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Energy-Efficient Resource Allocation for D2DCommunications in Cellular Networks

Tuong Duc Hoang,Student Member, IEEE,Long Bao Le,Senior Member, IEEE,and Tho Le-Ngoc,Fellow, IEEE

Abstract—In this paper, we study the energy-efficient resourceallocation problem for device-to-device (D2D) communicationunderlaying cellular networks which aims to maximize theminimum weighted energy-efficiency (EE) of D2D links whileguaranteeing the minimum data rates for cellular links. We firstcharacterize the optimal power allocation of the cellular links totransform the original resource allocation problem into the jointsubchannel and power allocation problem for D2D links. Wethen propose three resource allocation algorithms with differentcomplexity, namely Dual-Based, Branch-and-Bound (BnB), andRelaxation-Based Rounding (RBR) algorithms. While the Dual-Based algorithm solves the problem by using dual decompositionmethod, the BnB and RBR algorithms tackle the problem byemploying the relaxation approach. We establish the strongperformance guarantees for the proposed algorithms throughtheoretical analysis. Extensive numerical studies demonstratethat the proposed algorithms achieve superior performance andsignificantly outperform a conventional algorithm.

Index Terms—D2D communication, cellular networks, energy-efficiency, resource allocation, subchannel and power allocation

I. I NTRODUCTION

Deployment of device-to-device communications in thewireless cellular network has been expected to significantly in-crease the network throughput and reduce the traffic load in thecore network [1]–[7]. Efficient resource allocation algorithmsplay a critical role in acquiring these benefits while limitingnegative impacts on the performance of existing communica-tions between users and base-stations (BSs). In general, wecanperform orthogonal or non-orthogonal spectrum sharing forD2D and cellular communication links. Orthogonal spectrumsharing assumes cellular and D2D links using distinct partsof the spectrum, and consequently the system must reservededicated spectral resources for D2D links. On the other hand,the non-orthogonal spectrum sharing allows D2D links toreuse the resource of cellular links in order to improve thespectrum utilization and efficiency at the costs of co-channelinterference between cellular and D2D links. Since, in acellular system, the BS has more powerful processing capacitythan mobile terminals to deal with interference experienced atthe receiver’s side, it is more beneficial if the D2D links reusethe uplink resources of cellular links.

Copyright c©2015 IEEE. Personal use of this material is permitted. How-ever, permission to use this material for any other purposes must be obtainedfrom the IEEE by sending a request to [email protected].

Manuscript received January 21, 2015; revised August 7, 2015; acceptedSeptember 19, 2015.

T. D. Hoang and L. B. Le are with INRS-EMT, Universite du Quebec,Montreal, Quebec, Canada. Emails:{tuong.hoang,long.le}@emt.inrs.ca.

T. Le-Ngoc is with Department of Electrical and Computer Engineering,McGill University, Montreal, Quebec, Canada. Emails: [email protected]

Green communication has attracted a lot of attention inrecent years where maximization of EE has become an impor-tant design objective in engineering modern wireless systems[8]–[10]. In fact, development of energy-efficient resourceallocation algorithms has been considered for 3GPP LTEsystems [11], [12]. In general, downlink EE would be lesscritical than the uplink EE since BSs can have access to variousenergy sources while user devices are supported by energy-limited batteries [13], [14].

There are some existing works considering spectrum-efficient resource allocation for the D2D underlaying cellularnetwork [15]–[19] with various objectives and system con-straints. In [15], the authors develop a simple power controlalgorithm based on the signal to interference plus noise ratio(SINR) of the cellular link to guarantee its required perfor-mance and to maximize the sum-rate of D2D links. The au-thors of [16] propose a mode selection algorithm to maximizethe sum-rate where they develop power control algorithms toattain the optimal solution for each mode. Power allocationforcellular and D2D links to maximize the rate of a single D2Dlink while guaranteeing the required rate of each cellular linkis studied in [17]. In [18], the sum-rate optimization for D2Dand cellular links is considered where the system with multipleD2D and multiple cellular links is studied. Finally, the work[19] develops a fair resource allocation for D2D links whileassuring the required quality of service (QoS) of cellular links.In fact, to maximize the throughput or spectrum-efficiency,mobile devices would utilize large transmission powers, whichmay result in serious degradation of their EE. This motivatesus to consider energy-efficient resource allocation for D2Dcommunications in this current work.

There have been some initial efforts in developing energy-efficient resource allocation solutions for D2D underlaying cel-lular networks [20]–[24]. In [20], a resource allocation solutionbased on non-cooperative game theory is proposed where eachD2D link selfishly performs power and subchannel allocationto maximize its own EE considering the fixed resource al-location of other links. This design approach, however, maynot lead to efficient utilization of the spectral resources.Theauthors in [21] solve the energy-efficient resource allocationproblem by using the combinatorial auction game where thecellular BS acts as an auctioneer which sequentially decidesthe price of each resource and sells it to the set of D2D linksachieving the highest utility. In this design, each D2D linkis allowed to reuse the resource of only one cellular link,which may limit the achievable rates of D2D links. Moreover,a coalition game is employed to tackle the energy-efficientresource allocation problem in [22] where the authors address


the joint mode selection and resource allocation for D2Dand cellular links. Nevertheless, this work assumes that eachD2D link only achieves its minimum required rate, whichmight not fully exploit the advantage of short-range D2Dcommunications.

Other energy-efficient designs that aim to minimize thetotal power consumption are pursued in [25], [26]. However,resource allocation solutions in these papers may not fullyexploit the advantages of D2D communications to achieve theoptimum EE. In this paper, we study the joint subchannel andpower allocation that maximizes the minimum weighted EE ofD2D links and guarantees the minimum data rates of cellularlinks. Specifically, we make the following contributions.

• We formulate a general energy-efficient resource alloca-tion problem considering multiple cellular and D2D linkswhere each D2D link can reuse the spectral resourcesof multiple cellular links. This model is, therefore, moregeneral than most models studied in the literature [15]–[18]. We first characterize the optimal power allocationsolution for a cellular link as a function of the optimalpower of the co-channel D2D link. Based on this impor-tant result, we transform the original resource allocationproblem into the resource allocation problem for onlyD2D links.

• We propose the dual-based algorithm that solves theresource allocation problem in the dual domain. Partic-ularly, we adopt the max-min fractional programmingtechnique to iteratively transform the resource alloca-tion problem into a Mixed Integer Nonlinear Program-ming (MINLP) problem. Then, we solve the underly-ing MINLP problem by using the dual decompositionapproach. Theoretical analysis demonstrates that the al-gorithm converges to a feasible solution of the originalproblem. Moreover, the achieved solution is optimal if,at convergence, the duality gap of the underlying MINLPproblem is zero. In addition, a distributed implemen-tation with limited message exchange for the proposedalgorithm is described, which can potentially reduce thecomputational burden of the BS and the system signalingoverhead.

