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RESEARCH ARTICLE
Joint Relay Selection and Energy-Efficient Power Allocation
Strategies in Energy-Harvesting Cooperative NOMA Networks
Mohammed W. Baidas*1| Emad Alsusa2| Khairi A. Hamdi2
1Department of Electrical Engineering,
Kuwait University, Kuwait City, Kuwait
2School of Electrical and Electronic
Engineering, University of Manchester,
Manchester, United Kingdom
Correspondence
*Mohammed W. Baidas, Electrical
Engineering Department, Kuwait University,
PO Box: 5969, Safat, 13060, Kuwait City,
Kuwait. Email: m.baidas@ku.edu.kw
Summary
In this paper, the problem of joint relay selection and energy-efficient power alloca-
tion (J-RS-EE-PA) in energy-harvesting (EH) cooperative non-orthogonal multiple-
access (NOMA) networks is considered. Particularly, a base-station communicates
with multiple users via a selected EH relay, such that energy-efficiency (EE) is max-
imized, subject to quality-of-service (QoS) constraints. Four different EE power
allocation strategies are considered, namely global energy-efficiency (GEE), mini-
mum energy-efficiency (MEE), product energy-efficiency (PEE), and sum energy-
efficiency (SEE). For all the aforementioned strategies, the J-RS-EE-PA problem
happens to be non-convex, and thus is computationally-prohibitive. In turn, problem
J-RS-EE-PA is decoupled into two sub-problems: (1) relay selection, and (2) energy-
efficient power allocation. Different relay selection strategies are explored, and then
computationally-efficient algorithms are devised to optimally solve the different EE
power allocation strategies for the selected relay. In addition, a low-complexity solu-
tion procedure is proposed to solve the J-RS-EE-PA problem. Simulation results
are presented to evaluate the resulting EE of the different relay selection and power
allocation strategies in comparison to the formulated J-RS-EE-PA problem. More
importantly, the proposed solution procedure is shown to be superior to the different
explored relay selection strategies as well as yielding the global optimal joint relay
selection and energy-efficient power allocation solution of the J-RS-EE-PA problem
for the GEE, MEE, PEE, and SEE strategies, while satisfying QoS requirements.
KEYWORDS:
Cooperation, energy-efficiency, non-orthogonal multiple-access, power allocation, relay selection
1 INTRODUCTION
Non-orthogonal multiple-access (NOMA) has recently been proposed as a promising radio access technique for next-
generation cellular networks1. More specifically, NOMA can achieve high spectrum efficiency, low latency, massive connectivity
and improvement to cell-edge users by multiplexing users in the power-domain or in the code-domain2. By applying succes-
sive interference cancellation (SIC) at the receiver side, the different users’ signals can be decoded, yielding a practical trade-off
between network throughput, user fairness and receiver complexity. Although conventional orthogonal multiple-access (OMA)
techniques have been used extensively in the past, the increasing demands and challenges of 5G networks and beyond require sig-
nificant improvement in data rates, transmission reliability and coverage. Therefore, cooperative relay networks have also been
2M. W. Baidas ET AL
studied extensively to improve network capacity, exploit spatial diversity gains and extend network coverage3. More recently,
the application of cooperative relaying to NOMA networks has become more and more popular 4. In other words, relays can
fully exploit channel gains asymmetry for near and far NOMA network users so as to achieve significant performance gains.
Another important aspect of future generation cellular networks is energy-efficiency (EE). Particularly, energy consumption has
been considered the primary concern in the design of cellular networks, and energy-efficient transmission schemes and designs
have become more compelling than ever for 5G networks and beyond 5,6.
There have been several research works analyzing and investigating relay selection, resource allocation, and energy-efficiency
in cooperative NOMA networks. For instance, the impact of relay selection on the performance of cooperative NOMA networks
is studied in7. Specifically, a two-stage relay selection strategy is proposed, which is shown to achieve minimal outage prob-
ability and maximal diversity gain among different relay selection schemes. In 8, the authors analyze the outage probability of
a multi-relay amplify-and-forward (AF) cooperative NOMA network. In particular, the relays harvest energy from the signals
broadcasted by the source, and a relay selection scheme is proposed based on the channel state information and battery status of
relays. Moreover, approximate closed-form expressions for the outage probability are obtained and validated via numerical sim-
ulations. In9, the authors consider the outage and capacity performances over Rayleigh fading channels of a decode-and-forward
(DF) cooperative NOMA network with two users. Particularly, closed-form outage probability and approximate capacity expres-
sions are derived, where cooperative NOMA is shown to achieve superior performance over non-cooperative NOMA. In10 ,
two optimal DF relay selection schemes are proposed, namely weighted-max-min, and max-weighted-harmonic-mean. Outage
probabilities for the proposed schemes are analyzed and their diversity gains are obtained. It has been demonstrated that the pro-
posed schemes are superior in comparison to other schemes with fixed and adaptive power allocations. A max-min user-relay
assisted cooperative NOMA scheme is investigated in11. To be more specific, a closed-form expression for the outage probabil-
ity is derived, where the proposed DF scheme has been shown to outperform random relay selection. In12 , a novel simultaneous
wireless information and power transfer (SWIPT) protocol is applied to cooperative NOMA networks, where users nearby the
source act as energy harvesting (EH) DF relays to help farther users. Moreover, three user selection schemes have been proposed,
and their closed-form outage probability expressions derived. It has been demonstrated that SWIPT preserve diversity gains and
is superior in terms of throughput to a random selection scheme. The impact of power allocation on cooperative NOMA net-
works with SWIPT is studied in13 . Specifically, the authors consider a network of two users and an energy-harvesting DF relay,
and investigate two power allocation strategies, namely fixed power allocation NOMA (F-NOMA), and cognitive radio-inspired
NOMA (C-NOMA). Closed-form expressions for the outage probability and their high SNR approximations are derived. It has
been illustrated that the proposed NOMA schemes reduce the outage probability in comparison to conventional SWIPT relaying
networks with OMA. Enhanced transmission schemes for DF cooperative NOMA networks are proposed in14 . In particular, the
authors proposed a single signal transmission, where SIC is utilized to decode the transmitted signal by the relay at the destina-
tion. Additionally, an enhanced superposition coded signal transmission scheme with maximum ratio combining is devised to
improve the ergodic sum-rate, which has been shown to yield significant sum-rate performance improvement in comparison to
other schemes. Partial relay selection (PRS) in NOMA networks with AF relaying is investigated in15 . In particular, the authors
demonstrate that NOMA with PRS improves both the sum-rate and user-fairness in comparison to traditional OMA. In 16, power
control for multi-cell NOMA systems is considered. To be specific, the aim is to minimize the total transmit power of all base-
stations subject to user data rate requirements, where the feasibility and optimality properties are analyzed. Joint subcarrier and
power allocation for downlink multicarrier NOMA systems is studied in 17. Particularly, a three-step resource allocation frame-
work is devised for sum-rate maximization, which has been shown to achieve comparable sum-rate performance to existing
near-optimal solutions, but with reduced computational complexity. In18, optimal power allocation with individual quality-
of-service (QoS) constraints is studied. Specifically, the sum-rate maximization subject to minimum user rate requirements is
solved, and the optimal user decoding order is rigorously analyzed. The problems of sum-power minimization and sum-rate max-
imization in multi-cell NOMA networks is considered in19 . Particularly, closed-form solutions for the optimal power allocation
for sum-power minimization are obtained, while for the sum-rate maximization, an efficient distributed algorithm is proposed.
Energy-efficient resource allocation for machine-to-machine enabled energy-harvesting cellular networks is considered in 20. To
be specific, the aim is to minimize the total energy consumption via joint power control and time allocation in NOMA- and
TDMA-based networks. In the case of NOMA, a suboptimal iterative algorithm is proposed, while for TDMA, an equivalent
tractable problem is devised and solved iteratively. It has been demonstrated that NOMA consumes less energy than TDMA
in the low circuit power regime, and vice versa in the high circuit power regime. Physical layer security of a cooperative relay
NOMA network with two source-destination pairs with a DF relay and multiple eavesdroppers is studied in21 . In particular, a
two-stage secure relay selection (TSSRS) scheme is proposed to maximize the capacity of one source-destination pair, while
M. W. Baidas ET AL 3
guaranteeing successful communication for the other pair. Additionally, exact and asymptotic expressions for the secrecy outage
probability, where the TSSRS-NOMA has been shown to be superior to its counterpart TSSRS-OMA scheme.
This paper studies the problem of joint relay selection and energy-efficient power allocation (J-RS-EE-PA) in energy-
harvesting cooperative NOMA networks. Particularly, the scenario where a base-station communicates with multiple users via a
single selected EH AF relay is considered, with the aim of maximizing energy-efficiency, subject to QoS constraints*. Moreover,
four different EE power allocation strategies are studied, namely global energy-efficiency (GEE), minimum energy-efficiency
(MEE), product energy-efficiency (PEE), and sum energy-efficiency (SEE). The J-RS-EE-PA problem for each of the EE power
allocation strategies happens to be non-convex (i.e. computationally-prohibitive), and thus is decoupled into two sub-problems:
(1) relay selection, and (2) energy-efficient power allocation. For the first sub-problem, different relay selection strategies are
explored, while for the second sub-problem, computationally-efficient algorithms are devised to optimally solve the different EE
power allocation strategies for the selected EH relay. Additionally, a low-complexity solution procedure is proposed to solve the
J-RS-EE-PA problem based on the devised algorithms. Simulation results are presented to evaluate the resulting EE of the dif-
ferent relay selection and power allocation strategies in comparison to the formulated J-RS-EE-PA problem. More importantly,
the proposed solution procedure is shown to be superior to the different explored relay selection strategies as well as yielding
the global optimal joint relay selection and energy-efficient power allocation solution of the J-RS-EE-PA problem for the GEE,
MEE, PEE, and SEE strategies, while satisfying QoS requirements.
To the best of our knowledge, no prior work has considered joint relay selection and energy-efficient power allocation in
energy-harvesting cooperative NOMA networks, and provided globally optimal algorithmic solutions. Thus, the contributions
of this work can be summarized as:
•Formulated the joint relay selection and energy-efficient power allocation problem for the maximization of the GEE, MEE,
PEE and SEE.
•Explored several relay selection strategies, and devised computationally-efficient algorithms to optimally solve the
different energy-efficient power allocation strategies for the selected relay.
•Proposed a low-complexity solution procedure to solve the J-RS-EE-PA problem, which yields the global optimal joint
relay selection and energy-efficient power allocation solution for the GEE, MEE, PEE, and SEE strategies, while satisfying
QoS requirements.
