R
T
e
P
T
(8)
subject to
K
X
k=1
N
X
n=1
e
P
D,n
0,k
+
e
P
A,n
0,M(k)
+
e
P
A,n
M(k),k
≤ P
max
,
(9)
s
D,n
k
+ s
A,n
k
≤ 1, ∀k, n, (10)
K
X
k=1
s
D,n
k
+ s
A,n
k
≤ 1, ∀n, (11)
e
P
D,n
0,k
,
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
∈ R
+
, ∀k, n, (12)
0 ≤ s
D,n
k
, s
A,n
k
≤ 1, ∀k, n, (13)
where the objective function is the ratio between (14) and (15).
In this formulation, the variables to be optimized are s
D,n
k
,
s
A,n
k
,
e
P
D,n
0,k
,
e
P
A,n
0,M(k)
and
e
P
A,n
M(k),k
, ∀k, n. Physically, the
constraint (9) ensures that the sum of the power allocated
to variables
e
P
D,n
0,k
,
e
P
A,n
0,M(k)
and
e
P
A,n
M(k),k
does not exceed the
maximum power budget of the system. Constraint (10) ensures
that a single transmission protocol, either direct or AF, is
chosen for each user-subcarrier pair. The constraint (11) guar-
antees that each subcarrier is only allocated to at most one user,
thus intra-cell interference is avoided. The constraints (12)
and (13) describe the feasible region of the optimization
variables. The following is a proof that the OF of problem (P)
is quasi-concave [31].
A. Proving that the OF in problem (P) is quasi-concave
A function, f : R
n
→ R, is quasi-concave if its domain
is convex, and all its superlevel sets are convex, i.e. if
the domain S
α
= {x ∈ dom f | f (x) ≥ α} is convex
for α ∈ R [32]. For a fractional function, g(x)/h(x), the
inequality g(x)/h(x) ≥ α is equivalent to [g(x)−αh(x)] ≥ 0,
assuming h(x) > 0, ∀x. Therefore, in order to prove that (8)
is quasi-concave, it is sufficient to show that the numerator is
concave and the denominator is both affine and positive, whilst
the domain is convex. It is plausible that the denominator is
both affine and positive, since it is the linear combination of
multiple nonnegative variables and a positive constant. The
proof that the numerator is concave is as follows.
Firstly, the concavity of f
1
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
=
e
P
A,n
0,M(k)
G
n
0,M(k)
e
P
A,n
M(k),k
G
n
M(k),k
∆γN
0
W
e
P
A,n
0,M(k)
G
n
0,M(k)
+
e
P
A,n
M(k),k
G
n
M(k),k
is proven. This may
be accomplished by examining the Hessian matrix of
f
1
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
with respect to (w.r.t.) the variables
e
P
A,n
0,M(k)
and
e
P
A,n
M(k),k
[32]. The Hessian has the eigenvalues
e
1
= 0 and
e
2
= −
2
G
n
0,M(k)
G
n
M(k),k
2
e
P
A,n
0,M(k)
+
e
P
A,n
M(k),k
∆γN
0
W
e
P
A,n
0,M(k)
G
n
0,M(k)
+
e
P
A,n
M(k),k
G
n
M(k),k
3
,
(16)
which are non-positive, indicating that the Hessian is negative-
semidefinite. This indicates that f
1
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
is
concave w.r.t. the variables
e
P
A,n
0,M(k)
and
e
P
A,n
M(k),k
.
Examination of the composition f
2
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
=
log
2
h
1 + f
1
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
i
reveals that
f
2
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
is concave, since log
2
(·) is concave
as well as non-decreasing and 1 + f
1
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
is
concave [32].
The second term in the summation of (14) may be denoted
by (17). This may be obtained using the perspective transfor-
5
e
R
T
=
K
X
k=1
N
X
n=1
s
D,n
k
log
2
1 +
e
P
D,n
0,k
G
n
0,k
s
D,n
k
∆γN
0
W
!
+
s
A,n
k
2
log
2
1 +
e
P
A,n
0,M(k)
G
n
0,M(k)
e
P
A,n
M(k),k
G
n
M(k),k
s
A,n
k
∆γN
0
W
e
P
A,n
0,M(k)
G
n
0,M(k)
+
e
P
A,n
M(k),k
G
n
M(k),k
(14)
e
P
T
=
P
(B)
C
+ M · P
(R)
C
+
K
X
k=1
N
X
n=1
ξ
(B)
e
P
D,n
0,k
+
1
2
ξ
(B)
e
P
A,n
0,M(k)
+ ξ
(R)
e
P
A,n
M(k),k
(15)
f
3
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
, s
A,n
k
= s
A,n
k,n
· log
2
1 +
e
P
A,n
0,M(k)
G
n
0,M(k)
e
P
A,n
M(k),k
G
n
M(k),k
s
A,n
k
∆γN
0
W
e
P
A,n
0,M(k)
G
n
0,M(k)
+
e
P
A,n
M(k),k
G
n
M(k),k
. (17)
mation
6
of [32] yielding
f
3
e
P
A,n
0,M(k)
,
e
P
A,n
M(k),k
, s
A,n
k
= s
A,n
k
·f
2
e
P
A,n
0,M(k)
s
A,n
k
,
e
P
A,n
M(k),k
s
A,n
k
!