• We study the relaxation-based solution approach, whichtackles the resource allocation problem by relaxing thesubchannel allocation variables. In particular, we applythe branch-and-bound (BnB) approach to branch thesubchannel allocation vector space to smaller sub-spacesin which some subchannel allocation variables are de-termined and others are undetermined. An upper-boundis calculated by solving a max-min fractional programof the relaxed problem where all undetermined subchan-nel allocation variables are relaxed. In particular, wesequentially transform the relaxed problem into a convexproblem and solve it by using the interior-point methoduntil convergence. Moreover, we obtain a lower-bound ofthe objective value by rounding the fractional subchannelallocation solution acquired in the upper-bound calcula-tion. Motivated by the procedure to calculate the upper-bound in the BnB algorithm, we also propose a low-

complexity Relaxation-based Rounding (RBR) algorithm.In this RBR algorithm, we first solve the relaxed problemof the original resource allocation problem for D2D links.Then, based on the obtained solution for the relaxed prob-lem, we develop an efficient rounding procedure, whichaims at minimizing the performance loss and maximizethe design objective, to attain a feasible solution for theconsidered resource allocation problem.

• The computational complexity of the proposed algorithmsis analyzed. Moreover, extensive numerical results arepresented to evaluate the performance of the developedalgorithms. Specifically, it is shown that the objectivevalues achieved by the dual-based and RBR algorithmsare very close to that of the optimal BnB algorithm.In addition, the proposed algorithms significantly out-perform the conventional algorithm and the spectrum-efficient resource allocation design.

The remainder of the paper is organized as follows. SectionII presents the system model and problem formulation. Theproblem transformation is described in Section III. Sections IVand V develop the dual-based and relaxation-based algorithms,respectively. The computational complexity is analyzed inSection VI. Section VII presents illustrative results, followedby conclusions in Section VIII.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We consider uplink resource allocation scenario where cel-lular links share the same spectrum with multiple D2D links ina single macro-cell system. We assume thatK uplink cellularlinks in a setK = {1, · · · ,K} occupying K orthogonalsubchannels in the setN = {1, · · · ,K} in the consideredcell. Moreover, we assume that the setL = {1, · · · , L} ofD2D links transmits data using the same set of subchannels.1

In these notations,K = |K|, L = |L|, andN = |N | denotethe numbers of cellular links, D2D links, and subchannels,respectively, where|A| denotes the cardinality of set A.

Let hnkl denote the channel gain from the transmitter of

link l to the receiver of linkk on subchanneln. We assumethat the subchannel allocation for cellular links has been pre-determined and we are interested in allocating these subchan-nels to D2D links efficiently. Without loss of generality, weassume that cellular linkk has been allocated subchannelk.We introduce vectorρl = [ρ1l , · · · , ρKl ] to describe subchannelallocation decisions for D2D linkl whereρkl = 1 if subchannelk is allocated for D2D linkl and ρkl = 0, otherwise. Letρ = [ρ1, · · · ,ρL] denote the subchannel allocation variablesfor all D2D links.

We present the allocated power vectors asp = [pC ,pD]for all the links, wherepC = [p1C1, · · · , pKCK ] for K cellularlinks, pD = [pD1, · · · ,pDL], pDl= [p1Dl, · · · , pKDl], for D2Dlinks, andpkCk and pkDl denote the allocated transmit powerson subchannelk of cellular link k ∈ K and D2D link l ∈ L,

1The considered orthogonal subchannels can be sub-carriersor sub-channelsin the OFDMA system or simply channels in the FDMA system.


respectively. Then, the SINR achieved by cellular linkk onits allocated subchannelk can be expressed as

ΓkCk(p,ρ) =

pkCkhkkk

σkk +

∑

l∈Lρkl p

kDlh

kkl

, (1)

where∑

l∈L ρkl pkDlh

kkl represents the interference due to the

D2D link using subchannelk andσkk denotes the noise power

on subchannelk. Similarly, the SINR of D2D link l onsubchannelk can be written as

ΓkDl(p,ρ) =

ρkl pkDlh

kll

σkl + pkCkh

klk

. (2)

The data rates in b/s/Hz (i.e., normalized by the subchannelbandwidth) of cellular linkk ∈ K on its subchannelk, D2Dlink l ∈ L on subchannelk, and D2D link l on all thesubchannels can be calculated as

RkCk(p,ρ) = log2

(

1 + ΓkCk(p,ρ)

)

, (3)

RkDl(p,ρ) = log2

(

1 + ΓkDl(p,ρ)

)

, (4)

and

RDl(p,ρ) =∑

k∈K

ρkl RkDl(p,ρ), (5)

respectively. We assume that the totalconsumedpower of D2Dlink l can be expressed as [27], [28]

P total

Dl = 2P l0 + αl

∑

k∈K

ρkl pkDl, (6)

where 2P l0 represents the fixed circuit power of both trans-

mitter and receiver of D2D linkl, and αl > 1 is a factoraccounting for the transmit amplifier efficiency and feederlosses.

B. Problem Formulation

In this work, we consider the resource allocation designwith the following constraints. First, it is required to maintainthe minimum rate of each cellular linkk (on its allocatedsubchannelk), i.e.,

RkCk(p,ρ) ≥ Rmin

Ck , ∀k ∈ K. (7)

Second, the power constraints of individuallinks are givenas

pkCk ≤ Pmax

Ck , ∀k ∈ K, (8)∑

k∈K

ρkl pkDl ≤ Pmax

Dl , ∀l ∈ L, (9)

wherePmax

Ck andPmax

Dl are the maximum transmit powers ofcellular link k and D2D link l, respectively.

Third, the subchannel allocation variables are binary, i.e.,

ρkl ∈ {0, 1}, ∀k ∈ K, l ∈ L. (10)

Finally, similar to [16]–[18], [21], [22], [25], [29], werequire that each subchannel can be reused by at most oneD2D link to limit the interference from D2D links to cellular

links, and hence to guarantee the performance of the cellularlinks, i.e.,

∑

l∈L

ρkl ≤ 1, ∀k ∈ N . (11)

The objective of our resource allocation design is to maxi-mize the minimum weighted EE of the D2D links. Therefore,this design can be formulated as the following energy-efficientresource allocation problem to attain the max-min fairnessinweighed EE for D2D links:

maxp,ρ

minl∈L

wlRDl(p,ρ)P total

Dl

s.t. (7), (8), (9), (11), (10),

(12)

wherewlRDl(p,ρ)P total

Dl

represents the weighted EE of the D2D links.The weight parameterswl can be employed to control therelative priorities among different D2D links and

∑

l∈Lwl = L.

The resource allocation design in this paper allows eachD2D link to share spectral resources with multiple cellularlinks but the spectral resource of each cellular link can bereused by at most one D2D link. This model allows us to (i)achieve good balance between excellent performance for D2Dlinks and affordable interference management complexity,(ii)protect the QoS of cellular links efficiently, (iii) avoid largesignaling overhead due to the Channel State Information (CSI)estimation and feedback of interfering channels among D2Dlinks.

III. PROBLEM TRANSFORMATION

To solve problem (12), we first describe the optimal powerallocation of D2D link l ∈ L on subchannelk ∈ N in thefollowing proposition.

Proposition 1. If D2D link l ∈ L is allowed to reusesubchannelk ∈ N of cellular link k, then its power on

subchannelk, pkDl =1

hkkl

(

pkCkh

kkk

2Rmin

Ck−1− σk

k

)

∈ [0, Pmax

Dlk ], where

pkCk is the power of cellular linkk, andPmax

Dlk = min {Pmax

Dl ,

1hkkl

(

Pmax

Ck hkkk

2Rmin

Ck−1− σk

k

)}

.