The rest of this paper is organized as follows. The network model is presented in Section 2, while the problem formulation for
the joint relay selection and energy-efficient power allocation is given in Section 3. The relay selection strategies are outlined in
Section 4, while the different energy-efficient power allocation strategies, and their algorithmic solutions are studied in Section
5. The low-complexity solution procedure for solving the J-RS-EE-PA problem is given in Section 6. The simulation results are
presented in Section 7, while the conclusions are drawn in Section 8.
2 NETWORK MODEL
Consider a single-cell dual-hop cooperative NOMA network with a base-station (BS), 𝑁intended users 𝑈𝑖, for 𝑖∈
{1,2,…, 𝑁}, and 𝐾EH AF relays** 𝑅𝑘, for 𝑘∈ {1,2,…, 𝐾}. The BS wishes to communicate 𝑁data symbols 𝑥𝑖to each
intended user over a broadcast channel. The instantaneous BS-relay and relay-user channels are modeled as narrowband Rayleigh
fading with zero-mean 𝑁0-variance additive white Gaussian noise (AWGN). In particular, let ℎ𝐵𝑆,𝑅𝑘and ℎ𝑅𝑘,𝑈𝑖be the channel
coefficients between the BS and relay 𝑅𝑘, and between relay 𝑅𝑘and user 𝑈𝑖, which are defined as zero-mean complex Gaus-
sian random variables with variances 𝜎2
𝐵𝑆,𝑅𝑘=𝑑−𝜈
𝐵𝑆,𝑅𝑘and 𝜎2
𝑅𝑘,𝑈𝑖=𝑑−𝜈
𝑅𝑘,𝑈𝑖, respectively. In addition, 𝑑𝐵𝑆 ,𝑅𝑘and 𝑑𝑅𝑘,𝑈𝑖are the
BS-relay and relay-user distances, respectively, while 𝜈is the path-loss exponent. It is assumed that there is not a direct link
between the BS and the users. Moreover, a total transmit energy constraint per time-slot 𝐸𝑇is assumed.
The cooperative communication between the BS and the 𝑁users via the 𝐾AF relays follows the principle of NOMA and is
performed over two phases (each of one time-slot), namely the broadcasting phase, and the cooperation phase. Let 𝜏represent a
communication frame of two equal-length unit-duration time-slots (i.e. comprising the broadcasting and cooperation phases)†.
In each communication frame 𝜏, each relay harvests energy 𝜏
𝑅𝑘from the surrounding environment during the broadcasting
phase, and utilizes it for cooperative data transmission during the cooperation phase††. In this work, the total amount of harvested
*In this work, energy-efficiency is defined as a benefit-to-cost ratio in the form of the achievable rate to the power consumption 6.
**The relays could be other network users who do not have any data to transmit, but are willing to share their harvested energy for cooperative transmissions.
†Since the time-slots are assumed to be of unit-length, the terms “power” and “energy” can be used interchangeably.
††Due to hardware limitations, no energy is harvested during the time-slot of the cooperation phase.
4M. W. Baidas ET AL
energy at each relay is modeled as an independent uniform random variable, as given by 𝜏
𝑅𝑘∼𝑈0,max
𝑅𝑘, where max
𝑅𝑘is the
maximum value of the total amount of harvested energy per communication frame. Moreover, each relay 𝑅𝑘has a finite battery-
capacity 𝐵max,∀𝑘∈ {1,2,…, 𝐾 }. The total amount of harvested energy—up to the 𝜏𝑡ℎ communication frame—is given by
𝜏
Σ,𝑅𝑘=
𝜏
𝑡=1
𝑡
𝑅𝑘,(1)
where 𝜏
Σ,𝑅𝑘
≤𝐵max. In the cooperation phase of each communication frame 𝜏, if a relay 𝑅𝑘is selected for cooperative
transmission in communication frame 𝜏, then it utilizes 𝐸𝜏
𝑅𝑘amount of energy for relaying, where
𝐸𝜏
𝑅𝑘
≜min 𝜏
Σ,𝑅𝑘, 𝐸𝑇,(2)
while the leftover total harvested 𝜏
Σ,𝑅𝑘
←𝜏
Σ,𝑅𝑘−𝐸𝜏
𝑅𝑘is stored in the battery. On the other hand, if a relay is not selected in
a communication frame, then it stores all its harvested energy in the battery, while ensuring the battery constraint is satisfied.
From this point onwards, the superscript 𝜏is dropped for notational convenience while taking into account the battery dynamics.
Now, in the broadcasting phase, the transmitted signal by the BS can be expressed as
𝑥𝐵𝑆 =
𝑁
𝑖=1 𝑎𝑖𝐸𝐵𝑆 𝑥𝑖,(3)
where 𝐸𝐵𝑆 is the transmit energy at the BS, and 𝑁
𝑖=1 𝑎𝑖≤1is the sum of the power allocation coefficients. Thus, the received
signal at relay 𝑅𝑘is written as
𝑦𝐵𝑆,𝑅𝑘=ℎ𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖𝐸𝐵𝑆 𝑥𝑖+𝜂𝐵𝑆 ,𝑅𝐾,(4)
where 𝜂𝐵𝑆,𝑅𝑘is the AWGN sample at relay 𝑅𝑘.
In the cooperation phase, each relay amplifies-and-forwards its received signal as
𝑥𝑅𝑘=𝐺𝑅𝑘𝑦𝐵𝑆,𝑅𝑘,(5)
where 𝐺2
𝑅𝑘=𝐸𝑅𝑘
ℎ𝐵𝑆,𝑅𝑘2𝐸𝐵𝑆 𝑁
𝑖=1 𝑎𝑖+𝑁0
. Now, let the 𝑁network users be ordered in an ascending fashion accordingly to their channel
gains with respect to each relay 𝑅𝑘as ℎ𝑅𝑘,𝑈12<ℎ𝑅𝑘,𝑈22<⋯<ℎ𝑅𝑘,𝑈𝑁2. Thus, the received signal at the 𝑛𝑡ℎ ordered user
𝑈𝑛can be written as
𝑦𝑅𝑘,𝑈𝑛=𝐺𝑅𝑘ℎ𝐵𝑆,𝑅𝑘ℎ𝑅𝑘,𝑈𝑛𝑁
𝑖=1 𝑎𝑖𝐸𝐵𝑆 𝑥𝑖+𝐺𝑅𝑘ℎ𝑅𝑘,𝑈𝑛𝜂𝐵𝑆 ,𝑅𝑘+𝜂𝑅𝑘,𝑈𝑛,(6)
where 𝜂𝑅𝑘,𝑈𝑛is the AWGN sample at user 𝑈𝑛due to relay 𝑅𝑘’s transmission. In turn, by applying the principle of NOMA and
assuming perfect SIC‡, the end-to-end signal-to-interference-plus-noise ratio (SINR) of ordered user 𝑈𝑛can be shown to be15
Γ𝑅𝑘,𝑈𝑛=𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
,(7)
where 𝑎𝑛=𝑁
𝑗=𝑛+1 𝑎𝑗and 𝑎𝑁= 0. Moreover, 𝜉𝑅𝑘,𝑈𝑛is a product of the base-station-relay and relay-user SNRs, as given by
𝜉𝑅𝑘,𝑈𝑛=𝛾𝐵𝑆,𝑅𝑘𝛾𝑅𝑘,𝑈𝑛,(8)
where
𝛾𝐵𝑆,𝑅𝑘=𝐸𝐵𝑆 ℎ𝐵𝑆,𝑅𝑘2
𝑁0
and 𝛾𝑅𝑘,𝑈𝑛=𝐸𝑅𝑘ℎ𝑅𝑘,𝑈𝑛2
𝑁0
.(9)
Furthermore, 𝜁𝑅𝑘,𝑈𝑛is given by
𝜁𝑅𝑘,𝑈𝑛=𝛾𝑅𝑘,𝑈𝑛+ 1.(10)
Lemma 1: Γ𝑅𝑘,𝑈𝑛is a strictly concave fractional SINR function in 𝑎𝑛≥0,∀𝑛∈ {1,2,…, 𝑁}23.
Proof. Refer to Appendix I.
‡The case of imperfect SIC is beyond the scope of this paper22 .
M. W. Baidas ET AL 5
Remark 1: Since log2(⋅)is a non-linear and strictly monotonically increasing concave function in its parameter and Γ𝑅𝑘,𝑈𝑛is
strictly concave in 𝑎𝑛,∀𝑛∈ {1,2,…, 𝑁}, then 𝑈𝑛,𝑅𝑘(𝐚)is a jointly strictly concave function in 𝐚,∀𝑛∈ {1,2,…, 𝑁}24 .
The achievable rate of user 𝑈𝑛over relay 𝑅𝑘(in bits/s/Hz) can be expressed as
𝑈𝑛,𝑅𝑘(𝐚)=1
2log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛,(11)
where 𝐚=𝑎1,…, 𝑎𝑛,…, 𝑎𝑁. In this work, four different energy-efficiency strategies are considered 6, namely global energy-
efficiency (GEE), minimum energy-efficiency (MEE), product energy-efficiency (PEE), and sum energy-efficiency (SEE).
Particularly, the GEE (in bits/J/Hz) is defined over relay 𝑅𝑘as
GEE𝑘(𝐚)=
1
2𝑁
𝑛=1 log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐸𝑅𝑘
=𝑁
𝑛=1 𝑈𝑛,𝑅𝑘(𝐚)
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐸𝑅𝑘
,(12)
while the MEE is given by
MEE𝑘(𝐚)= min
1≤𝑛≤𝑁
1
2log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
= min
1≤𝑛≤𝑁
𝑈𝑛,𝑅𝑘(𝐚)
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
.(13)
Moreover, the PEE is expressed as
PEE𝑘(𝐚)=
𝑁
𝑛=1
1
2log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
=
𝑁
𝑛=1
𝑈𝑛,𝑅𝑘(𝐚)
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,(14)
whereas the SEE is written as
SEE𝑘(𝐚)=
𝑁
𝑛=1
1
2log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
=
𝑁
𝑛=1
𝑈𝑛,𝑅𝑘(𝐚)
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
.(15)
Specifically, the GEE is defined as the ratio of sum-of-rates to the total energy consumption of all users, while the MEE defines
the minimum rate to energy consumption ratio, and aims to achieve absolute fairness. Additionally, the PEE aims to achieve
users’ proportional fairness in terms of energy-efficiency, while the SEE is focussed on the sum of energy-efficiency of all
users23,25.
Remark 2: The GEE is considered a network-centric EE metric, while the MEE, PEE and SEE are user-centric metrics 25.
Remark 3: Since the numerator of GEE𝑘(𝐚)is a sum of non-linear strictly concave functions in 𝐚, while the denominator is
a linear function in 𝐚, then GEE𝑘(𝐚)is a non-linear fractional function in 𝐚,∀𝑘∈ {1,2,…, 𝐾}. Similarly, the energy-efficiency
term for each user under the MEE, PEE and SEE strategies is a ratio of a non-linear strictly concave function to a linear function
in 𝐚,∀𝑛∈ {1,2,…, 𝑁 }and ∀𝑘∈ {1,2,…, 𝐾}.
