IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 5, MAY 2013 2193
Relay Selection and Resource Allocation for
Multi-User Cooperative OFDMA Networks
Md Shamsul Alam, Student Member, IEEE, Jon W. Mark, Life Fellow, IEEE,
and Xuemin (Sherman) Shen, Fellow, IEEE
Abstract—The resource allocation problem is investigated
for relay-based multi-user cooperative Orthogonal Frequency
Division Multiple Access (OFDMA) uplink system, considering
heterogeneous services. A quality of service (QoS) aware optimal
relay selection, power allocation and subcarrier assignment
scheme under a total power constraint is proposed. The relay
selection, power allocation and subcarrier assignment problem is
formulated as a joint optimization problem with the objective of
maximizing the system throughput, which is solved by means of a
two level dual decomposition and subgradient method. To further
reduce the computational cost, two low-complexity suboptimal
schemes are also proposed. The performance of the proposed
schemes is demonstrated through computer simulations based
on LTE-A network. Numerical results show that the proposed
schemes support heterogeneous services while guaranteeing each
user’s QoS requirements with slight total system throughput
degradation.
Index Terms—OFDMA networks, cooperative relaying, relay
selection, resource allocation, joint optimization, QoS, LTE-A.
I. INTRODUCTION
W
ITH the rapid development in broadband wireless
access technology and explosive growth in demand for
new wireless cellular services, it is expected that the next
generation cellular network will support a wide variety of
communication services with diverse QoS requirements. Ac-
cording to the performance and technical requirements for the
4G networks defined by the International Telecommunication
Union (ITU), future International Mobile Telecommunications
(IMT)-Advanced mobile system will support very h igh peak
data rates f or mobile users, up to 1 Gb/s in static and pedes-
trian environments, and up to 100 Mb/s in high-speed mobile
environment [2]. In order to meet this increasing demand,
high-spectral-efficiency schemes are required in conjunction
with aggressive resource reuse strategies to ensure prudent
use of the scarce radio resources. OFDMA is accepted as
the most appropriate air interface for the 4G networks due
to it’s inherent ability to combat frequency-selective fad-
ing and hig her spectral efficiency. In OFDMA, multi-user
Manuscript received May 8, 2012; revised November 22, 2012; accepted
January 31, 2013. The associate editor coordinating the review of this paper
and appro ving it for publication was Y. Chen.
The authors are with the Department of Electrical and Computer Engineer-
ing, University of Waterloo, Waterloo, Canada, (e-mail: {ms3alam, jwmark,
sshen}@uwaterloo.ca).
This work has been supported by the Natural Science and Engineering Re-
search Council (NSERC) of Canada under Grant No. RGPIN7779. Part of the
paper has been presented in the International Conference on Communications
(ICC), 2012, Ottawa, Canada, June 2012 [1].
Digital Object Identifier 10.1109/TWC.2013.032113.120652
diversity (MUD) can be achieved by allowing subcarriers
to be shared among multiple users. Additionally, different
number o f subcarriers can be allocated to users depending
on their QoS requirements. OFDMA is popularly used in 4G
wireless systems of broadband communications such as Third
Generation Partnership Project (3GPP) Long Term Evolution
(LTE) [3], LTE-Advanced [4], Worldwide Interoperability for
Microwave Access (WiMAX) [5], and so on.
One of the main challenges facing the 4G network ing
community is the provision of high throughput for mobiles
at the cell edge. Users at the cell edge often suffer from
bad channel conditions. Moreover, in an urban environment,
shadowing by various obstacles can degrade the signal quality
significantly. Cooperative relaying is a very promising solution
to tackle this problem as it provides throughput gains as
well as coverage extension [6]. Combining OFDMA and
cooperative relaying assures high throughput requirements,
particularly f or users at the cell edge. A dditionally, relaying
is considered as a cost effective throughput enhancement in
both IEEE 802.16j and LTE-A standards. However, to fully
exploit the benefits of relaying in 4G networks, efficient relay
selection and resource allocation are crucial in multi-user and
multi-relay environment.