Proof. The proof is given in Appendix A. �

From Proposition 1, the data rate of D2D linkl on subchan-nel k given in (4) can be re-written as

RkDl(pD,ρ)

= ρkl log2

1 +

pkDlhkll

σkl +

(2Rmin

Ck−1)hklk

hkkk

(σkk + pkDlh

kkl)

. (13)

For convenience, let us define

akl ,σkl

hkll

+(2R

min

k − 1)hklkσ

kk

hkkkh

kll

(14)

bkl ,(2R

min

k − 1)hklkh

kkl

hkkkh

kll

. (15)


Then, the data rate of D2D linkl on subchannelk ∈ N ,Rk

Dl(pD,ρ) in (13), and the total rate over all subchannels,RDl(pD,ρ) in (5), can be rewritten, respectively, as

RkDl(pD,ρ) = ρkl log2

(

1 +pkDl

akl + bklpkDl

)

, (16)

and

RDl(pD,ρ) =∑

k∈N

RkDl(pD,ρ), (17)

where the allocated transmit power must satisfy

pkDl ≤ Pmax

Dlk , ∀k ∈ N , ∀l ∈ L. (18)

Therefore, problem (12) is equivalent to the following

max(pD,ρ)

minl∈L

wlRDl(pDl,ρ)

P total

Dl (pDl,ρ)

s.t. (9), (10), (11), (18).

(19)

In order to solve problem (19), we consider the followingoptimization problem

maxpD,ρ

η(ζ,pD,ρ) , minl∈L

[

wlRDl(pD,ρ)− ζP total

Dl (pD,ρ)]

s.t. (9), (10), (11), (18).(20)

Suppose thatη∗(ζ) = η(ζ,p∗D,ρ∗) where (p∗

D,ρ∗) is theoptimal solution of problem (20), andD denotes the set offeasible solutions of problem (19). Then, we can characterizethe optimal solution of problem (20) in the following theorem,which is adopted from [30].

Theorem 1. η∗(ζ) is a decreasing function ofζ. In addition,if we have

max(pD,ρ)∈D

minl∈L

[wlRDl(pD,ρ)− ζ∗P total

Dl (pD,ρ)]

= minl∈L

[

wlRDl(p∗D,ρ∗)− ζ∗P total

Dl (p∗,ρ∗)]

= 0(21)

thenζ∗ = minl∈L

wlRDl(p∗D,ρ∗)

P total

Dl(p∗

D,ρ∗)

is the optimal solution of (19).

It is worth noting that the main theorem in [30] states thenecessary and sufficient condition forζ∗ to be the optimalsolution of the fractional programming problem. In Theorem1, we only present the sufficient condition forζ∗ to be theoptimal solution of problem (19). However, the theorem of[30] requires the set of feasible solutions of the fractionalprogramming problem to be continuous, which is not requiredin our theorem. Importantly, Theorem 1 allows us to transforma general max-min fractional problem (19) to a non-fractionaloptimization problem with the parameterζ. In addition, the op-timal solution of problem (19),ζ∗, can be found ifη∗(ζ∗) = 0.Sinceη∗(ζ) is a decreasing function ofζ, it can be seen thatζ∗

can be indeed determined by the gradient or bisection method.A general algorithm solving problem (19) based on the

solution of problem (20) is described in Algorithm 1, whichiteratively solves problem (20) for givenζ and updatesζ untilconvergence. Therefore, the remaining challenge is how tosolve problem (20) for a givenζ. In general, problem (20)is NP-hard, which implies that solving this problem optimallyrequires exponential complexity.

Algorithm 1 General Algorithm

1: Initialization: Setǫ = 10−6, ζ = 0, ζt = ǫ

2: while (ζ − ζt) ≥ ǫ do3: Solve problem (20) for givenζ to obtain (p∗

D,ρ∗) =argmax(pD,ρ)

η(ζ,pD,ρ)

4: ζt = ζ, ζ = minl∈L

wlRDl(p∗D,ρ∗)

P total

Dl(p∗,ρ∗)

5: end while6: Output (p∗

D,ρ∗), andζ

IV. D UAL -BASED ALGORITHM

In this section, we propose a dual-based algorithm tosolve problem (19). Then, we will present the distributedimplementation for this algorithm.

A. Algorithm Development

Algorithm 2 Dual-Based Algorithm

1: Initialization: ζmax, ζmin

2: repeat3: Initialization: Chooseζ = 1

2 (ζmin+ ζmax), λ(0), µ(0)

l =1L

, step sizeθ(0), andκ(0)

4: repeat5: Step 1: For allk ∈ K, l ∈ L, calculatepk

∗

Dl accordingto (29)

6: Step 2: For allk ∈ K, perform subchannel allocationfollowing (37)

7: Step 3: Update dual variablesλ, andµ as in (38)and (43)

8: until Convergence9: Outputz∗ = min

l∈L

[

wlRDl(p∗D,ρ∗)− ζP total

Dl (p∗D,ρ∗)

]

10: If z∗ > 0, ζmin = ζ; otherwiseζmax = ζ

11: until Convergence ofζ12: Outputp∗

Dl, ρ∗, andζ∗ = min

l∈L

wlRDl(p∗Dl,ρ

∗)

P total

Dl (p∗Dl

,ρ∗)

The dual-based resource allocation algorithm is summa-rized in Algorithm 2. The algorithm comprises two iterativeloops. In the outer loop, we adopt the max-min fractionalprogramming technique described in Algorithm 1 to attainthe optimal value ofζ for problem (19). In the inner loop,we solve problem (20) for a givenζ by employing the dualdecomposition method.

Algorithm 2 performs two main tasks, which are executedsequentially. Specifically, we solve problem (20) for a givenζ (lines 4-8) in the first task while we update the value ofζ based on the results of the first task by using the bisectionmethod in the second task (line 10). In the following, we showhow to solve problem (20) for a given value ofζ. First, it canbe observed that problem (20) is equivalent to the followingproblem

maxz,pD,ρ

z

s.t. wlRDl(pDl,ρ)− ζP total

Dl (pD,ρ) ≥ z, ∀l ∈ L(9), (10), (11), (18).

(22)


To tackle problem (22), we consider its Lagrangian as follows:

LD(pD,ρ, z, ζ,λ,µ)

= z +∑

l∈L

µl

[


Dl (pD,ρ)− z]

+∑

l∈L

λl(Pmax

Dl −∑

k∈N

ρkl pkDl)

= z(1−∑

l∈L

µl) +∑

l∈L

µl

[


Dl (pD,ρ)]

+∑

l∈L

λl(Pmax

Dl −∑

k∈N

ρkl pkDl), (23)

whereλ = [λ1, · · · , λL]T andµ = [µ1, · · · , µL]

T representthe Lagrange multipliers. Then, the dual function can bewritten as

LD(ζ,λ,µ) , maxz,pD∈X ,ρ∈C

LD(pD,ρ, z, ζ,λ,µ), (24)

whereX = {pD|pkDl ≤ Pmax

Dlk , ∀k ∈ N , ∀l ∈ L}, and C ={ρ|∑l∈L ρkl ≤ 1, ∀k ∈ N ,andρkl ∈ {0, 1}, ∀k ∈ K, l ∈ L}.Then, the dual problem can be stated as

LD(ζ) , minλ,µ≥0

LD(ζ,λ,µ). (25)

In order to solve the dual problem (25), we investigate problem(24) for the givenλ andµ. In particular, we have

LD(ζ,λ,µ) = maxz,pD∈X ,ρ∈C

LD(pD,ρ, z, ζ,λ,µ)

= maxz,pD∈X ,ρ∈C∑

k∈N

∑

l∈L

ρkl

[

µlwlRkDl(pD,ρ)− (ζαlµl + λl)p

kDl

]

+ z(1−∑

l∈L

µl) +∑

l∈L

(

λlPmax

Dl − 2ζµlPl0

)

. (26)

Note that z is an uncontrolled variable in problem (26).Thus, to obtain the nontrivial optimal solution of the dualproblem (25),

∑

l∈L µl = 1 must hold. Moreover, problem(26) can be decomposed intoN individual resource allocationproblems forN subchannels where the resource allocationproblem for subchannelk ∈ N can be stated as

LkD(ζ,λ,µ)

= maxpD∈X ,ρ∈C

∑

l∈L

ρkl

[

µlwlRkDl(pD,ρ)− (ζαlµl + λl)p

kDl

]

.