3 JOINT RELAY SELECTION AND ENERGY-EFFICIENT POWER ALLOCATION
The joint relay selection and energy-efficient power allocation (J-RS-EE-PA) problem subject to target minimum rate 𝑟𝑇can
be formulated as
J-RS-EE-PA:
max EE (𝐚,
)
s.t.
𝐾
𝑘=1
𝑅𝑘= 1,(16a)
𝑁
𝑛=1
𝑎𝑛≤1,(16b)
𝑈𝑛(𝐚,
)≥𝑟𝑇,∀𝑛∈ {1,2,…, 𝑁 },(16c)
6M. W. Baidas ET AL
𝑎1≥𝑎2≥⋯≥𝑎𝑁,(16d)
𝑎𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 },(16e)
𝑅𝑘∈ {0,1},∀𝑘∈ {1,2,…, 𝐾},(16f)
where
EE (𝐚,
)=
𝑁
𝑛=1 𝑈𝑛(𝐚,
)
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐾
𝑘=1 𝑅𝑘𝐸𝑅𝑘
,GEE,
min
1≤𝑛≤𝑁
𝑈𝑛(𝐚,
)
𝑎𝑛𝐸𝐵𝑆 +𝐾
𝑘=1 𝑅𝑘𝐸𝑅𝑘
,MEE,
𝑁
𝑛=1
𝑈𝑛(𝐚,
)
𝑎𝑛𝐸𝐵𝑆 +𝐾
𝑘=1 𝑅𝑘𝐸𝑅𝑘
,PEE,
𝑁
𝑛=1
𝑈𝑛(𝐚,
)
𝑎𝑛𝐸𝐵𝑆 +𝐾
𝑘=1 𝑅𝑘𝐸𝑅𝑘
,SEE.
(17)
Moreover, 𝑈𝑛(𝐚,
)is given by
𝑈𝑛(𝐚,
)=1
2log21 +
𝐾
𝑘=1
𝑅𝑘
𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛,(18)
where
=𝑅1,…,𝑅𝑘,…,𝑅𝐾, and 𝑅𝑘is a binary decision variable, defined as
𝑅𝑘=1,if relay 𝑅𝑘is selected,
0,otherwise.(19)
In (16), the first constraint ensures that only one relay is selected, while the second constraint ensures that the sum of the power
allocation coefficients does not exceed one. The third constraint ensures that the target minimum rate 𝑟𝑇is satisfied for all users,
while the fourth constraint enforces the SIC decoding order. The last two constraints define the range of values the decision
variables take.
Remark 4: Each of the formulated joint relay selection and energy-efficient power allocation problems is a mixed-integer
non-linear fractional programming (MINLFP) problem, which is non-convex. Specifically, each problem can be classified a
mixed-integer non-linear programming problem, and hence is generally NP-hard 26.
Based on Remark 4, solving the J-RS-EE-PA problem for each of the energy-efficiency strategies is computationally-
prohibitive. Hence, problem J-RS-EE-PA can be decoupled into two sub-problems: (1) relay selection, and (2) energy-efficient
power allocation, as detailed in the following sections.
Remark 5: Decoupling the J-RS-EE-PA problem into two sub-problems does not necessarily guarantee the global optimal
solution to the original problem.
4 RELAY SELECTION
Based on the derived SINR expression in (7), it is clear that Γ𝑅𝑘,𝑈𝑛is dependent upon 𝐸𝑅𝑘,ℎ𝐵𝑆,𝑅𝑘2and ℎ𝑅𝑘,𝑈𝑛2. In turn,
the following relay selection schemes are explored.
4.1 Maximum Harvested Energy
The maximum harvested energy (MHE) relay selection scheme aims at selecting the relay with the greatest amount of
harvested energy, as
𝑅𝑀𝐻𝐸 = arg max
𝑘∈{1,2,…,𝐾}Σ,𝑅𝑘.(20)
4.2 Best Base-Station-Relay SNR
The best base-station-relay SNR (B-BSR-SNR) relay selection scheme is expressed as
𝑅𝐵−𝐵𝑆𝑅−𝑆𝑁𝑅 = arg max
𝑘∈{1,2,…,𝐾}𝛾𝐵 𝑆,𝑅𝑘,(21)
where 𝛾𝐵𝑆,𝑅𝑘is given in (9).
M. W. Baidas ET AL 7
4.3 Best Relay-User Sum-Channel Gains
The best relay-user sum-channel gains (B-RU-S-CG) relay selection scheme aims to select the relay with the best sum-channel
gains with the users, as given by
𝑅𝐵−𝑅𝑈−𝑆−𝐶𝐺 = arg max
𝑘∈{1,2,…,𝐾}
𝑁
𝑛=1 ℎ𝑅𝑘,𝑈𝑛2.(22)
4.4 Best Relay-User Sum-SNR
In this scheme, the relay with the best relay-user sum-SNR (B-RU-S-SNR) is selected, as
𝑅𝐵−𝑅𝑈−𝑆−𝑆𝑁𝑅 = arg max
𝑘∈{1,2,…,𝐾}
𝑁
𝑛=1
𝛾𝑅𝑘,𝑈𝑛,(23)
where 𝛾𝑅𝑘,𝑈𝑛is given in (9). This scheme is similar to the previous one, except that the transmit energy 𝐸𝑅𝑘—in (2)—and noise
variance are also taken into account, ∀𝑘∈ {1,2,…, 𝐾}.
4.5 Best Base-Station-Relay-User Sum-Channel Gains
This scheme selects the relay with the best base-station-relay-user sum-channel gains (B-BSRU-S-CG), as
𝑅𝐵−𝐵𝑆𝑅𝑈−𝑆−𝐶 𝐺 = arg max
𝑘∈{1,2,…,𝐾}
𝑁
𝑛=1 ℎ𝐵𝑆,𝑅𝑘2ℎ𝑅𝑘,𝑈𝑛2.(24)
4.6 Best Base-Station-Relay-User Sum-SNR
The best base-station-relay-user sum-SNR (B-BSRU-S-SNR) relay selection is expressed as
𝑅𝐵−𝐵𝑆𝑅𝑈−𝑆−𝑆𝑁𝑅 = arg max
𝑘∈{1,2,…,𝐾}
𝑁
𝑛=1
𝜉𝑅𝑘,𝑈𝑛,(25)
where 𝜉𝑅𝑘,𝑈𝑛is expressed by (8). Lastly, this scheme is similar to the (B-BSRU-S-CG) except that the noise variance, transmit
energy at the base-station, and transmit energy at each potential relay are incorporated.
Remark 6: Once a relay 𝑅𝑘is selected (i.e. 𝑅𝑘= 1), then the J-RS-EE-PA problem reduces to a power allocation problem
under each of the four EE strategies, and computationally-efficient algorithmic solutions can be devised.
5 ENERGY-EFFICIENT POWER ALLOCATION STRATEGIES
5.1 Global Energy-Efficiency (GEE)
The maximization of the GEE (MGEE) problem over relay 𝑅𝑘is expressed as
MGEE𝑘(𝐚) ∶
max GEE𝑘(𝐚)=
1
2𝑁
𝑛=1 log21+ 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐸𝑅𝑘
s.t.
𝑁
𝑛=1
𝑎𝑛≤1,(26a)
𝑈𝑛,𝑅𝑘(𝐚)≥𝑟𝑇,∀𝑛∈ {1,2,…, 𝑁 },(26b)
𝑎1≥𝑎2≥⋯≥𝑎𝑁,(26c)
𝑎𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 }.(26d)
8M. W. Baidas ET AL
Remark 7: Since the nominator of GEE𝑘(𝐚)is a non-negative and differentiable sum of strictly concave functions in 𝐚, while
the denominator is a linear function in 𝐚, then GEE𝑘(𝐚)is a strictly pseudo-concave function in 𝐚. Hence, if
𝐚is a stationary
point of GEE𝑘(𝐚), then it is also the unique global maximizer23.
Problem MGEE𝑘(𝐚)can be efficiently solved using a suitable lower-bound for the objective function, and then iteratively
tightening the lower-bound, such that the global optimal solution that satisfies Karush-Kuhn-Tucker (KKT) conditions can be
obtained27,23. To that end, the inequality log2(1 + 𝛾)≥𝛼log2(𝛾) + 𝛽for 𝛾, 𝛾 > 0is employed, where28
𝛼≜𝛾
1 + 𝛾 ,(27)
and
𝛽≜log2(1 + 𝛾)−𝛾 log2(𝛾 )∕(1 + 𝛾),(28)
with the inequality being tight for 𝛾=𝛾. Consequently, the GEE𝑘(𝐚)objective function can be lower-bounded as
GEE𝑘(𝐚)≥𝑁
𝑛=1
1
2𝛼𝑛log2𝑎𝑛𝜉𝑅𝑘,𝑈𝑛− log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 𝑎𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐸𝑅𝑘
,(29)
where the bound can be made tight if 𝛼𝑛, 𝛽𝑛𝑁
𝑛=1 are repeatedly updated, as per (27) and (28). Now, by applying the variable
substitution 𝑎𝑛= 2𝜒𝑛, the lower-bound can be re-written as
GEE𝑘(𝐚)≥𝑁
𝑛=1
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛− log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 𝑁
𝑛=1 2𝜒𝑛+𝐸𝑅𝑘
≜𝑓𝑘(𝜒
𝜒
𝜒)
𝑔𝑘(𝜒
𝜒
𝜒),(30)
where 𝜒
𝜒
𝜒≜𝜒1,…, 𝜒𝑛,…, 𝜒𝑁.
Remark 8: In the lower-bound of GEE𝑘(𝐚), it is well-known that log-sum-exp function is convex 29, which in turn makes
the − log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛term concave. Therefore, the lower-bounded objective function has a
numerator 𝑓𝑘(𝜒
𝜒
𝜒)that is a sum of concave functions, while the denominator 𝑔𝑘(𝜒
𝜒
𝜒)is convex.
Note that the rate constraint in (26b) can also be reformulated as
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛− log2𝜉𝑅𝑘,𝑈𝑛
𝑁
𝑗=𝑛+1
2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘
𝑁
𝑖=1
2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛≥𝑟𝑇,(31)
which can be verified to be concave in 𝜒
𝜒
𝜒,∀𝑛∈ {1,2,…, 𝑁 }. Hence, the maximization of the lower-bounded GEE (M-LB-GEE)
problem can be written as‡‡
M-LB-GEE𝑘(𝜒
𝜒
𝜒)∶
max 𝑁
𝑛=1
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛−log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 𝑁
𝑛=1 2𝜒𝑛+𝐸𝑅𝑘
s.t.