Resource allocation in OFDMA-based cellular networks
without relay has been studied [7]–[9]. Choosing the best
relay and allocating the resources in an OFDMA relay network
with single user and multiple relays are straightforward a nd
have been well investigated [10], [11]. In the presence of
multiple users and multiple relays, relay selection and resource
allocation are complicated due to the interactions among the
users. An isolated relay assignment and power allocation
scheme for cooperative networks considering homogeneous
traffic is proposed in [12]. A heuristic algorithm is presented
to find a near optimal relay assignmen t and power allocation
where each user is supported by a single relay. However, this
scheme can not achieve the optimal solution because of the
isolated design approach and the relay selection criterion only
based on the maximum allocated power.
There have been numerous research works considering
the downlink OFDMA systems [13]–[15]. However, resource
allocation schemes designed for the downlink may not be
applicable for the uplink due to the distributive nature of
power constraints [16]. A multi-user joint distributed resource
allocation scheme for uplink cooperative OFDMA system
is proposed in [17]. Using the primal-dual decomposition
method, the authors have provided an optimal solution and
1536-1276/13$31.00
c
2013 IEEE
2194 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 5, MAY 2013
distributed implementation. However, relay selection is per-
formed for each user and all subcarriers allocated to that
user use the same relay, and each user’s QoS requirement
is not considered in the joint design. A most recent work
[18] proposed a low complexity suboptimal algorithm for
subcarrier assignment, power allocation and partner selection
for amplify-forward cooperative multicarrier systems. How-
ever, homogeneous users with same service and demand are
considered. Resource allocation supporting each user’s QoS
requirements has been considered in several works [16], [19],
[20]. In [20], a rate adaptive joint subcarrier and power
allocation algorithm under interference and QoS constraints is
proposed for cooperative OFDMA based broadband wireless
access networks. However, the problem is solved heuristically.
A cross layer approach for uplink OFDMA based cellular
networks supporting heterogeneous services is introduced in
[16]. The authors formulated two different optimization prob-
lems to support two types of uplink flows and determined
cross-layer trade-off between uplink service rate and power
consumption of u sers. Finally, they solved the problem using
dual decomposition.
In this paper, we investigate the joint relay selection and
resource allocation problem for the uplink OFDMA-based
system. We develop both optimal and suboptimal schemes for
relay selection, subcarrier assignment and power allocation
with fixed relays, considerin g service differentiation. The re-
source allocation problem is formulated as a maximization of
the total system throughput by satisfying the individual users’
QoS requirements subject to a total power constraint. We
consider two types of users, Guaranteed Bit Rate (GBR) users
and Aggregate Maximum Bit Rate (AMBR) users. The users
are differentiated on the basis of minimum required data rate.
GBR users have a specific rate requirement (e.g., real-time
gaming) and AMBR users have a flexible service rate (e.g.,
best-effort and non-real-time service). By relaxing the integer
constraints, we derive an optimal solution for this relaxed
problem via a two level dua l decomposition with reduced
computational complexity. We also p resent two suboptimal
schemes based on equal power allocation, with and without
power refinement to reduce computational complexity. Nu-
merical results reveal that our proposed schemes significantly
outperforms the traditional unconstrained scheme [21] in terms
of both services support and QoS satisfaction.
The remainder of this paper is organized as follows. Section
II introduces the system model. The problem formulation and
analytical framework for the optimal solution are presented
in Section III. Suboptimal schemes based on equal power
allocation with power refin ement are described in Section
IV. The computational complexity is discussed in Section
V. Numerical re sults are shown in Section VI, followed by
concluding remarks in Section VII.