(27)

Then,

LD(ζ,λ,µ)

=∑

k∈N

LkD(ζ,λ,µ) +

∑

l∈L

(

λlPmax

Dl − 2ζµlPl0

)

. (28)

We now define

fkl (p

kDl) , µlwlR

kDl(pD,ρ)− (ζαlµl + λl)p

kDl. (29)

For problem (27), suppose that D2D linkl is allocatedsubchannelk ∈ N then we have

pk∗

Dl = argmaxpkDl

∈Xl

fkl (p

kDl). (30)

Note that we must haveµl > 0. This is because ifµl = 0, wehavepk∗Dl = 0, ∀k ∈ N , which cannot be the optimal solutionof problem (26). In addition, problem (30) can be addressedby solving ∂fk

l (pkDl)

∂pkDl

= 0, where ∂fkl (pk

Dl)

∂pkDl

is the first order

derivative offkl (p

kDl), which can be written as

∂fkl

∂pkDl

=µlwlakllog2e

(akl+bklpkDl)[akl+(bkl+1)pkDl]

−(ζαlµl+λl). (31)

Then, it can be verified that solving∂fkl (pk

Dl)

∂pkDl

= 0 is equivalent

to solvingAkl(pkDl)

2 + 2BklpkDl + Ckl = 0 where

Adkl ,

(

ζαl +λl

µl

)

bkl(bkl + 1) (32)

Bdkl ,

(

ζαl +λl

µl

)

(aklbkl + 0.5akl) (33)

Cdkl ,

(

ζαl +λl

µl

)

a2kl − wlakllog2e (34)

∆dkl , (Bd

kl)2 −Ad

klCdkl. (35)

Consequently, the optimal solution of D2D linkl that maxi-mizesfk

l (pkDl) is given by

pk∗

Dl =

−Bdkl +

√

∆dkl

Adkl

Pmax

Dl

0

, (36)

where [x]ba = b if x > b, [x]ba = a if x < a, otherwise[x]ba = x.

In summary, by solving problem (26) we can obtain theoptimal power allocation for any D2D link on subchannelk ∈ N . Recall that we have assumed that each subchannelcan be allocated to at most one D2D link; therefore, for allsubchannelsk ∈ N , we have

ρk∗

l =

1 if l = argmaxl∈L

fkl (p

k∗

Dl)

0 otherwise.(37)

So far we have presented the resource allocation solutionfor given λ, µ. Therefore, the remaining task is to solveproblem (25), which can be completed by the sub-gradientmethod as described in the following. In the initial iterations = 0, we solve problem (24) with the initial value ofλ(0)

andµ(0). Then, in iterations+1, we update the dual variablesλ(s+1) andµ(s+1) based on the solution in iterations, thenwe solve problem (24) with the updated value ofλ and µ.The procedure to updateλ andµ by using the sub-gradientmethod can be expressed as follows

λ(s+1)l =

[

λ(s)l + θ

(s)l

(

∑

k∈N

ρk∗

l pk∗(s)Dl − Pmax

Dl

)]+

, ∀l ∈ L

(38)

µ′(s+1)l =

[

µ(s)l − κ

(s)l

(

z(s)l − z

(s)min

)]+

, ∀l ∈ L, (39)


where[x]+ = max{x, 0}, and

z(s)l = wlRDl(p

(s)D ,ρ(s))− ζ(2P l

0 + αl

∑

k∈N

ρk∗

l pk∗(s)Dl ) (40)

z(s)min

= minl∈L

[

wlRDl(p(s)D ,ρ(s))− ζ(2P l

0 + αl

∑

k∈N

ρk∗

l pk∗(s)Dl )

]

(41)

andθ(s)l , κ(s)l are step sizes, which can be chosen appropriately

to ensure the convergence of the underlying iterative updates.Note thatp(s)

D , ρ(s) are, respectively, the transmit power andsubchannel allocation solutions for givenλ(s), µ(s). Recallthat µ must satisfy the constraint

∑

l∈L

µl = 1. Therefore, we

normalizeµ(s+1) as follows:

µ(s+1)l =

µ′(s+1)l

∑

l∈L

µ′(s+1)l

, ∀l ∈ L. (42)

It is shown in [35] that the dual decomposition procedureconverges to the optimal solution of problem (25) for appro-priately chosenθ(s)l andκ

(s)l . Therefore, the iterative loop in

the first task of Algorithm 2 always converges to the dualsolution of problem (20) for any value ofζ. On the otherhand, the performance achieved by Algorithm 2, which solvesproblem (19), is stated in the following proposition

Proposition 2. Algorithm 2 returns a feasible solution ofproblem (19) withζ∗, p∗

D, ρ∗ λ∗, and µ∗ at the end ofits first phase. Moreover, if

∑

k∈Nρk

∗

l pk∗

Dl ≤ Pmax

Dl , λ∗l (P

max

D −∑

k∈Nρk

∗

l pk∗

Dl) = 0, and RDl(p∗D,ρ∗) − ζ∗P total

Dl (p∗D,ρ∗) =

0, ∀l ∈ L, this feasible solution is the optimal solution ofproblem (19).

Proof. The proof is provided in Appendix C. �

B. Distributed Implementation with Limited Message Passing

The distributed implementation with limited message ex-change to execute Algorithm 2 is now described. In this im-plementation, instead of performing all the necessary tasks, theBS assigns some to the D2D links to reduce the computationalburden on the BS. Due to the QoS requirements of cellularlinks and the strong interference coupling among wirelesslinks, certain coordination among the BS and mobile devicesvia message passing deems necessary to achieve efficientspectrum sharing for D2D and cellular links.2

We can modify the procedure to update the dual-variableµ, which is introduced to adjustz(s)l in each iteration, with∑

l∈Lµl = 1 as

µ(s+1)l = µ

(s)l + κ

(s)l , (43)

2Distributed resource allocation algorithms can also be developed by usingadvanced game-theory and learning techniques [32], [33]. Wewill explorethese solution approaches in our future works.

whereκ(s)l > 0 if z

(s)l = z

(s)min

and κ(s)l < 0, otherwise, and

κ(s)l is chosen to satisfy

∑

l∈Lκ(s)l = 0. A typical update ofµ

with a fixed step-size can be implemented as

κ(s)l =

{

κ, if z(s)l = z

(s)min

−κL−1 , otherwise,

(44)

whereκ is a small value to guarantee the convergence of theupdates [35]. The distributed procedure for Algorithm 2 canbe described as follows.