𝑁
𝑛=1
2𝜒𝑛≤1,(32a)
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛− log2𝜉𝑅𝑘,𝑈𝑛
𝑁
𝑗=𝑛+1
2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘
𝑁
𝑖=1
2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛≥𝑟𝑇,∀𝑛∈ {1,2,…, 𝑁 },(32b)
𝜒1≥𝜒2⋯≥𝜒𝑁,(32c)
𝜒𝑛≤0,∀𝑛∈ {1,2,…, 𝑁 }.(32d)
Clearly, all the constraints in problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)are convex. Thus, it can be globally solved via the generalized
Dinkelback’s algorithm 30 . Specifically, define the auxiliary function 31
𝐹𝑘(𝜆)≜max
𝜒
𝜒
𝜒𝑓𝑘(𝜒
𝜒
𝜒)−𝜆𝑔𝑘(𝜒
𝜒
𝜒),(33)
where 𝜆≥0is the unique root of 𝐹(𝜆). Hence, problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)—for fixed values of 𝛼𝑛and 𝛽𝑛,∀𝑛∈ {1,2,…, 𝑁}—
can be solved to obtain the global optimal solution via the Generalized Dinkelbach’s algorithm, as outlined in Algorithm 1.
‡‡Note that 𝜒𝑛≤0,∀𝑛∈ {1,2,…, 𝑁 }, since 2𝜒𝑛must be less than or equal to one.
M. W. Baidas ET AL 9
Algorithm 1 : Generalized Dinkelbach’s Algorithm for Solving Problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)
Set 𝜖∈ (0,1),𝑙= 0, and 𝜆(0) = 0.
1 WHILE 𝐹𝑘𝜆(𝑙)> 𝜖
2
𝜒
𝜒
𝜒(𝑙)= ar g max
𝜒
𝜒
𝜒𝑓𝑘(𝜒
𝜒
𝜒)−𝜆(𝑙)𝑔𝑘(𝜒
𝜒
𝜒);
3𝐹𝑘𝜆(𝑙)= max 𝑓𝑘
𝜒
𝜒
𝜒(𝑙)−𝜆(𝑙)𝑔𝑘
𝜒
𝜒
𝜒(𝑙);
4𝜆(𝑙+1) =𝑓𝑘(
𝜒
𝜒
𝜒(𝑙))
𝑔𝑘(
𝜒
𝜒
𝜒(𝑙));
5𝑙=𝑙+ 1;
6 END WHILE
Lemma 2: The succession of 𝜆(𝑙)𝑙in Algorithm 1 is increasing, ultimately converging to the global optimal solution
𝜒
𝜒
𝜒of
problem M-LB-GEE𝑘(𝜒
𝜒
𝜒).
Proof. This is a direct result of the fact that—for fixed values 𝛼𝑛and 𝛽𝑛,∀𝑛∈ {1,2,…, 𝑁 }—the functions 𝑓𝑘(𝜒
𝜒
𝜒)and 𝑔𝑘(𝜒
𝜒
𝜒)
are respectively concave and convex in 𝜒
𝜒
𝜒, while all the constraints are convex32,6.
Remark 9: In each iteration of Algorithm 1, a concave maximization problem is solved, subject to a convex set of constraints.
Hence, Algorithm 1 is solved within polynomial-time complexity, and with linear convergence rate 6.
Upon convergence of Algorithm 1, the solution
𝜒
𝜒
𝜒—based on fixed values of 𝛼𝑛and 𝛽𝑛—is obtained, which is then converted
to the original power allocation coefficients as
𝐚= 2
𝜒
𝜒
𝜒. Thus, the value of the objective function GEE𝑘(𝐚)is calculated based
on
𝐚(obtained via Algorithm 1), and denoted as
GEE𝑘(
𝐚)=𝑁
𝑛=1 𝑈𝑛,𝑅𝑘(
𝐚)
𝐸𝐵𝑆 𝑁
𝑛=1 𝑎𝑛+𝐸𝑅𝑘
.(34)
Since Algorithm 1 maximizes the lower-bound of the MGEE𝑘(𝐚), then the lower-bound value in (34) must be repeatedly
improved by updating 𝛼𝑛and 𝛽𝑛,∀𝑛∈ {1,2,…, 𝑁}, every time Algorithm 1 is executed until convergence, and this can be
achieved via Algorithm 2.
Algorithm 2 : Solution Procedure for Solving Problem MGEE𝑘(𝐚)
Set 𝜖∈ (0,1),𝑙= 0, select a feasible
𝐚(0), and set 𝛼(0)
𝑛= 1 and 𝛽(0)
𝑛= 0,∀𝑛∈ {1,2,…, 𝑁 }.
1 WHILE GEE𝑘
𝐚(𝑙+1)−GEE𝑘
𝐚(𝑙)> 𝜖
2𝑙=𝑙+ 1;
3 Evaluate GEE𝑘
𝐚(𝑙);
4 Update 𝛼(𝑙)
𝑛and 𝛽(𝑙)
𝑛,∀𝑛∈ {1,2,…, 𝑁 };
5 Determine
𝜒
𝜒
𝜒(𝑙)by solving problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)via Algorithm 1;
6 Set
𝐚(𝑙)= 2
𝜒
𝜒
𝜒(𝑙);
7 END WHILE
Lemma 3: The sequence of GEE𝑘
𝐚(𝑙)𝑙
values obtained from Algorithm 2 is monotonically increasing, ultimately
converging in a finite number of iterations to the global optimal point satisfying the KKT conditions of problem MGEE𝑘(𝐚).
Proof. The proof can be obtained by noting that in the 𝑙𝑡ℎ iteration of Algorithm 2, the obtained power allocation coefficients
𝐚(𝑙)= 2
𝜒
𝜒
𝜒(𝑙)maximize problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)based on the same of constraints of problem MGEE𝑘(𝐚). More importantly,
by applying the KKT constraints of Proposition 4.2 in23 , the following inequalities hold24
GEE𝑘
𝐚(𝑙)≥GEE𝑘
𝐚(𝑙)≥GEE𝑘
𝐚(𝑙−1)=GEE𝑘
𝐚(𝑙−1),(35)
10 M. W. Baidas ET AL
where the first inequality holds since GEE𝑘
𝐚(𝑙)is a lower-bound value of GEE𝑘
𝐚(𝑙). Moreover, the second inequality holds
as
𝐚(𝑙)maximizes GEE𝑘(𝐚)in the 𝑙𝑡ℎ iteration. Additionally, the last equality holds since GEE𝑘
𝐚(𝑙−1)is the value of GEE𝑘(𝐚)
when 𝐚=
𝐚(𝑙−1). In turn, the value of GEE𝑘(𝐚)increases in each iteration until convergence, at which point the value GEE𝑘(𝐚)
for 𝐚=
𝐚. Additionally, since problem M-LB-GEE𝑘(𝜒
𝜒
𝜒)is derived from the original MGEE𝑘(𝐚)problem (and hence the same
set of KKT conditions apply to both problems5), and due to the convexity of the constraints, and also the update of the 𝛼𝑛and
𝛽𝑛values, then upon convergence GEE𝑘(
𝐚)=GEE𝑘(
𝐚).
Remark 10: The MGEE𝑘(𝐚)is solved via Algorithm 2 to obtain the global optimal solution over relay 𝑅𝑘,∀𝑘∈
{1,2,…, 𝐾}.
5.2 Minimum Energy Efficiency (MEE)
The maximization of the MEE (MMEE) problem—when relay 𝑅𝑘is selected—is written as
MMEE𝑘(𝐚) ∶
max MEE𝑘(𝐚)= min
1≤𝑛≤𝑁
1
2log21+ 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
s.t. Constraints (26𝑎) − (26𝑑).(36)
Using a similar approach to the MGEE𝑘(𝐚), problem MMEE𝑘(𝐚)can be solved by lower-bounding the objective function
MEE𝑘(𝐚)for each user 𝑈𝑛(for 𝑛∈ {1,2,…, 𝑁 }) as
1
2𝛼𝑛log2𝑎𝑛𝜉𝑅𝑘,𝑈𝑛− log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 𝑎𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 𝑎𝑛+𝐸𝑅𝑘
,(37)
which after applying the variable change 𝑎𝑛= 2𝜒𝑛can be re-written as
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛− log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 2𝜒𝑛+𝐸𝑅𝑘
≜
𝑓𝑘(𝜒
𝜒
𝜒)
𝑔𝑘(𝜒
𝜒
𝜒).(38)
Hence, the maximization of the lower-bounded MMEE (M-LB-MMEE) problem can be written as
M-LB-MMEE𝑘(𝜒
𝜒
𝜒)∶
max min
1≤𝑛≤𝑁
1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛−log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑗=𝑛+1 2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
𝐸𝐵𝑆 2𝜒𝑛+𝐸𝑅𝑘
s.t. Constraints (32𝑎) − (32𝑑).(39)
As before, the auxiliary function can be written as
𝐹𝑘(𝜆)≜max
𝜒
𝜒
𝜒min
1≤𝑛≤𝑁
𝑓𝑘(𝜒
𝜒
𝜒)−𝜆 𝑔𝑘(𝜒
𝜒
𝜒).(40)
Thus, the Generalized Dinkelbach’s algorithm (i.e. Algorithm 1) can be applied to solve problem M-LB-MMEE𝑘(𝜒
𝜒
𝜒), and
obtain the power allocation coefficients
𝐚= 2
𝜒
𝜒
𝜒. The corresponding value of the objective function of MEE𝑘(𝐚)is determined as
MEE𝑘(
𝐚)= min
1≤𝑛≤𝑁
𝑈𝑛,𝑅𝑘(
𝐚)
𝐸𝐵𝑆 𝑎𝑛+𝐸𝑅𝑘
.(41)
Algorithm 2 is then executed to update the lower-bounded objective function value by repeatedly updating 𝛼𝑛and 𝛽𝑛until
convergence§.
Remark 11: The global optimal solution of the MMEE𝑘(𝐚)is obtained via Algorithm 2 over relay 𝑅𝑘,∀𝑘∈ {1,2,…, 𝐾}.
§Lemmas 2 and 3 can be easily applied to problem MMEE𝑘(𝐚)24.
M. W. Baidas ET AL 11
5.3 Product Energy Efficiency (PEE)
The maximization of the PEE (MPEE) problem over relay 𝑅𝑘is given by
MPEE𝑘(𝐚) ∶
max PEE𝑘(𝐚)=𝑁
𝑛=1
1
2log21+ 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
s.t. Constraints (26𝑎) − (26𝑑).(42)
To simplify problem MPEE𝑘(𝐚), and since the ln (⋅)is concave and strictly monotonically increasing in its parameter, then
applying the logarithmic function to the objective function PEE𝑘(𝐚)yields
ln (PEE (𝐚)) =
𝑁
𝑛=1
ln 1
2log21 + 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛−
𝑁
𝑛=1
ln 𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘.(43)
Now, using the inequality log2(1 + 𝛾)≥𝛼log2(𝛾) + 𝛽, the lower-bound of ln PEE𝑘(𝐚)is obtained as
ln PEE𝑘(𝐚)≥
𝑁
𝑛=1
ln 1
2𝛼𝑛log2𝑎𝑛𝜉𝑅𝑘,𝑈𝑛− log2𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘
𝑁
𝑖=1
𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛−
𝑁
𝑛=1
ln 𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘.