II. S
YSTEM MODEL
Consider a single cell relay enhanced OFDMA-based uplink
system with K users (UE) (1 ≤ k ≤ K) and N fixed relays
(1 ≤ n ≤ N), where relays are shared by all users. The
cell is divided into two ring shaped boundary regions and
users are distributed between inner and outer boundaries. The
reason is that the users located between inner boundary and
Timeslot 1
UE
eNodeB
Relay
Timeslot 1
Timeslot 2
Fig. 1. System model.
outer boundary may require relays in most cases due to heavy
blockage and long distance transmission [22]. Users located
inside the inner boundaries are not considered because they do
not require relays in most cases due to good channel condition
since they are closer to the eNodeB. Resource allocation for
these users may be done separately with simple algorithm.
The distance of the relays from the base station (eNodeB)
is δR and the relay’s angle relative to the base station is
uniformly distributed in [0, 2π],whereR is the radius of the
cell and δ is the distance factor. The cell spectrum is divided
into subbands, each supported by a subcarrier. The subcarriers
are grouped into resource blocks (RBs). The total number of
subcarriers used in the system is M (1 ≤ m ≤ M ).The
transmit power of the kth user in the mth subcarrier is P
m
s,k
,
and the transmit power of the nth relay in the mth subcarrier is
P
m
r,n
. Assume that each node is equipped with a single antenna
and the relays operate in a half duplex mode. The broadband
channel is assumed to be frequency-selective Rayleigh fading
and the destination node (eNodeB) has perfect channel state
information (CSI) of all links. The noise variances of the
source-to-relay (SR) links, relay-to-destination (RD) links and
source-to-destination (SD) links per subcarrier are denoted by
σ
2
k,n
, σ
2
n,D
,andσ
2
k,D
, respectively. The system model is shown
in Fig. 1. The network model described here can be used to
model the uplink system of relay-based LTE-A
1
and I EEE
802.16j networks.
There are two types of users: user class κ
1
,theGBRusers,
which have specific rate requirements (called rate constrained
(RC) user) and the AMBR u sers under the user class κ
2
,
which have a flexible service rate requirements. The traffic
class of a user is determined based o n the applications. The
1
LTE-A networks adopt Single Carrier Frequency Division Multiple Access
(SC-FDMA) in it’s uplink, considering the power consumption issue of mobile
handsets. Similar to OFDMA downlink, the uplink supports multiple users
simultaneously. One prominent advantage of SC-FDMA over OFDMA is that
SC-FDMA has significantly lower peak-to-average power ratio (PAPR) [3].
Our network model can be used in LTE-A uplink system under the assump-
tion that subcarriers are not constrained to be consecutive or equidistantly
distributed for a higher degree of freedom as considered in [23].
ALAM et al.: RELAY SELECTION AND RESOURCE ALLOCATION FOR MULTI-USER COOPERATIVE OFDMA NETWORKS 2195
minimum QoS requirement of the kth user is denoted by Q
k
.
Based on QoS requirement, a user can transmit directly to the
destination or transmit using cooperative communication. In
cooperative scenario, the communication between the user and
the eNodeB is carried out in two phases. In the first phase,
the user tra nsmits to the eNodeB w hich is overheard by the
selected relay as well. In the second phase, the selected relay
forwards to the eNodeB using the regenerate-and-forward
cooperative protocol. The data received in both time slots
are combined together by the eNodeB using maximal ratio
combining (MRC). The achievable rate in bits/sec/Hz for the
regenerate-and-forward scheme for the kth user in the mth
subcarrier when the nth relay is selected is given by
R
m
k,n
=
⎧
⎪
⎨
⎪
⎩
1
2
min
log
2
(1 + P
m
s,k
α
m
k,n
) ,
log
2
(1 + P
m
s,k
α
m
k,D
+ P
m
r,n
α
m
n,D
)
, cooperative mode
log
2
(1 + P
m
s,k
α
m
k,D
), non-cooperative mode
(1)
where α
m
k,D
=
|
h
m
k,D
|
2
σ
2
k,D
, α
m
k,n
=
|
h
m
k,n
|
2
σ
2
k,n
and α
m
n,D
=
|
h
m
n,D
|
2
σ
2
n,D
and
h
m
k,D
2
,
h
m
k,n
2
and
h
m
n,D
2
are the channel coefficients
between the kth user and the destination, the kth user and the
nth relay and the nth relay and the destination in the mth
subcarrier, respectively.