Initialization:Each D2D linkl initializes the following system parameters:ζmax, ζmin, ζ = 1

2 (ζmin + ζmax), λ(0)

l = 0, µ(0)l = 1

L.

Step 1 (D2D):For given ζ, λl, µl, each D2D link l calculates

pk∗

Dl =

[

−Bdkl+√

∆dkl

Adkl

]Pmax

Dl

0

, ∀k ∈ K and broadcasts

the value offkl (p

k∗

Dl), which is defined in (29).

Step 2 (BS):The BS after collecting all valuesfk

l (pk∗

Dl), ∀k ∈ K, ∀l ∈ Lbroadcastsfmax

k = maxl∈L

fkl (p

k∗

Dl). to all users.

Step 3 (D2D):Each D2D link l performs subchannel allocation by usingthe following rule

ρk∗

l =

{

1 if fkl (p

k∗

Dl) ≥ fmax

k

0 otherwise.(45)

Moreover, it calculateszl = wlRDl(pDl,ρ∗l ) − ζ(2P l

0 +αl

∑

k∈N

ρk∗

l pk∗

Dl), then broadcastszl. By using the received

information, each D2D linkl updatesλl andµl accordingto (38) and (43), and return to Step 1 until convergence.

Step 4 (BS):The BS collects information ofzl, calculateszmin = min

l∈Lzl,

then broadcastszmin. If zmin > ζ, it updatesζ ← zmin andbroadcastsζ. This procedure continues (go back to Step 1)and only terminates if there is no further increase inζ. TheBS then calculates the power allocation for each cellularlink as

pk∗

Ck =σkk + Ik

hkkk

(2Rmin

Ck − 1), (46)

where Ik is the estimated interference caused by theco-channel D2D link on channelk. Then it broadcastspk

∗

Ck

to other cellular links.

The main tasks performed by different network entitiescan be explained as follows. The BS collects the informationregardingfk

l (pk∗

Dl), ∀k ∈ K, ∀l ∈ L then it broadcasts themaximum valuefmax

k = maxl∈L

fkl (p

k∗

Dl) for each subchannel

(step 2). The BS is also responsible for calculating andbroadcasting the updated value ofzmin = min

l∈Lzl and ζ (step


4). Moreover, as the algorithm converges, the BS calculatesthe transmit powers of all cellular links and broadcasts theresults to the cellular links. Each D2D linkl is responsible forcalculating the possible power allocation in each subchanneland performing subchannel allocation based on the obtainedinformation. Moreover, it broadcast the values offk

l (pk∗

Dl) andzl (step 1 and step 3).

V. RELAXATION -BASED ALGORITHMS

The dual-based Algorithm 2 has polynomial time complex-ity; however, it may not achieve the optimal solution. In thissection, we propose the optimal BnB algorithm and the low-complexity Relaxation-Based Rounding (RBR) algorithm.

A. Optimal Branch-and-Bound Algorithm

In this section, we apply the Branch-and-Bound (BnB)approach [38] to develop an algorithm that attains the op-timal solution of the original problem. Although the BnBalgorithm may not achieve the polynomial time complexity,it can significantly reduce the complexity compared to theexhaustive search algorithm. Since any feasible subchannelallocation variable is binary, we propose the BnB algorithmbybranching the feasible set of the subchannel allocation vectorswhere each branching iteration is executed by setting anundetermined subchannel allocation variable to a binary value0 or 1. Specifically, the algorithm determines the optimal pathin the search tree that corresponds to the optimal subbchannelallocations for all D2D links. In addition, this optimal path isdecided by iteratively visiting potential nodes in the searchtree where each nodem is associated with some alreadydetermined subchannel allocation variables (corresponding topart of the underlying path connecting nodem with the rootnode) and other undetermined subchannel allocation variables.

Let Qm be the set of all feasible subchannel allocationvectorsρ related to nodem where each vectorρ ∈ Qm con-tains corresponding determined and undetermined subchannelallocation variables associated with nodem. For convenience,we usem to indicate the iteration index of the searchingprocedure, and hencem = 1 indicates the root node (i.e., westart our search from the root node). Note that each elementof Q1 contains all the undetermined subchannel allocationvariables.

In each iteration with the corresponding parent node, weconsider one of its two child nodes by choosing one unde-termined elementρkl of subchannel assignment vectorρ andset it to a binary value 0 or 1 (called nodem). In nodem,the local upper-bound,BUm, and lower-bound,BLm, must becalculated. We also maintain the global upper-bound,BU

∗, andlower-bound,BL∗, which are, respectively, the highest localupper-bound and local lower-bound of all active nodes in thesearching procedure. In a particular nodem, if the calculatedlocal upper-bound satisfiesBUm < BL

∗ then we can removethis node from future consideration because it cannot lead tothe optimal solution. On the other hand, if the calculated locallower-bound satisfiesBLm > BL

∗, we can updateBL∗ = BLm.Furthermore, if the global lower-boundBL∗ is sufficientlyclose to the global upper-boundBU∗, we can terminate the

algorithm, and outputBL∗. In the following, we present theprocedures to find the local upper-bound and lower-bound ineach nodem of the algorithm.

1) Upper-bound Calculation:To obtain the upper-boundof nodem, we take the following procedure. First, we definethe setQm corresponding to setQm but any undeterminedsubchannel allocation variableρkl for each element ofQm

is relaxed asρkl ∈ [0, 1]. Then, we consider the followingproblem

maxpD,ρ∈Qm

minl∈L

wlRDl(pD,ρ)

P total

Dl

s.t. (9), (11), (18),

(47)

whose optimal objective value provides the local upper-boundBUm of the resource allocation solution in nodem. Thedifference-form of problem (47) can be expressed as

maxz,pD,ρ∈Qm

z

s.t. wlRDl(pD,ρ)− ζP total

Dl (pD,ρ) ≥ z, ∀l ∈ L,and (9), (11), (18).

(48)

We now introduce a new vectorsD that corresponds to thepower vector of D2D linkspD and consider the followingoptimization problem

maxz,sD,ρ∈Qm

z (49a)

s.t. wlRDl(sDl,ρ)− ζP total

Dl (sDl,ρ) ≥ z, ∀l ∈ L (49b)∑

k∈K

skDl ≤ Pmax

D , ∀l ∈ L (49c)

∑

l∈L

ρkl ≤ 1, ∀k ∈ N (49d)

skDl ≤ ρkl Pmax

Dlk , ∀k ∈ N , ∀l ∈ L (49e)

where

RDl(sDl,ρ) =∑

k∈N

ρkl log2

(

1 +skDl

aklρkl + bkls

kDl

)

(50)

P total

Dl = 2P l0 + αl

∑

k∈K

skDl. (51)

We state one important result in the following proposition.

Proposition 3. Problem (48) and (49) are equivalent, andproblem (49) is convex.

Proof. The proof can be found in Appendix D. �

Proposition 3 implies that the optimum solution of (48) canbe obtained by solving the convex problem (49) using thestandard interior point method [34]. We propose Algorithm 3to solve problem (47), in which we iteratively solve problem(49) for a given ζ (line 3) and updateζ (line 4) untilconvergence. If(ζ∗, s∗D,ρ∗) is the solution obtained fromAlgorithm 3 then the components ofp∗

D are given in (73)detailed in Appendix D. We now state another important resultin the following proposition.