(44)
Using the substitution 𝑎𝑛= 2𝜒𝑛, yields
ln PEE𝑘(𝐚)≥
𝑁
𝑛=1
ln 1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛− log2𝜉𝑅𝑘,𝑈𝑛
𝑁
𝑗=𝑛+1
2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘
𝑁
𝑖=1
2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛
−
𝑁
𝑛=1
ln 2𝜒𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘.
(45)
As before, the log-sum-exp is convex, and the − log2𝜉𝑅𝑘,𝑈𝑛𝑁
𝑖=𝑛+1 2𝑞𝑖+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 2𝜒𝑖+𝜁𝑅𝑘,𝑈𝑛term is concave. Also,
the log-exp term (i.e. ln 2𝜒𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘) is convex, and thus − ln 2𝜒𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘is concave29 . In turn, the lower-bounded
ln PEE𝑘(𝐚)function is concave. Hence, the maximization of the lower-bounded MPEE (M-LB-MPEE) problem is given by
M-LB-MPEE𝑘(𝜒
𝜒
𝜒)∶
max
𝑁
𝑛=1
ln 1
2𝛼𝑛log2𝜉𝑅𝑘,𝑈𝑛+𝜒𝑛−log2𝜉𝑅𝑘,𝑈𝑛
𝑁
𝑗=𝑛+1
2𝜒𝑗+𝛾𝐵𝑆,𝑅𝑘
𝑁
𝑖=1
2𝜒𝑖++𝜁𝑅𝑘,𝑈𝑛+1
2𝛽𝑛−
𝑁
𝑛=1
ln 2𝜒𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
s.t. Constraints (32𝑎) − (32𝑑),(46)
which is a concave maximization problem, and thus can be solved efficiently via any standard convex optimization software
package29 to determine the optimal power allocation coefficients
𝐚= 2
𝜒
𝜒
𝜒for fixed values of 𝛼𝑛and 𝛽𝑛. Thus, the corresponding
objective function value PEE𝑘(𝐚)can be evaluated at
𝐚as
PEE𝑘(
𝐚)=
𝑁
𝑛=1
𝑈𝑛,𝑅𝑘(
𝐚)
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
.(47)
Lastly, Algorithm 2 is used to repeatedly update the value of 𝛼𝑛and 𝛽𝑛until convergence§§ .
Remark 12: Upon convergence of Algorithm 2, the global optimal solution of problem MPEE𝑘(𝐚)is obtained over each
relay 𝑅𝑘,∀𝑘∈ {1,2,…, 𝐾}.
5.4 Sum Energy-Efficiency (SEE)
The maximization of the SEE (MSEE) problem over relay 𝑅𝑘is given by
§§Lemma 3 can be straightforwardly be extended for problem MPEE𝑘(𝐚).
12 M. W. Baidas ET AL
MSEE𝑘(𝐚) ∶
max SEE𝑘(𝐚)=𝑁
𝑛=1
1
2log21+ 𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1 𝑎𝑖+𝜁𝑅𝑘,𝑈𝑛
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
s.t. Constraints (26𝑎) − (26𝑑).(48)
Remark 13: The objective function SEE𝑘(𝐚)is a sum of ratios of concave functions to linear functions. However, the sum-
of-ratios is generally not necessarily pseudo-concave or quasi-concave23 , which makes problem MSEE𝑘(𝐚)non-convex.
Alternatively, problem MSEE𝑘(𝐚)can be reformulated as33
R-MSEE𝑘(𝐚) ∶
max 𝑁
𝑛=1 𝜇𝑛
s.t. 𝑈𝑛,𝑅𝑘(𝐚)≥𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘,∀𝑛∈ {1,2,…, 𝑁 },(49a)
Constraints (26𝑎) − (26𝑑).(49b)
Lemma 4: If (
𝐚,
𝜇
𝜇
𝜇)is the optimal solution to problem R-MSEE𝑘(𝐚), then there exists
𝜇
𝜇
𝜇=𝜇1, 𝜇2,…, 𝜇𝑁and
𝜆
𝜆
𝜆=
𝜆1,
𝜆2,…,
𝜆𝑁, such that
𝐚is the optimal solution to the parametric problem
P-R-MSEE𝑘(𝐚) ∶
max 𝑁
𝑛=1 𝜆𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
s.t. Constraints (26𝑎) − (26𝑑),(50)
with 𝜇
𝜇
𝜇=
𝜇
𝜇
𝜇and 𝜆
𝜆
𝜆=
𝜆
𝜆
𝜆, such that the following conditions are satisfied,
𝜆𝑛=1
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,(51)
and
𝑈𝑛,𝑅𝑘(𝐚)=𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘,(52)
∀𝑛∈ {1,2,…, 𝑁 }.
Proof. Refer to Appendix II.
Remark 14: It is straightforward to verify that problem P-R-MSEE𝑘(𝐚)is a concave maximization problem for given values
of 𝜇𝑛≥0and 𝜆𝑛>0,∀𝑛∈ {1,2,…, 𝑁 }. Additionally, Slater’s regularity condition can be shown to be satisfied34,33. Hence,
by applying Lemma 4 to problem P-R-MSEE𝑘(𝐚), the optimal solution to problem R-MSEE𝑘(𝐚)(and hence MSEE𝑘(𝐚)) can
be obtained.
To efficiently solve problem P-R-MSEE𝑘(𝐚), the conditions in (51) and (52) must be satisfied33 . In turn, the values of 𝜇
𝜇
𝜇and
𝜆
𝜆
𝜆must be iteratively updated until convergence, which can be achieved via the modified Newton’s method 35,36. To that end, and
for notational convenience, let 𝜋
𝜋
𝜋≜(𝜇
𝜇
𝜇, 𝜆
𝜆
𝜆), and also let
𝜓𝑛(𝜋
𝜋
𝜋)≜−𝑈𝑛,𝑅𝑘(𝐚(𝜋
𝜋
𝜋)) +𝜇𝑛𝑎𝑛(𝜋
𝜋
𝜋)𝐸𝐵𝑆 +𝐸𝑅𝑘,and
𝜓𝑛(𝜋
𝜋
𝜋)≜−1 + 𝜆𝑛𝑎𝑛(𝜋
𝜋
𝜋)𝐸𝐵𝑆 +𝐸𝑅𝑘,∀𝑛∈ {1,2,…, 𝑁 },(53)
with the aim of making 𝜓𝑛(𝜋
𝜋
𝜋)= 0 and
𝜓𝑛(𝜋
𝜋
𝜋)= 0,∀𝑛∈ {1,2,…, 𝑁 }. Define the diagonal matrix 𝜓
𝜓
𝜓(𝜋
𝜋
𝜋), such that 𝜓
𝜓
𝜓𝑛(𝜋
𝜋
𝜋)=
𝜓𝑛(𝜋
𝜋
𝜋)and 𝜓
𝜓
𝜓𝑛+𝑁(𝜋
𝜋
𝜋)=
𝜓𝑛(𝜋
𝜋
𝜋),∀𝑛∈ {1,2,…, 𝑁 }. Thus, for the 𝑙𝑡ℎ iteration, the modified Newton’s method is defined by
𝜋
𝜋
𝜋(𝑙+1) =𝜋
𝜋
𝜋(𝑙)−𝛿(𝑙)𝜓
𝜓
𝜓′𝜋
𝜋
𝜋(𝑙)−1 𝜓
𝜓
𝜓𝜋
𝜋
𝜋(𝑙),(54)
where 𝜓
𝜓
𝜓′𝜋
𝜋
𝜋(𝑙)−1 is the inverse of the Jacobian matrix of 𝜓
𝜓
𝜓(𝜋
𝜋
𝜋). Hence, 𝜇𝑛and 𝜆𝑛can be updated at each iteration ∀𝑛∈
{1,2,…, 𝑁}as
𝜇(𝑙+1)
𝑛=𝜇(𝑙)
𝑛−𝛿(𝑙)
𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘𝜇(𝑙)
𝑛𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘−𝑈𝑛,𝑅𝑘𝐚(𝑙)
=1 − 𝛿(𝑙)𝜇(𝑙)
𝑛+𝛿(𝑙)𝑈𝑛,𝑅𝑘𝐚(𝑙)
𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,
(55)
M. W. Baidas ET AL 13
and
𝜆(𝑙+1)
𝑛=𝜆(𝑙)
𝑛−𝛿(𝑙)
𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘𝜆(𝑙)
𝑛𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘− 1
=1 − 𝛿(𝑙)𝜆(𝑙)
𝑛+𝛿(𝑙)1
𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,
(56)
respectively. Lastly, 𝛿(𝑙)is the greatest 𝜀𝑖that satisfies¶
𝜓
𝜓
𝜓𝜋
𝜋
𝜋(𝑙)−𝜀𝑖𝜓
𝜓
𝜓′𝜋
𝜋
𝜋(𝑙)−1 𝜓
𝜓
𝜓𝜋
𝜋
𝜋(𝑙)
≤1 − 𝜖𝜀𝑖
𝜓
𝜓
𝜓𝜋
𝜋
𝜋(𝑙)
,(57)
where 𝑖∈ {0,1,…},𝜀∈ (0,1), and 𝜖∈ (0,1). The solution procedure for solving problem R-MSEE𝑘(𝐚)is outlined in
Algorithm 3.
Algorithm 3 : Solution of Problem R-MSEE𝑘(𝐚)
Set 𝜖∈ (0,1),𝜀∈ (0,1),𝑙= 0, and select a feasible 𝐚(0).
1 Determine 𝜆(0)
𝑛=1
𝑎(0)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
and 𝜇(0)
𝑛=𝑈𝑛,𝑅𝑘(𝐚(0))
𝑎(0)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,∀𝑛∈ {1,2,…, 𝑁 }.
2 WHILE max
𝐚(𝑙)𝑁
𝑛=1 𝜆(𝑙)
𝑛𝑈𝑛,𝑅𝑘𝐚(𝑙)−𝜇(𝑙)
𝑛𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘> 𝜖
3 Solve 𝐚(𝑙+1) = ar g max
𝐚(𝑙)𝑁
𝑛=1 𝜆(𝑙)
𝑛𝑈𝑛,𝑅𝑘𝐚(𝑙)−𝜇(𝑙)
𝑛𝑎(𝑙)
𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘;
4 Update 𝜆(𝑙)
𝑛and 𝜇(𝑙)
𝑛,∀𝑛∈ {1,2,…, 𝑁 }via the modified Newton’s method;
5 Set 𝑙=𝑙+ 1;
6 END WHILE
Output:
𝐚.