,
(18)
which preserves concavity. Using similar arguments,
s
D,n
k
log
2
1 +
e
P
D,n
0,k
G
n
0,k
s
D,n
k
∆γN
0
W
is proven to be concave w.r.t.
the variables s
D,n
k
and
e
P
D,n
0,k
.
Finally, the numerator is shown to be concave w.r.t the
variables s
A,n
k
, s
D,n
k
,
e
P
D,n
0,k
,
e
P
A,n
0,k
,
e
P
A,n
0,M(k)
and
e
P
A,n
M(k),k
,
∀k, n, since it is the non-negative sum of multiple concave
functions. Thus, the OF in problem (P) has a numerator that
is concave, while its denominator is affine. Hence, the OF of
problem (P) is quasi-concave.
B. Problem solution methods
Quasi-concavity may be viewed as a type of generalized
concavity, since it can describe discontinuous functions as
well as functions that have multiple stationary points. This
means that a local maximum is not guaranteed to be a global
maximum and thus standard convex optimization techniques,
such as interior-point or ellipsoid methods, cannot be readily
applied for finding the optimal solution [32]. However, a
quasi-concave function has convex superlevel sets, hence the
bisection method [18] may be used for iteratively closing the
gap between an upper and lower bound solution, until the
difference between the two becomes lower than a predefined
tolerance. The drawback of this method is that there is no exact
method of finding the initial upper as well as lower bounds.
Additionally, a convex feasibility problem [32] must be solved
in each iteration, which may become computationally undesir-
able. In light of these discussions, the method detailed in [23]
is employed, which allows the quasi-concave problem to be
solved as a sequence of parameterized concave programming
problems. For clarity, the algorithm is summarized in Fig. 2,
which is discussed in the following section.
IV. DINKELBACH’S METHOD FOR SOLVING PROBLEM (P)
A. Introduction to Dinkelbach’s method
Dinkelbach’s method [23], [31] is an iterative algorithm that
can be used for solving a quasi-concave problem in a parame-
terized concave form. The algorithm is summarized in Table I.
6
The perspective transformation of the function f (x) is given by tf (x/t).
Table I: Dinkelbach’s method for energy efficiency maximiza-
tion.
Algorithm 1 Dinkelbach’s method for energy efficiency maximization
Input: I
D
outer
(maximum number of iterations)
D
outer
> 0 (convergence tolerance)
1: q
0
← 0
2: i ← 0
3: do while q
i
− q
i−1
>
D
outer
and i < I
D
outer
4: i ← i + 1
5: Solve max.
P,S
R
T
(P, S) − q
i−1
P
T
(P, S) to obtain the
optimal solution P
∗
i
and S
∗
i
(inner loop)
6: q
i
← R
T
(P
∗
i
, S
∗
i
)/P
T
(P
∗
i
, S
∗
i
)
7: end do
8: return
The concave form of the fractional program (P) is formed by
denoting the OF value as q so that a subtractive form of the OF
may be written as F (q) = R
T
(P, S) − qP
T
(P, S), which is
concave. Since the parameter q now acts as a negative weight
on the total power consumption of system, it may be intuitively
viewed as the ’price’ of the system’s power consumption. At
the optimal OF value of q
∗
, the following holds true
max.
P,S
{F (q
∗
)} = max.
P,S
{R
T
(P, S) − q
∗
P
T
(P, S)} = 0. (19)
Explicitly, the solution of F (q
∗
) is equivalent to the solution
of the fractional problem (P). Dinkelbach [23] proposed an
iterative method to find increasing q values, which are feasible,
by solving the parameterized problem of max
P,S
{F (q
i−1
)}
at each iteration. Hence, it can be shown that the method
produces an increasing sequence of q values, which converges
to the optimal value at a superlinear convergence rate. As
shown in Table I, each outer iteration corresponds to solving
max
P,S
{F (q
i−1
)}, where q
i−1
is a given value of the
parameter q, to obtain P
∗
i
and S
∗
i
, which at the optimal
power and subcarrier values at the ith iteration of Dinkelbach’s
method. For further details and a proof of convergence, please
refer to [23].
B. Solving the inner loop maximization problem
Dinkelbach’s method relies on solving max
P,S
F (q
i−1
), in
each iteration, which will henceforth be referred to as (P
q
i−1
).
Since it has been shown that R
T
(P, S) is concave whilst
P
T
(P, S) is affine, then (P
q
i−1
) is concave w.r.t. the variables
P and S. Assuming the existence of an interior point (Slater’s
condition), there is a zero duality gap between the dual
problem of (P
q
i−1
) and the primal problem of (P) [32]. Thus