Proposition 4. The obtained solution(p∗D,ρ∗) in Algorithm 3

is the optimal solution of (47).

Proof. See proof in Appendix E. �


Proposition 4 implies thatζ obtained from Algorithm 3 isan upper-bound of the resource allocation problem associatedwith nodem.

Algorithm 3 Upper-bound Calculation

1: Initialization: Setǫ = 10−6, ζt = 0, t = 0, ζ = ǫ

2: while |ζt − ζ| ≥ ǫ do3: Solve problem (49) asζ = ζt by interior point method

to get (z(t), s(t)D ,ρ(t))

4: ζ = ζt, ζt = minl∈L

wlRDl(s(t)Dl

,ρ(t))

P total

Dl(s

(t)Dl

,ρ(t)).

5: t← t+ 16: end while7: Output (ζ, s∗D,ρ∗) = (ζ, s

(t)D ,ρ(t))

8: Perform power allocation for all D2D links

pk∗

Dl =

0, if ρk∗

l = 0

sk∗

Dl

ρk∗

l

, otherwise(52)

9: Output (ζ,p∗D,ρ∗)

2) Lower-bound Calculation:Note that the local lower-bound in a particular nodem can be the objective valueachieved by a feasible solution. In nodem, while determiningthe local upper-bound, we obtain(s∗D,ρ∗) and (p∗

D,ρ∗) inlines 7 and 9 of Algorithm 3, respectively. Sinceρ∗ cancontain fractional components, it might not be a feasiblesolution of problem (47). The local lower-bound in nodem, BLm, can be obtained by rounding off the values of thefractional subchannel allocation variables. The new feasibleresource allocation vector(pD, ρ) can be obtained by thefollowing rules

ρkl =

{

1, if ρk∗

l = maxl∈L

ρk∗

l

0, otherwise,(53)

pkDl =

{

0, if ρkl = 0

sk∗

Dl, if ρkl = 1.(54)

Specifically, subchannelk is assigned to D2D linkl withhighest value ofρk

∗

l . Moreover, the power allocated to sub-channelk is equal tosk

∗

Dl to ensure the feasibility of theresulting solution. This feasible solution(pD, ρ) is then usedto calculate the local lower-bound.

B. Relaxation-Based Rounding Algorithm

The BnB algorithm may require, in some cases, to visita large number of nodes. In the following, we propose theRelaxation-Based-Rounding (RBR) algorithm (Algorithm 4),which requires to solve only one relaxed problem and exe-cute the rounding procedure only once. Specifically, we runAlgorithm 3 for the root node, which is employed by theBnB algorithm, to obtain the initial solution in line 1. Basedon the obtained result, we perform subchannel allocationsfor all subchannelk and D2D link l with ρk

∗

l = 1 thenwe execute the rounding procedure (lines 5-10), which isdesigned to minimize the performance loss as follows. LetSl = {k ∈ N|ρk∗

l = 1}, Sfl = {k ∈ N|ρk∗

l ∈ (0, 1)} be

the sets of exclusive and shared subchannels allocated to D2Dlink l, respectively, andUk = {l ∈ L|ρk∗

l > 0} be the setof D2D links with positive subchannel allocation variablesonsubchannelk. In line 2, we calculate the EE of D2D linkl ∈ Lcontributed by its exclusive subchannels in setSl as

ζl =

∑

k∈Sl

log2(

1 +sk

∗

Dl

akl+sk∗

Dl

)

2P0 + αl

∑

k∈Slsk

∗

Dl

. (55)

In lines 8-9, we allocate each shared subchannel to a uniqueD2D link. First, we calculate the possible EE improvement ofeach D2D linkl ∈ L over its shared subchannelk ∈ Cf as

∆kl (Sl) =

∑

n∈Sl∪{k} wllog2(

1 +sk

∗

l

akl+bklsk∗

l

)

2P0 + αl

∑

k∈Sl∪{k} sk∗

Dl

−∑

n∈Slwllog2

(

1 +sk

∗

l

akl+bklsk∗

l

)

2P0 + αl

∑

k∈Slsk

∗

Dl

, (56)

where Sl is the set of subchannels assigned to D2D linkl before we consider subchannelk and Cf is the set ofunallocated subchannels defined in Algorithm 4.

Now defineS = {S1, · · · ,SL} whereSl denotes the setof subchannels allocated to D2D linkl, and AS

k = {l ∈N|∆k

l (Sl) > 0} as the set of D2D links which can improve itsEE if these links are assigned subchannelsk ∈ Cf for a givenset S. In the proposed rounding procedure, we sequentiallyallocate one subchannelk ∈ Cf to the D2D link inAS

k that hasthe minimum EE (line 8). After each assignment, we updatethe set of assigned subchannels and EE for each D2D link (line9). The rounding procedure is terminated when all subchannelsare allocated.

Algorithm 4 Relaxation-Based Rounding Algorithm1: Run Algorithm 3 for the root node of BnB algorithm to

obtain (s∗D,ρ∗).2: Perform subchannel allocations for all subchannelk and

D2D link l with ρk∗

l = 13: Define following setsCf = {k ∈ N|ρk∗

l ∈ [0, 1), ∀l ∈ L}Sl = {k ∈ N|ρk

∗

l = 1}, S = {S1, · · · ,SL}4: Calculate EE for D2D linkl:

ζl =

∑

n∈Sl

wl log2

(

1+sk

∗

l

akl+bklsk∗

l

)

2P0+αl

∑

k∈Slsk

∗

Dl

, ∀l ∈ L5: while Cf 6= ∅ do6: Select subchannelk ∈ Cf

Uk = {l ∈ L|ρk∗

l > 0}7: Calculate∆k

l (Sl), ∀l ∈ Uk, according to (56)AS

k = {l ∈ L|∆kl (Sl) > 0}

8: l∗ = argminl∈AS

k

ζl.

Assign subchannelk to D2D link l∗, update EE of D2Dlink l∗, ζl ← ζl +∆k

l (Sl)9: UpdateSl∗ ← Sl∗ ∪ {k}, sk

∗

Dl ← 0, ∀l ∈ L\{l∗}, andCf ← Cf − {k}

10: end while11: Outputp∗

Dl = s∗D, andS


TABLE ISIMULATION PARAMETERS

Description Parameter ValueNumber of D2D links L 2 or 4

Number of cellular links K 20Number of subchannels N 20

Maximum distance between Tx and Rx ofD2D links

dmax 50m

Maximum transmit power of cellular link Pmax

Ck0.5W

Maximum transmit power of D2D link Pmax

Dl0.5W

Circuit power P0 0.5WScaling factor αl 1.5

Minimum required rate of cellular links Rmin

Ck2 b/s/Hz

Weighting parameter wl 1Noise power σk 10

−12

VI. COMPLEXITY ANALYSIS

In this section, we analyze the complexity of the proposedalgorithms in term of the number of required arithmetic oper-ations. Since in BnB algorithm, the number of visited nodes isnot fixed, its complexity cannot be exactly determined. On theother hand, the dual-based algorithm requires to solve problem(20) iteratively by the dual decomposition method for givenζ

with complexity ofO(NL). Number of iterations required toupdateζ has complexity ofO(1). Therefore, the complexityof the dual-based algorithm isO(NL).