Remark 15: Since Algorithm 3 is based on the modified Newton’s method, and problem P-R-MSEE𝑘(𝐚)is a concave
maximization problem, then it is guaranteed to converge in a finite number of iterations to the global optimal solution of problem
MSEE𝑘(𝐚), and with a linear convergence rate 33,36.
6 SOLUTION PROCEDURE FOR JOINT RELAY SELECTION AND ENERGY-EFFICIENT
POWER ALLOCATION
Based on the obtained algorithmic solutions for problems MGEE𝑘(𝐚),MMEE𝑘(𝐚),MPEE𝑘(𝐚), and MSEE𝑘(𝐚), a simple
solution procedure can be devised to jointly solve the relay selection and energy-efficient power allocation (SP-J-RS-EE-PA),
as outlined in Algorithm 4. Basically, the idea is to iterate over each relay, and optimally solve the energy-efficiency power
allocation problem. Then, the relay that yields the highest objective function value (i.e. GEE, MEE, PEE, or SEE) is selected.
Remark 16: It can be easily seen that SP-J-RS-EE-PA requires 𝐾iterations, and in each of which Algorithm 2 is solved
efficiently to obtain the global optimal solution of problems MGEE𝑘(𝐚),MMEE𝑘(𝐚), and MPEE𝑘(𝐚), while Algorithm 3 is
utilized to obtain the global optimal solution for problem MSEE𝑘(𝐚).
7 SIMULATION RESULTS
In this section, the different relay selection schemes when combined with EE power allocation strategies are compared to
the J-RS-EE-PA and SP-J-RS-EE-PA schemes¶¶. The simulations assume a network of 𝑁= 3 users and 𝐾= 3 EH AF
relays, as per the network topology shown in Fig. 1, where the path-loss exponent is set as 𝜈= 3. The maximum values of
¶𝜀is a properly selected step-size37 .
¶¶All optimization problems are solved via MIDACO38 , with tolerance set to 0.001.
14 M. W. Baidas ET AL
Algorithm 4 : Solution Procedure for the joint relay selection and energy-efficient power allocation (SP-J-RS-EE-PA)
Stage 1: (Optimal Power Allocation Per Relay)
1 FOR each 𝑘∈ {1,2,…, 𝐾 }
2 Set 𝑅𝑘= 1,𝑅𝑙= 0,∀𝑙∈ {1,2,…, 𝐾}and 𝑙≠𝑘in J-RS-EE-PA;
3 IF GEE
4 Solve problem MGEE𝑘(𝐚)via Algorithm 2 to obtain
𝐚;
5 Evaluate GEE𝑘(
𝐚);
6 ELSE IF MEE
7 Solve problem MMEE𝑘(𝐚)via Algorithm 2 to obtain
𝐚;
8 Evaluate MEE𝑘(
𝐚);
9 ELSE IF PEE
10 Solve problem MPEE𝑘(𝐚)via Algorithm 2 to obtain
𝐚;
11 Evaluate PEE𝑘(
𝐚);
12 ELSE IF SEE
13 Solve problem MSEE𝑘(𝐚)via Algorithm 3 to obtain
𝐚;
14 Evaluate SEE𝑘(
𝐚);
15 END IF
16 END FOR
Stage 2: (Optimal Relay Selection)
17 IF GEE
18 Determine 𝑅∗= arg max
𝑘∈{1,2,…,𝐾}GEE𝑘(
𝐚), and GEE∗= max
𝑘∈{1,2,…,𝐾}GEE𝑘(
𝐚);
19 ELSE IF MEE
20 Determine 𝑅∗= arg max
𝑘∈{1,2,…,𝐾}MEE𝑘(
𝐚), and MEE∗= max
𝑘∈{1,2,…,𝐾}MEE𝑘(
𝐚);
21 ELSE IF PEE
22 Determine 𝑅∗= arg max
𝑘∈{1,2,…,𝐾}PEE𝑘(
𝐚), and PEE∗= max
𝑘∈{1,2,…,𝐾}PEE𝑘(
𝐚);
23 ELSE IF SEE
24 Determine 𝑅∗= arg max
𝑘∈{1,2,…,𝐾}SEE𝑘(
𝐚), and SEE∗= max
𝑘∈{1,2,…,𝐾}SEE𝑘(
𝐚);
25 END IF
the energy arrivals are set as max
𝑅𝑘= 50,40,30mJ, for 𝑘= 1,2,3, respectively. The transmit power of the base-station is set as
𝐸𝐵𝑆 = 250mW, while the noise variance is set to 𝑁0= 10−6𝑊. The total transmit energy per time-slot is set to 𝐸𝑇= 250mW,
and the battery capacity is set to 𝐵max = 1000mW. The target minimum rate is set to 𝑟𝑇= 0.75 bits/s/Hz, while the tolerance
for Algorithms 1—3 is set to 𝜖= 0.001, while 𝜀= 0.9. Moreover, the simulation results are averaged over 103independent
network instances with randomly generated channel coefficients that remain constraint during each communication frame, but
vary slowly from one communication frame to another, and over 10 communication frames per network instance.
Remark 17: For notational convenience, the results of the J-RS-EE-PA and SP-J-RS-EE-PA schemes with the GEE power
allocation strategy are referred to as J-RS-GEE-PA and SP-J-GEE-PA, respectively. This also applies to the other EE power
allocation strategies.
7.1 Global Energy-Efficiency
The average rate of each user under the different relay selection schemes with GEE power allocation is demonstrated in Fig.
2a. It is evident that all users satisfy the target minimum rate of 𝑟𝑇= 0.75 bits/s/Hz under the different relay selection schemes.
Moreover, it is clear that the average rate per user under the SP-J-RS-GEE-PA and J-RS-GEE-PA schemes are identical. Fig.
M. W. Baidas ET AL 15
FIGURE 1 Simulated Network Topology
(a) Average Rate Per User (Bits/s/Hz)
U1U2U3
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average Rate Per User (Bits/s/Hz)
MHE (GEE) B-BSR-SNR (GEE) B-RU-S-CG (GEE) B-RU-S-SNR (GEE) B-BSRU-S-CG (GEE) B-BSRU-S-SNR (GEE) SP-J-RS-GEE-PA J-RS-GEE-PA
(b) Average User Rate (Bits/s/Hz)
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average User Rate (Bits/s/Hz)
(c) Average Power Allocation Coefficient Per User
U1U2U3
0
0.0025
0.005
0.0075
0.01
0.0125
0.015
0.0175
0.02
0.0225
0.025
Average Power Allocation Coefficient Per User
(d) Average User Power Allocation Coefficient
0
0.0025
0.005
0.0075
0.01
0.0125
0.015
0.0175
0.02
0.0225
0.025
Average User Power Allocation Coefficient
(e) Average Energy-Efficiency Per User (Bits/J/Hz)
U1U2U3
0
25
50
75
100
125
150
175
200
225
250
Average Energy-Efficiency Per User (Bits/J/Hz)
(f) Average User Energy-Efficiency (Bits/J/Hz)
0
25
50
75
100
125
150
175
200
225
250
Average User Power Allocation Coefficient
FIGURE 2 GEE Results: (a) Average Rate Per User, (b) Average User Rate, (c) Average Power Allocation Coefficient Per User,
(d) Average User Power Allocation Coefficient, (e) Average Energy-Efficiency Per User, and (f) Average User Energy-Efficiency
2b illustrates that average user rate for the different schemes, where it is clear that the SP-J-RS-GEE-PA and J-RS-GEE-PA
schemes achieve the lowest values. In order to properly quantify the resulting energy-efficiency, it is essential that the average
power allocation coefficients (PACs) be considered. Specifically, in Fig. 2c, it is clear that for user 𝑈1, the average PAC per user
for the SP-J-RS-GEE-PA and J-RS-GEE-PA scheme are comparable to the B-BSR-SNR (GEE) and B-BSRU-S-CG (GEE)
schemes, but lower than the other schemes. More importantly, for users 𝑈2and 𝑈3, the average PACs are lower than the other
schemes. Nevertheless, their achievable rates satisfy the target minimum rate of 0.75 bits/s/Hz. Fig. 2d illustrates the average
user PAC, where it is evident that the SP-J-RS-GEE-PA and J-RS-GEE-PA schemes have the least average user PAC values.
By keeping in mind that the GEE is proportional to the sum-of-rates, but inversely proportional to the total consumed energy,
the resulting GEE values (as per (12)) are evaluated for the different schemes. Fig. 2e illustrates the average EE value per user,
where it can be seen that the SP-J-RS-GEE-PA and J-RS-GEE-PA schemes achieve the highest values among the different
16 M. W. Baidas ET AL
Average Global Energy-Efficiency (Bits/J/Hz)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-GEE-PA J-RS-GEE-PA
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500
Average Global Energy-Efficiency (Bits/J/Hz)
FIGURE 3 Average Global Energy-Efficiency
relay selection schemes, as well as the highest values of average user EE values, as shown in Fig. 2f. Lastly, the network GEE
values under the different relay selection schemes are shown in Fig. 3, which reveals that the SP-J-RS-GEE-PA completely
agrees with the J-RS-GEE-PA, which indicates that the solution procedure (i.e. Algorithm 4) achieves the global optimal
solution of the J-RS-GEE-PA scheme. More importantly, it is evident that the SP-J-RS-GEE-PA and J-RS-GEE-PA schemes
yield significantly higher average GEE values than all the other relay selection schemes. This is attributed to the fact that the SP-
J-RS-GEE-PA and J-RS-GEE-PA schemes not only select the optimal relay, but also “jointly” maximize the network global
energy-efficiency. Additionally, although the other relay selection schemes are combined with optimal global energy-efficient
power allocation, they may not necessarily select the optimal relay, which severely degrades their average GEE values.