Consider binary relay selection and subcarrier allocation
characterized by the parameter ρ
m
k,n
,whereρ
m
k,n
=1means
that relay node n performs as a relay for user k in the mth
subcarrier. Otherwise, it is equal to 0. We assume that each
user can have only one relay, but each relay can support several
users and a subcarrier is allocated to only one source and one
relay, so that there is no interference between sources. The
same subcarrier will be used by the relay in the second time
slot. Even if it is decided that relay will not transmit in the
second time slot (i.e. non-cooperative mode), the user is not
allowed to use this idle time slot [21].
III. P
ROBLEM FORMULATION AND SOLUT ION APPROACH
Our objective is to maximize the total system throughput
subject to a set of constraints. The relay selection and subcar-
rier assignment constraints are as follows:
K
k=1
N
n=0
ρ
m
k,n
=1,ρ
m
k,n
∈{0, 1}, ∀m (2)
where n =0, it means user k utilize subcarrier m in non-
cooperative mode. The total power allocated to the mth sub-
carrier of the kth user in both time slots is P
m
t,k
= P
m
s,k
+ P
m
r,n
[10], [24] and the total power constraint can be expressed as
K
k=1
M
m=1
N
n=0
ρ
m
k,n
P
m
t,k
≤ P
T
(3)
where P
T
is the sum of the power available for all users plus
relays in the network. Although individual power constraints
will lead more accurate power allocation, however, our goal
is to maximize the total system throughput subject to a joint
total power constraint, considering the simplicity of the prob-
lem formulations and lower computational complexity under
the sum power constraint. The computational complexity is
lower in the studied model since we only need to update
one dual variable using subgradient method under the total
power constraint compared to updating K + N dual variables
simultaneously until all of them are converged when individual
power constraints are used. Similar assumptions on the total
power constraint are taken in previous studies [10], [24]–[26].
Maximization of the rate in (1) using cooperative commu-
nication under total power constraint has advantageous only if
α
m
k,n
>α
m
k,D
and α
m
n,D
>α
m
k,D
[25], [27]. First, consider the
case when the user to relay channel is weaker (lower Signal-
to-Noise Ratio (SNR) due to bad channel condition) than the
direct link channel, i.e., α
m
k,n
<α
m
k,D
. In such case, from
equation (1), any power increment will be more beneficial if
allocated to the direct link and the use of relay will not be
advantageous. Second, consider the case when the user to relay
channel is stronger (higher SNR due to good channel condi-
tion) than the direct link channe l, .i.e., α
m
k,n
>α
m
k,D
.Thentwo
cases may happen: 1) if α
m
k,D
>α
m
n,D
, the rate benefit will
be greater if the p ower is allocated to the direct link; and 2)
if α
m
k,D
<α
m
n,D
, the allocation of power to the relay is better
under the constraint of P
m
s,k
α
m
k,D
+P
m
r,n
α
m
n,D
≤ P
m
s,k
α
m
k,n
.This
means that any power increment has to be shared between
the user and relay, and th e rate will be maximized when the
constraint is saturated,.i.e., P
m
s,k
α
m
k,D
+ P
m
r,n
α
m
n,D
= P
m
s,k
α
m
k,n
.