The Relaxation-Based Rounding algorithm comprises twophases. In the relaxation phase, we iteratively solve problem(49) for givenζ by the interior-point method with complexityof O

(

m12 (m+ n)n2

)

, wherem is the number of inequalityconstraints andn is number of variables [36]. Therefore,the complexity of solving problem (49) and also of therelaxation phase isO(N3.5). In addition, the rounding phasehas complexity ofO(L2). Finally, the complexity of the RBRalgorithm isO(N3.5).

VII. N UMERICAL RESULTS

We consider the simulation setting shown in Fig. 1 withthe base-station located at the center,K = 20 cellular users,andL = 2 or 4 D2D links randomly placed in 500m x 500marea, andN = 20 subchannels for uplink communications.The summary of parameter settings used in the simulations ispresented in Table I.

The subchannel power gain is modeled ashnkl =

(

dkl

d0

)−3

δ

whered0 = 1 m is the reference distance, anddkl > d0 is thedistance between the receiver of linkk and the transmitter oflink l, andδ represents the Rayleigh fading coefficient, whichfollows the exponential distribution with the mean value of1.We set the noise power equal to10−12 W for every link. Thecircuit power of each cellular linkP0 is 0.5 W, the factorαl is1.5 for each D2D link, and the maximum transmit powers ofeach cellular linkk and cellular linkl arePmax

Ck = Pmax

Dl = 0.5W, ∀k ∈ K, ∀l ∈ L. In addition, the weighting parameters ofD2D links are set aswl = 1, ∀l ∈ L, the maximum distanceof D2D links dmax is 50 m, and the minimum required rate ofeach cellular linkk is Rmin

Ck = Rmin

C = 2 b/s/Hz,∀k ∈ K.We evaluate the performance of the proposed algorithms

and that [20] with minor modification (called “conventional”

−500 0 500−500

−400

−300

−200

−100

0

100

200

300

400

500

X axis (m)

Y a

xis

(m)

BSCellular userD2D RxD2D Tx

Fig. 1. Simulation setting

algorithm ) since our work and [20] consider the similarnetwork settings, non-orthogonal spectrum sharing betweencellular and D2D links, and link EE maximization objective.However, there are some differences between two works. In[20], each D2D or cellular link performs power allocation forall subchannels to maximize the EE while in this work, weconsider the joint subchannel assignment and power allocationto maximize the minimum weighted EE of D2D links. Sincewe focus on maximizing the minimum weighted EE of D2Dlinks while satisfying the minimum cellular link data rates,we modify the algorithm developed in [20] so as to maximizethe EE of D2D links, and minimizing the total transmit powerof cellular links while maintaining the minimum cellular-linkrate requirement.

We also consider the spectrum-efficient solution as a ref-erence, which is obtained by solving problem (48) for thecase where all subchannel allocation variables are undeter-mined andζ = 0. In addition, to verify the efficiency ofour algorithms, we compare the objective values achievedby our algorithms with their corresponding upper-boundsachieved by solving the relaxed version of problem (19).All numerical results are acquired by averaging over 1000random realizations of D2D and cellular locations, and chan-nel gains. The EE of D2D links corresponding to the BnBalgorithm, dual-based algorithm, RBR algorithm, upper boundas well as the spectrum-efficiency (SE) maximization solutionare indicated by “BnB Alg.”, “Dual Alg.”, “Rounding Alg.”,“Upper-bound”, and “SE solution”, respectively. In all figuresin the following, we show the minimum achieved EE ofall D2D links (i.e., the design objective) versus differentparameters. For brevity, the minimum EE of all D2D linksis simply referred to as EE of D2D links in the figures andfollowing discussions.

Figs. 2 and 3 show the achieved EE of D2D links versusdmax and Rmin

C , respectively, forL = 2, which allows usto obtain the optimal solution of problem (12) through theBnB algorithm described in Section V.A within reasonabletime. It can be seen that the conventional algorithm and


SE-maximization solution achieve much lower EE than ourproposed energy-efficient algorithms. Moreover, the EE gapbetween the proposed and conventional algorithms signifi-cantly increases asdmax increases. Therefore, it confirms thatthe proposed algorithms can effectively manage the co-channelinterference among the links and make efficient subchannelassignments for D2D links. On the other hand, in the conven-tional algorithm, since each D2D link selfishly optimizes itsEE, the interference among the links may not be well managed.As a result, some D2D links could achieve low EE values,which explains the inferior performance of the conventionaldesign.

It is remarkable that the RBR algorithm performs extremelywell with its achieved EE very close to that of the optimal BnBalgorithm. In fact, when the number of D2D links is small,the number of shared subchannels is small as compared tothe number of available subchannels, which leads to smallperformance loss during the rounding phase. Figs. 2 and3 also indicate that the Dual-Based Algorithm 2 can offerperformance close to that of the BnB algorithm since witha small number of D2D links, the duality gap of problem (22)for given ζ is also small.

10 30 50 70 90 110 130 15060

100

140

180

220

240

D2D link distance (m)

EE

of D

2D li

nks

(b/J

/Hz)

Rounding Alg.Dual Alg.BnB Alg.SE solutionConv. Alg.

Fig. 2. Minimum EE of D2D links versusdmax for L = 2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5100

120

140

160

180

200

Minimum rate of each cellular link (b/s/Hz)

EE

of D

2D li

nks

(b/J

/Hz)

Rounding Alg.Dual Alg.BnB Alg.Conv. Alg.

Fig. 3. Minimum EE of D2D links versus minimum rate of cellular links forL = 2

The characteristics and performance of different algorithmsare further investigated with larger number of D2D links,

L = 4. Figs. 4 and 5 show the convergence of the dual-based Algorithm 2 and the initial relaxation phase of the RBRAlgorithm 4 for dmax = 10 m and100 m, respectively. Theyconfirm that the gradient-based method used in Algorithm 4to determine the optimal solution converges faster than thebisection method employed by Algorithm 2.

2 4 6 8 10 12 14 16 18 20

100

150

200

250

Iteration index

EE

of D

2D li

nks

(b/J

/Hz)

d2dmax = 10md2dmax = 100m

Fig. 4. Convergence behavior of dual-based Algorithm 2

1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

100

110

120

Iteration index

EE

of D

2D li

nks

(b/J

/Hz)

d2dmax = 10md2dmax = 100m

Fig. 5. Convergence behavior of the relaxation phase of RBR Algorithm 4

Fig. 6 indicates that the EE of D2D links achieved byAlgorithms 2 and 4 are significantly higher than that of theconventional algorithm, e.g., atdmax = 150 m, Algorithms 2and 4 can achieve more than90% of the upper-bound EE,which is about300% that of the conventional algorithm andabout130% that of the SE-maximization solution.

The achieved EE of D2D links versus the minimum requiredrate of cellular links, plotted in Fig. 7, indicates that asthe required cellular-link rate increases, the achieved EEofD2D links is reduced. This observation can be explainedas follows. As the minimum required rate of each cellularlink increases, each cellular user has to increase its transmitpower to maintain the required rate, which results in strongerinterference for the co-channel D2D links. Moreover, sinceD2D links are relatively robust against interference due totheir short communication distances, the minimum EE ismoderately impacted as the minimum rate of cellular linksincreases.