7.2 Minimum Energy-Efficiency
(a) Average Rate Per User (Bits/s/Hz)
U1U2U3
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average Rate Per User (Bits/s/Hz)
MHE (MMEE) B-BSR-SNR (MMEE) B-RU-S-CG (MMEE) B-RU-S-SNR (MMEE) B-BSRU-S-CG (MMEE) B-BSRU-S-SNR (MMEE) SP-J-RS-MMEE-PA J-RS-MMEE-PA
(b) Average User Rate (Bits/s/Hz)
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average User Rate (Bits/s/Hz)
(c) Average Power Allocation Coefficient Per User
U1U2U3
0
0.01
0.02
0.03
0.04
0.05
0.06
Average Power Allocation Coefficients
(d) Average User Power Allocation Coefficient
0
0.01
0.02
0.03
0.04
0.05
0.06
Average User Power Allocation Coefficient
(e) Average Energy-Efficiency Per User (Bits/J/Hz)
U1U2U3
0
25
50
75
100
125
150
175
200
225
250
Average Energy-Efficiency Per User (Bits/J/Hz)
(f) Average User Energy-Efficiency (Bits/J/Hz)
0
25
50
75
100
125
150
175
200
225
250
Average User Energy-Efficiency (Bits/J/Hz)
FIGURE 4 MMEE Results: (a) Average Rate Per User, (b) Average User Rate, (c) Average Power Allocation Coefficient Per
User, (d) Average User Power Allocation Coefficient, (e) Average Energy-Efficiency Per User, and (f) Average User Energy-
Efficiency
In Fig. 4a, the average rate per user under the different schemes is illustrated, where it can be seen that all users satisfy
the target minimum rate. More importantly, the SP-J-RS-MMEE-PA and J-RS-MMEE-PA schemes yield identical achievable
M. W. Baidas ET AL 17
Average Minimum Energy-Efficiency (Bits/J/Hz)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-MMEE-PA J-RS-MMEE-PA
0
25
50
75
100
125
150
175
200
225
250
Average Minimum Energy-Efficiency (Bits/J/Hz)
FIGURE 5 Average Minimum Energy-Efficiency
rates per user, which also happen to be relatively lower than the other relay selection schemes. This also applies to the average
user rate in Fig. 4b. Fig. 4 c demonstrates the average PAC per user, where it is evident that for users 𝑈2and 𝑈3, the PACs are
lower than the other schemes, whereas for user 𝑈1, its average PAC is slightly higher than the B-BSR-SNR (MMEE) and B-
BSRU-S-CG (MMEE) schemes, but lower than the other schemes. However, the average user PAC is still lower than all the
other schemes, as can be seen from Fig. 4d. In Fig. 4 e, the average EE per user is demonstrated, where it can be easily verified
that under the different schemes scheme, all users have identical average EE values. This is turn proves that the MMEE power
allocation strategy achieves absolute fairness in EE, irrespective of the relay selection strategy. Also, the EE per user under the
SP-J-RS-MMEE-PA and J-RS-MMEE-PA schemes are significantly higher than the different relay selection schemes. This is
also evident from 4f, which illustrates the average user EE values. Fig. 5 illustrates the average minimum EE, where it is evident
that SP-J-RS-MMEE-PA and J-RS-MMEE-PA schemes are superior to the other relay selection schemes.
7.3 Product Energy-Efficiency
(a) Average Rate Per User (Bits/s/Hz)
U1U2U3
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average Rate Per User (Bits/s/Hz)
MHE (PEE) B-BSR-SNR (PEE) B-RU-S-CG (PEE) B-RU-S-SNR (PEE) B-BSRU-S-CG (PEE) B-BSRU-S-SNR (PEE) SP-J-RS-PEE-PA J-RS-PEE-PA
(b) Average User Rate (Bits/s/Hz)
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
Average User Rate (Bits/s/Hz)
(c) Average Power Allocation Coefficient Per User
U1U2U3
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Average Power Allocation Coefficient Per User
(d) Average User Power Allocation Coefficient
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Average User Power Allocation Coefficient
(e) Average Energy-Efficiency Per User (Bits/J/Hz)
U1U2U3
0
25
50
75
100
125
150
175
200
225
250
Average Energy-Efficiency Per User (Bits/J/Hz)
(f) Average User Energy-Efficiency (Bits/J/Hz)
0
25
50
75
100
125
150
175
200
225
250
Average User Energy-Efficiency (Bits/J/Hz)
FIGURE 6 PEE Results: (a) Average Rate Per User, (b) Average User Rate, (c) Average Power Allocation Coefficient Per User,
(d) Average User Power Allocation Coefficient, (e) Average Energy-Efficiency Per User, and (f) Average User Energy-Efficiency
18 M. W. Baidas ET AL
Average Product Energy-Efficiency (Bits/J/Hz)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-PEE-PA J-RS-PEE-PA
104
105
106
107
108
Average Product Energy-Efficiency (Bits/J/Hz)
FIGURE 7 Average Product Energy-Efficiency
Similar observations can be made in Fig. 6 under the PEE power allocation strategy for the different relay selection schemes.
Evidently, all users satisfy the target minimum rate (see Fig. 6a). Also, the SP-J-RS-PEE-PA and J-RS-PEE-PA schemes
achieve lower average user PAC, as illustrated in Fig. 6d. More importantly, the average EE per user under the aforementioned
schemes are superior to the other schemes, which also applies to the average user EE (see Figs. 6e and 6f). Additionally, Fig.
7 shows the average product EE, where it is evident that the SP-J-RS-PEE-PA and J-RS-PEE-PA schemes are superior to all
the other relay selection schemes.
7.4 Sum Energy-Efficiency
(a) Average Rate Per User (Bits/s/Hz)
U1U2U3
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
Average Rate Per User (Bits/s/Hz)
MHE (SEE) B-BSR-SNR (SEE) B-RU-S-CG (SEE) B-RU-S-SNR (SEE) B-BSRU-S-CG (SEE) B-BSRU-S-SNR (SEE) SP-J-RS-SEE-PA J-RS-SEE-PA
(b) Average User Rate (Bits/s/Hz)
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
Average User Rate (Bits/s/Hz)
(c) Average Power Allocation Coefficient Per User
U1U2U3
0
0.025
0.05
0.075
0.1
0.125
0.15
Average Power Allocation Coefficient Per User
(d) Average User Power Allocation Coefficient
0
0.025
0.05
0.075
0.1
0.125
0.15
Average User Power Allocation Coefficient
(e) Average Energy-Efficiency Per User (Bits/J/Hz)
U1U2U3
0
25
50
75
100
125
150
175
200
225
250
Average Energy-Efficiency Per User (Bits/J/Hz)
(f) Average User Energy-Efficiency (Bits/J/Hz)
0
25
50
75
100
125
150
175
200
225
250
Average User Energy-Efficiency (Bits/J/Hz)
FIGURE 8 SEE Results: (a) Average Rate Per User, (b) Average User Rate, (c) Average Power Allocation Coefficient Per User,
(d) Average User Power Allocation Coefficient, (e) Average Energy-Efficiency Per User, and (f) Average User Energy-Efficiency
As in the previous power allocation strategies, the SP-J-RS-SEE-PA and J-RS-SEE-PA schemes yield identical average
rate, PAC and EE values, as can be seen from Fig. 8. More importantly, Fig. 9 reveals that the average SEE values of the
aforementioned schemes are superior to the other relay selection schemes.
M. W. Baidas ET AL 19
Average Sum Energy-Efficiency (Bits/J/Hz)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-SEE-PA J-RS-SEE-PA
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
Average Sum Energy-Efficiency (Bits/J/Hz)
FIGURE 9 Average Sum Energy-Efficiency
(a) SP-J-RS-GEE-PA
23.2%
36.1%
40.7%
R1R2R3
(b) J-RS-GEE-PA
23.2%
36.1%
40.7%
(c) SP-J-RS-MMEE-PA
42.2%
22.9%
34.9%
(d) J-RS-MMEE-PA
42.2%
22.9%
34.9%
(e) SP-J-RS-PEE-PA
15.5%
37.2%
47.3%
(f) J-RS-PEE-PA
15.5%
37.2%
47.3%
(g) SP-J-RS-SEE-PA
20.8%
35.9%
43.3%
(h) J-RS-SEE-PA
20.8%
35.9%
43.3%
FIGURE 10 Average Relay Selections for the SP-J-RS-EE-PA and J-RS-EE-PA Schemes Under the Different Energy-
Efficiency Power Allocation Strategies
Fig. 10 illustrates the average relay selections of the SP-J-RS-EE-PA and J-RS-EE-PA schemes under the different energy-
efficiency power allocation strategies, where it is clear that both schemes yield identical average relay selections. This proves
that the solution procedure in Algorithm 4 (i.e. SP-J-RS-EE-PA) optimally solves problem J-RS-EE-PA.
In Fig. 11, the average number of iterations for the different algorithms under the different energy-efficiency power allocation
strategies is shown. Particularly, under the GEE, the M-LB-GEE𝑘(𝜒
𝜒
𝜒)power allocation problem requires—on average—less
than 5 iterations of Algorithm 1 (see Fig. 11 a). Moreover, for the different schemes, Algorithm 2 requires—on average—less
than 6 iterations to solve problem MGEE𝑘(𝐚), as can be seen from Fig. 11b. On the other hand, the MMEE requires a higher
average number of iterations for Algorithms 1 and 2than under the GEE. This is attributed to the fact that the MMEE aims
at achieving absolute fairness in terms of energy-efficiency, and thus requires a greater number of iterations than the GEE to
achieve that. As for the PEE, the different relay selection schemes require—on average—less than 9 iterations of Algorithm 2
to optimally solve problem MPEE𝑘(𝐚), as evident from Fig. 11e. Lastly, Fig. 11f illustrates the average number of iterations
of Algorithm 3, which solves problem R-MSEE𝑘(𝐚)to obtain the global optimal solution to problem MSEE𝑘(𝐚). Clearly,
the different relay selection schemes require—on average—less than 25 iterations. Also, recall that the solution procedure of
Algorithm 4 requires 𝐾= 3 iterations (as per Remark 16), and in each of which any of the energy-efficiency power allocation
strategies can be solved efficiently. Based on all the above, it is concluded that the global optimal solution to the different energy-
efficiency power allocation strategies can be obtained efficiently and with minimal computational complexity. This is in contrast
to problem J-RS-EE-PA problem, which is NP-hard (i.e. computationally-expensive due to the exponential running time) as
per Remark 4.
20 M. W. Baidas ET AL
(a) Algorithm 1 (GEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-GEE-PA
0
1
2
3
4
5
6
7
8
9
10
Average Number of Iterations
(b) Algorithm 2 (GEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-GEE-PA
0
1
2
3
4
5
6
7
8
9
10
Average Number of Iterations
(c) Algorithm 1 (MMEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-MMEE-PA
0
1
2
3
4
5
6
7
8
9
10
Average Number of Iterations
(d) Algorithm 2 (MMEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-MMEE-PA
0
1
2
3
4
5
6
7
8
9
10
Average Number of Iterations
(e) Algorithm 2 (PEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-PEE-PA
0
1
2
3
4
5
6
7
8
9
10
Average Number of Iterations
(f) Algorithm 3 (SEE)
MHE B-BSR-SNR B-RU-S-CG B-RU-S-SNR B-BSRU-S-CG B-BSRU-S-SNR SP-J-RS-SEE-PA
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
Average Number of Iterations
FIGURE 11 Average Number of Iterations: (a) Algorithm 1 (GEE), (b) Algorithm 2 (GEE), and (c) Algorithm 1 (MMEE), (d)
Algorithm 2 (MMEE), (e) Algorithm 2 (PEE), and (f) Algorithm 3 (SEE)
8 CONCLUSIONS AND FINAL REMARKS
This paper has studied the problem of joint relay selection and energy-efficient power allocation in energy-harvesting coop-
erative NOMA networks. In particular, four different EE power allocation strategies have been considered, namely GEE, MEE,
PEE, and SEE. The J-RS-EE-PA for the aforementioned power allocation strategies has been shown be non-convex, and thus
is computationally-prohibitive. Alternatively, the J-RS-EE-PA has been split into two sub-problems, where several relay selec-
tion strategies have been explored, and computationally-efficient algorithmic solutions have been proposed to optimally solve
the different EE power allocation strategies. Specifically, the different algorithmic designs have been shown to efficiently obtain
the optimal energy-efficient power allocation solution for the different EE strategies. Additionally, a low-complexity solution
procedure has been proposed to solve the J-RS-EE-PA problem. More importantly, the proposed solution procedure has been
shown to be superior to the different explored relay selection strategies as well as yielding the global optimal joint relay selec-
tion and energy-efficient power allocation solution of the J-RS-EE-PA problem for the GEE, MEE, PEE, and SEE strategies,
while satisfying QoS requirements.