Then the source power allocation is given by
P
m
s,k
=
α
m
n,D
α
m
k,n
+α
m
n,D
−α
m
k,D
P
m
t,k
, cooperative mode
P
m
t,k
, non-cooperative mode
(4)
and the relay power allocation is given by
P
m
r,n
=
α
m
k,n
−α
m
k,D
α
m
k,n
+α
m
n,D
−α
m
k,D
P
m
t,k
, cooperative mode
0, non-cooperative mode
(5)
The computation of the source and relay power can be ex-
plained as follows. First, for any subcarrier, when the channel
gains are known, the transmission mode can be determined
for the given total power on that subcarrier. Second, for the
selected transmission mode, the optimal source and relay
power are computed. If cooperative mode is selected, P
m
t,k
will be divided in two time slots depending on the channel
condition and the optimal source and r elay power are given
by (4) and (5) [10], [25]. In case of non-cooperative mode,
P
m
r,n
=0,andP
m
s,k
= P
m
t,k
from (4). For the non-cooperative
transmission mode, there may be two scenarios: transmission
can be held in two time slots by dividing the total power, P
m
t,k
in two time slots, or use only one time slot with power P
m
t,k
.
In this work, we assume that the user only transmits in the
first time slot using the total power, P
m
t,k
when non-cooperative
transmission mode is selected [21]. Thus, substituting (4) and
(5) into (1), the rate expression can be unified as
R
m
k,n
=
1
2
[log
2
(1 + P
m
t,k
α
m
k,eq
)] (6)
where α
m
k,eq
is the equivalent channel gain given by
α
m
k,eq
=
α
m
k,n
α
m
n,D
α
m
k,n
+α
m
n,D
−α
m
k,D
, cooperative mode
α
m
k,D
, non-cooperative mode.
(7)
2196 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 5, MAY 2013
The total achievable rate of the kth user for all subcarriers
allocated to the kth user is given by
R
k
=
M
m=1
N
n=0
ρ
m
k,n
R
m
k,n
. (8)
We formulate the joint resource allocation and relay selection
problem subject to a minimum data rate constraint for each
GBR user. The optimization p roblem can be formulated as
(P 1) maximize
ρ,P
t
K
k=1
M
m=1
N
n=0
ρ
m
k,n
R
m
k,n
subject to c1:ρ
m
k,n
∈{0, 1}, ∀k, m, n
c2:
K
k=1
N
n=0
ρ
m
k,n
=1, ∀m
c3:R
k
≥ Q
k
, ∀k ∈ κ
1
c4:
K
k=1
M
m=1
N
n=0
ρ
m
k,n
P
m
t,k
≤ P
T
c5:P
m
t,k
≥ 0, ∀k, m, n
(9)
where constraints c1 and c2 represent the relay selection and
subcarrier allocation and indicate that each user can have
one relay to cooperate and can utilize multiple subcarriers to
transmit; however, a subcarrier can not be shared by different
users. Constraint c3 applies minimum QoS requirements for
the GBR users in terms of data rate requirement. Finally, the
source and the relay power allocation are constrained by c4
and c5.
The optimization problem in (9) is a mixed integer nonlinear
programming (MINP) problem. One challenging aspect of this
problem in the context of OFDMA uplink is the discrete
nature of subcarrier assignment, which, when coupled with
QoS constraint, makes the problem even harder to solve.
Therefore, finding the optimal solution for this non-convex
problem requires searching through all the possible u ser, relay
and subcarrier allocations, which is prohibitively complex
to employ in large system . However, to make the problem
tractable, we relax the integer constraints, ρ
m
k,n
to take any
real value between 0 and 1 via time-sharing condition which
allows time sharing of each subcarrier. The duality gap of any
optimization problem satisfying the time sharing condition is
negligible as the number of subcarriers becomes sufficiently
large [28]. Since our optimization problem obviously satisfies
the time-sharing condition, it can be solved by using the dual
method and the solution is optimal [16], [28].