It is note-worthy that this research work can be extended in several directions. For instance, the cases of imperfect channel state
information and imperfect successive interference cancellation and their effect on joint relay selection and energy-efficient power
allocation are worth-pursuing39. In addition, this work has considered single relay selection; however, it would be interesting
to consider the case of joint multi-relay selection and energy-efficient power allocation. Furthermore, the scenario where the
base-station and/or relays have multiple transmit/receive antennas is definitely challenging but equally important to pursue, as
this would entail optimal antenna selection or beamforming40. Lastly, it would be interesting to extend this work for multi-cell
scenarios, while taking into account inter-cell interference.
APPENDIX I: PROOF OF LEMMA 1
Proof. For Γ𝑅𝑘,𝑈𝑛in (7), note that it can be re-written and re-arranged as
Γ𝑅𝑘,𝑈𝑛=𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆,𝑅𝑘𝑁
𝑖=1,𝑖≠𝑛𝑎𝑖+𝛾𝐵𝑆,𝑅𝑘𝑎𝑛+𝜁𝑅𝑘,𝑈𝑛
.(I.1)
M. W. Baidas ET AL 21
For notational convenience, define 𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛≜𝜉𝑅𝑘,𝑈𝑛𝑎𝑛+𝛾𝐵𝑆 ,𝑅𝑘𝑁
𝑖=1,𝑖≠𝑛𝑎𝑖, which is independent of 𝑎𝑛. Hence, the Γ𝑅𝑘,𝑈𝑛in
(I.1) becomes
Γ𝑅𝑘,𝑈𝑛=𝑎𝑛𝜉𝑅𝑘,𝑈𝑛
𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛+𝛾𝐵 𝑆,𝑅𝑘𝑎𝑛+𝜁𝑅𝑘,𝑈𝑛
.(I.2)
Now, it is straightforward to show that the first derivative is
𝜕Γ𝑅𝑘,𝑈𝑛
𝜕𝑎𝑛
=𝜉𝑅𝑘,𝑈𝑛𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛+𝜁𝑅𝑘,𝑈𝑛
𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛+𝛾𝐵 𝑆,𝑅𝑘𝑎𝑛+𝜁𝑅𝑘,𝑈𝑛2>0,(I.3)
which is strictly positive, and hence Γ𝑅𝑘,𝑈𝑛is strictly increasing in 𝑎𝑛,∀𝑎𝑛≥0. On the other hand, the second derivative is
obtained as
𝜕2Γ𝑅𝑘,𝑈𝑛
𝜕𝑎2
𝑛
= − 2𝛾𝐵 𝑆,𝑅𝑘𝜉𝑅𝑘,𝑈𝑛𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛+𝜁𝑅𝑘,𝑈𝑛
𝑎𝐵𝑆,𝑅𝑘,𝑈𝑛+𝛾𝐵 𝑆,𝑅𝑘𝑎𝑛+𝜁𝑅𝑘,𝑈𝑛3<0,(I.4)
which is strictly negative. Hence, Γ𝑅𝑘,𝑈𝑛is strictly concave in 𝑎𝑛,∀𝑎𝑛≥0, and ∀𝑛∈ {1,2,…, 𝑁 }.
APPENDIX II: PROOF OF LEMMA 4
Proof. For problem R-MSEE𝑘(𝐚), the Lagrangian function can be expressed as✠
𝐚,𝜆𝑛𝑁
𝑛=1 , 𝜏, 𝜔𝑛𝑁
𝑛=1 ,𝜈𝑛𝑁
𝑛=1 ,𝜌𝑛𝑁−1
𝑛=1 =
𝑁
𝑛=1
𝜇𝑛+
𝑁
𝑛=1
𝜆𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘−𝜏𝑁
𝑛=1
𝑎𝑛− 1+
𝑁
𝑛=1
𝜔𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝑟𝑇+
𝑁−1
𝑛=1
𝜌𝑛𝑎𝑛−𝑎𝑛+1+
𝑁
𝑛=1
𝜈𝑛𝑎𝑛.
(II.1)
Then, the KKT conditions are obtained as 𝜕𝐚,{𝜆𝑛}𝑁
𝑛=1,𝜏 ,{𝜔𝑛}𝑁
𝑛=1,{𝜈𝑛}𝑁
𝑛=1,{𝜌𝑛}𝑁−1
𝑛=1
𝜕𝑎𝑛
, yielding
𝜕
𝜕𝑎𝑛
𝑁
𝑛=1
𝜆𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘−𝜏+𝜕
𝜕𝑎𝑛
𝑁
𝑛=1
𝜔𝑛𝑈𝑛,𝑅𝑘(𝐚)+𝜕
𝜕𝑎𝑛
𝑁−1
𝑛=1
𝜌𝑛𝑎𝑛−𝑎𝑛+1+𝜈𝑛= 0,∀𝑛∈ {1,2,…, 𝑁 },
𝜆𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘= 0,∀𝑛∈ {1,2,…, 𝑁 },
𝜏1 −
𝑁
𝑛=1
𝑎𝑛= 0,
𝜔𝑛𝑈𝑛,𝑅𝑘(𝐚)−𝑟𝑇= 0,∀𝑛∈ {1,2,…, 𝑁 },
𝜌𝑛𝑎𝑛−𝑎𝑛+1= 0,∀𝑛∈ {1,2,…, 𝑁 − 1},
𝜈𝑛𝑎𝑛= 0,∀𝑛∈ {1,2,…, 𝑁 },
𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘≥0,∀𝑛∈ {1,2,…, 𝑁 },
1≥
𝑁
𝑛=1
𝑎𝑛,
𝑈𝑛,𝑅𝑘(𝐚)≥𝑟𝑇,∀𝑛∈ {1,2,…, 𝑁 },
𝑎𝑛−𝑎𝑛+1 ≥0,∀𝑛∈ {1,2,…, 𝑁 − 1},
𝑎𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 },
𝜆𝑛≥0
𝜏≥0,
𝜔𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 },
𝜌𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 − 1},
𝜈𝑛≥0,∀𝑛∈ {1,2,…, 𝑁 },
(II.2)
where 𝜆𝑛𝑁
𝑛=1,𝜏,𝜔𝑛𝑁
𝑛=1,𝜈𝑛𝑁
𝑛=1 and 𝜌𝑛𝑁−1
𝑛=1 are the corresponding Lagrange multipliers. By complimentary slackness,
𝜈𝑛= 0 implies 𝑎𝑛>0,∀𝑛∈ {1,2,…, 𝑁}. Similarly, 𝜌𝑛= 0, implies 𝑎𝑛> 𝑎𝑛+1,∀𝑛∈ {1,2,…, 𝑁 − 1}, while 𝜔𝑛= 0
✠Note that 𝜌𝑁= 0, since 𝑎𝑁is less than any other power allocation coefficient.
22 M. W. Baidas ET AL
implies that 𝑈𝑛,𝑅𝑘(𝐚)> 𝑟𝑇,∀𝑛∈ {1,2,…, 𝑁 }. More importantly, it can be verified that 𝜆𝑛>0implies that 𝑈𝑛,𝑅𝑘(𝐚)=
𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘,∀𝑛∈ {1,2,…, 𝑁 }33. Additionally, 𝜕𝐚,{𝜆𝑛}𝑁
𝑛=1,𝜏 ,{𝜔𝑛}𝑁
𝑛=1,{𝜈𝑛}𝑁
𝑛=1,{𝜌𝑛}𝑁−1
𝑛=1
𝜕𝜇𝑛
gives
1 − 𝜆𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘= 0 ⇒𝜆𝑛=1
𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘
,∀𝑛∈ {1,2,…, 𝑁 },(II.3)
whereas 𝜕𝐚,{𝜆𝑛}𝑁
𝑛=1,𝜏 ,{𝜔𝑛}𝑁
𝑛=1,{𝜈𝑛}𝑁
𝑛=1,{𝜌𝑛}𝑁−1
𝑛=1
𝜕𝜆𝑛
yields
𝑈𝑛,𝑅𝑘(𝐚)−𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘= 0 ⇒𝑈𝑛,𝑅𝑘(𝐚)=𝜇𝑛𝑎𝑛𝐸𝐵𝑆 +𝐸𝑅𝑘,∀𝑛∈ {1,2,…, 𝑁 }.(II.4)
Clearly, (II.3) and (II.4) are equivalent to conditions (51) and (52), respectively, since 𝜆𝑛>0,∀𝑛∈ {1,2,…, 𝑁 }. By setting 𝜇𝑛=
𝜇𝑛and 𝜆𝑛=
𝜆𝑛,∀𝑛∈ {1,2,…, 𝑁 }, then the KKT conditions in (II.2) can be verified to be of those of problem P-R-MSEE𝑘(𝐚),
which is a concave optimization problem (as per Remark 14). Hence, for 𝜆𝑛>0and 𝜇𝑛≥0(∀𝑛∈ {1,2,…, 𝑁}), the KKT
conditions are also sufficient optimality conditions, and thus
𝐚is the optimal solution to problem P-R-MSEE𝑘(𝐚)when 𝜇
𝜇
𝜇=
𝜇
𝜇
𝜇
and 𝜆
𝜆
𝜆=
𝜆
𝜆
𝜆.
Financial disclosure
This work is partially supported by the Kuwait Foundation for the Advancement of Sciences (KFAS), under project code
PN17-15EE-02.
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How to cite this article: M. W. Baidas, Emad Alsusa, and K. A. Hamdi (2019), Joint Relay Selection and Energy-Efficient Power
Allocation Strategies in Energy-Harvesting Cooperative NOMA Networks, Transactions on Emerging Telecommunications
Technologies,2019;00:1–6.
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