A. Dual Problem
The Lagrangian function of p roblem in ( 9) can be written
as
L(ρ, P
t
,λ,μ)
=
K
k=1
M
m=1
N
n=0
ρ
m
k,n
R
m
k,n
+
k∈ κ1
λ
k
(
M
m=1
N
n=0
ρ
m
k,n
R
m
k,n
−Q
k
)+μ(P
T
−
K
k=1
M
m=1
N
n=0
ρ
m
k,n
P
m
t,k
)
=
M
m=1
K
k=1
N
n=0
ρ
m
k,n
R
m
k,n
+
k∈ κ1
λ
k
N
n=0
ρ
m
k,n
R
m
k,n
−μ
K
k=1
N
n=0
ρ
m
k,n
P
m
t,k
−
k∈ κ1
λ
k
Q
k
+ μP
T
(10)
where λ =[λ
1
,λ
2
, ......λ
κ1
]
T
is the vector of the dual
variables associated with the individual QoS constraints and μ
is the dual variable for the power constraint. The Lagrangian
dual fu nction can therefore be written as
g(λ, μ)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
max
ρ,P
t
L(ρ, P
t
,λ,μ)
s.t.
K
k=1
N
n=0
ρ
m
k,n
=1, ∀m
0 ≥ ρ
m
k,n
≤ 1,P
m
t,k
≥ 0.
(11)
Then the dual optimization pro blem is given by
min
λ,μ0
g(λ, μ). (12)
The coupling between subcarriers via Lagrangian relaxation
can be removed and (11) can be decomposed into M subprob-
lems at each subcarrier, which can be solved independently
given λ, μ with low complexity. The subproblem at each
subcarrier is given by
max
ρ,P
t
L
m
(ρ
m
,P
m
t
)
=max
ρ,P
t
K
k=1
N
n=0
ρ
m
k,n
R
m
k,n
+
k∈ κ1
λ
k
N
n=0
ρ
m
k,n
R
m
k,n
−μ
K
k=1
N
n=0
ρ
m
k,n
P
m
t,k
(13)
s.t.
K
k=1
N
n=0
ρ
m
k,n
=1, 0 ≥ ρ
m
k,n
≤ 1,P
m
t,k
≥ 0, ∀k, n
where ρ
m
,P
m
t
are the vectors of ρ
m
k,n
,P
m
t,k
on the mth
subcarrier, respectively. The subproblem can be further de-
composed through a second level p rimal decomposition. The
decomposition hierarchy of the du al problem is shown in Fig .
2. Thus, we have two subproblems which will be solved in
two phases: optimal power allocation and joint relay selection
and subcarrier allocation.
Proposition 1 : Considering the convex optimization
problem in (12), the subgradients of g(λ, μ) denoted by Δλ
k
,
and Δμ are given by
Δλ
k
=
M
m=1
N
n=0
ρ
m
k,n
R
m
k,n
− Q
k
, ∀k ∈ κ
1
Δμ = P
T
−
K
k=1
M
m=1
N
n=0
ρ
m
k,n
P
m
t,k
where ρ
m
k,n
,R
m
k,n
and P
m
t,k
are the optimal solution of the dual
objective function in (12).
The proof of proposition 1 is given in the Appendix.
ALAM et al.: RELAY SELECTION AND RESOURCE ALLOCATION FOR MULTI-USER COOPERATIVE OFDMA NETWORKS 2197
Master Dual
Problem (12)
1
st
Subproblem
(13)
mth Subproblem
(13)
Subproblem
(14)
Subproblem
(17)
. . . . . . . . .
Subproblem
(14)
Subproblem
(17)
Fig. 2. Hierarchy of the decomposed dual problem.
B. Optimal Power Allocation for a Given Relay Assignment
and Subcarrier Allocation
Let subcarrier m be allocated to user k and relay n in a
frame of transmission time and ρ
m
k,n
=1. Then optimal power
allocation over this subcarrier and relay assignment can be
determined by solving the following problem
max
P
m
t,k
L
m
, ∀k, n
s.t. P
m
t,k
≥ 0.
(14)
Substituting (6) into (14) and differentiating L with respect to
P
m
t,k
we have
∂L
∂P
m
t,k
=
(1 +
¯
λ
k
)α
m
k,eq
2ln(2)(1 + P
m
t,k
α
m
k,eq
)
− μ (15)
where
¯
λ
k
=
λ
k
, ∀k ∈ κ
1
0, Otherwise.
Applying the Karush-Kuhn-Tucker (KKT) [29] condition,
we can deduce the optimal power allocation as follows:
P
m
t,k
∗
=
1+
¯
λ
k
2μ ln(2)
−
1
α
m
k,eq
+
(16)
where [x]
+
= max [x, 0].
C. Joint Optimal Relay Selection and Subcarrier Allocation
By eliminating the power variables in (14) and then substi-
tuting into (10), we have an alternative expression of the dual
function as
g(λ, μ)=max
ρ
M
m=1
K
k=1
N
n=0
ρ
m
k,n
H
m
k,n
(λ, μ) (17)
−
k∈ κ1
λ
k
Q
k
+ μP
T
s.t.
K
k=1
N
n=0
ρ
m
k,n
=1, ∀m, 0 ≥ ρ
m
k,n
≤ 1
where the function H
m
k,n
(λ, μ) is d e fined as follows
H
m
k,n
=
1
2
(1 +
¯
λ
k
)[log
2
(1 + P
m
t,k
∗
α
m
k,eq
)] − μP
m
t,k
∗
. (18)
An intuitive explanation for each term in (18) is as follows.
The first term can be viewed as the rate obtained by selecting
subcarrier m by user k and relay n and the second term is
the price for the power consumption. Therefore, H
m
k,n
can be
interpreted as the gain of transmitting over subcarrier m by
user k and r elay n and H =[H
m
k,n
] can be represented as a
K × N profit matrix at each subcarrier m.Inotherwords,
the profit matrix H is different for different value of m.
The objective function in (17) can be maximized by picking
exactly one element of matrix H for each subcarrier so that
the sum of profit is as large as possible. Finally, optimal relay
selection and subcarrier allocation should be the one having
the maximum value of H
m
k,n
(λ, μ) in (18) and is given by
ρ
m
k,n
=
⎧
⎨
⎩
1, (n
,k
)=arg max
n,k
H
m
k,n
0, otherwise.
(19)
In the operation, first, the power allocation for each subcarrier
using both transmission modes is computed using (16). Then,
these power allocation values are used in (18) to compute
H
m
k,n
. After that, for each subcarrier, the user and relay pair
is determined using ( 19) that gives the largest H
m
k,n
. Non-
cooperative mode is the case that no relay is selected, i.e.,
n =0.
D. Variable Update
Since a dual function is always convex by definition,
subgradient method can be used to minimize g(λ, μ).Dual
variables λ and μ are updated in p arallel as follows
λ
k
(t +1)=
λ
k
(t)+η(t)
Q
k
−
N
n=0
M
m=1
ρ
m
k,n
(t)R
m
k,n
(t)
+
μ(t +1)=
μ(t)+θ(t)
K
k=1
M
m=1
N
n=0
ρ
m
k,n
(t)P
m
t,k
(t) − P
T
+
(20)
where η(t) and θ(t) are diminishing stepsizes and t is the
iteration index. The subgradient method above is guaranteed
to converge to the optimal dual variables if the stepsizes are
chosen following the diminishing stepsize policy [29]. Based
on the mathematical formulations and derivations, the optimal
relay selection, subcarrier assignment and power allocation
can be computed algorithmically. The pseudocode of the
proposed optimal scheme is outlined in Algorithm 1.
IV. S
UBOPTIMAL SCHEMES
The computational complexity of the proposed optimal
scheme may still be too high for practical implementation. In
this section, we present two suboptimal schemes which have
lower computational cost compared to the optimal one.
A. Equal Power Allocation (EPA) Scheme
In this scheme, we determine relay selection and subcarrier
allocation assuming that the power is equally distributed over
all subcarriers. First, relay selection and subcarrier allocation
are performed for the GBR users in two steps considering that
AMBR users are absent. In step 1, to ensure fairness among
the users, we select the user whose current achievable rate is