10 30 50 70 90 110 130 15020

40

60

80

100

120

140

D2D link distance (m)

EE

of D

2D li

nks

(b/J

/Hz)

Upper−boundRounding Alg.Dual Alg.SE solutionConv. Alg.

Fig. 6. Minimum EE of D2D links versus D2D link distance whenL = 4

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5100

120

140

160

180

200

Minimum rate of each cellular link (b/s/Hz)

EE

of D

2D li

nks

(b/J

/Hz)

Upper−boundRounding Alg.Dual Alg.Conv. Alg.

Fig. 7. Minimum EE of D2D links versus minimum required cellular-linkrate withL = 4

Fig. 8 demonstrates the achieved EE of D2D links as afunction of the circuit power. Both Algorithm 2 and Algorithm4 offer excellent performance, which is very close to theupper bound. For small circuit power, their achieved EE isabout 150% of the EE due to the conventional algorithm.However, when the circuit power increases, the performancegap between the proposed and the conventional algorithms isreduced since the total consumed power is dominated by thecircuit power.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

D2D−link circuit power (W)

EE

of D

2D li

nks

(b/J

/Hz)


Fig. 8. Minimum EE of D2D links versus circuit power

The achieved EE of D2D links versus the noise power,plotted in Fig. 9 indicates that, for lower noise power, bothAlgorithms 2 and 4 achieve higher EE, much better than theconventional algorithm. This is because for the proposed algo-rithms, whenσ reduces, each D2D or cellular link decreasesits transmit power. As a result, the co-channel interferencebetween D2D and cellular links also decreases, which leadsto the improvement in the EE. In contrast, in the conventionalalgorithm, all links would operate in the high-interferenceregime; therefore, the noise power is more negligible comparedto the interference and its variation does not impact the EEachieved by D2D links.

−150 −140 −130 −120 −110 −10050

70

90

110

130

σ (dB)

EE

of D

2D li

nks

(b/J

/Hz)


Fig. 9. Minimum EE of D2D links versus the noise power

Finally, Fig. 10 shows that the achieved EE of D2D linksdecreases as the number of D2D links increases. The perfor-mance gap between the proposed and the conventional algo-rithms also decreases as the number of D2D links increases.This is because as the system supports more D2D links, theavailable resources for each D2D link becomes smaller, whichresults in the decrease in the achieved EE of D2D links. Fig. 10also illustrates that as the number of D2D links increases, thegap between the proposed dual-based algorithm and the upper-bound becomes larger since the duality gap of the underlyingMINLP at convergence under Algorithm 2 becomes higher,which results in larger performance loss for this algorithm.

2 3 4 5 6 7 8 9 100

50

100

150

200

Number of D2D links

EE

of D

2D li

nks

(b/J

/Hz)


Fig. 10. Minimum EE of D2D links versus number of D2D links


VIII. C ONCLUSIONS

In this paper, we have developed efficient resource al-location algorithms for D2D underlaying cellular systems,which maximizes the minimum weighted EE of D2D linkswhile guaranteeing the QoS of cellular links. In particular,we have proposed the optimal BnB algorithm based on thenovel branching and bounding procedures, and proposed twolow-complexity algorithms: (i) the dual-based Algorithm 2solves the resource allocation problem in the dual domain,and (ii) the Relaxed-Based Rounding (RBR) Algorithm 4solves the relaxed version first and then applies a roundingprocedure to obtain a feasible solution for the consideredresource allocation problem. We have studied the theoreticalperformance of the proposed low-complexity algorithms andanalyzed their computational complexity. Numerical resultshave confirmed that both proposed Algorithms 2 and 4 canachieve excellent performance, which is close to that due tothe optimal BnB algorithm and the upper bound.

APPENDIX APROOF OF PROPOSITION1

We first show that the min-rate constraints of cellular linkk must be met at equality as follows:

log2

(

1 +pkCkh

kkk

σkk + pkDlh

kkl

)

= Rmin

Ck . (57)

Note that if σkk

hkkk

(2Rmin

Ck−1) > Pmax

Ck , the required minimum rateof cellular link k cannot be supported; hence, problem (12) isinfeasible. On the other hand, ifσ

kk

hkkk

(2Rmin

Ck − 1) ≤ Pmax

Ck , thecellular link k can allow D2D link l to reuse its resource.Therefore, the power of cellular linkk can be expressed asthe power of D2D linkl on subchannelk as follows:

pkCk =σkk + pkDlh

kkl

hkkk

(2Rmin

Ck − 1) (58)

pkDl =1

hkkl

(

pkCkhkkk

2Rmin

Ck − 1− σk

k

)

(59)

Moreover, the power of cellular linkk must satisfy themaximum power constraint, which can be expressed as

pkDl ≤1

hkkl

(

Pmax

Ck hkkk

2Rmin

Ck − 1− σk

k

)

. (60)

Let us now define

Pmax

Dlk , min

{

Pmax

Dl ,1

hkkl

(

Pmax

Ck hkkk

2Rmin

Ck − 1− σk

k

)}

(61)

Then, it can be verified that if D2D linkl ∈ L reuses theresource of cellular linkk, we havepkDl ∈ [0, Pmax

Dlk ], and

pkDl =1

hkkl

(

pkCkh

kkk

2Rmin

Ck−1− σk

k

)

.

APPENDIX BPROOF OFTHEOREM 1

First, we prove thatη∗(ζ) is a decreasing function ofζ.Suppose we haveζ1 > ζ2, and η∗(ζ1) and η∗(ζ2) are theoptimal objective value of problem (20) corresponding to

the parametersζ1 and ζ2, respectively. Now, we define thefollowing

(poD,ρo) , argmax

(pD,ρ)∈Dminl∈L

[wlRDl(pD,ρ)− ζ1Ptotal

Dl (pD,ρ)].

(62)Then, we have

wlRDl(poD,ρo)− ζ1P

total

Dl (poD,ρo) ≥ η∗(ζ1), ∀l ∈ L. (63)

BecauseP total

Dl (poD,ρo) ≥ 0 and ζ1 > ζ2, we have

ζ1Ptotal

Dl (poD,ρo) ≥ ζ2P

total

Dl (poD,ρo), ∀l ∈ L. Conse-

quently, we havewlRDl(poD,ρo) − ζ1P

total

Dl (poD,ρo) ≤

wlRDl(poD,ρo)− ζ2P

total

Dl (poD,ρo), ∀l ∈ L. Finally, we arrive

at the following

η∗(ζ2) = max(pD,ρ)∈D

minl∈L

[wlRDl(pD,ρ)− ζ2Ptotal

Dl (pD,ρ)]

≥minl∈L

[wlRDl(poD,ρo)− ζ2P

total

Dl (poD,ρo)]

≥minl∈L

[wlRDl(poD,ρo)− ζ1P

total

Dl (poD,ρo)]

=η∗(ζ1).(64)

From these results, we haveη∗(ζ2) ≥ η∗(ζ1), ∀ζ1 > ζ2, whichimplies thatη∗(ζ) is a decreasing function ofζ.

Assume that (21) holds, we need to prove thatζ∗ isthe optimal solution of problem (19). We prove this bycontradiction as follows. Ifζ∗ is not the optimal solutionof problem (19), then we have∃ζo = min

l∈L

wlRDl(poD,ρo)

P total

Dl(po

D,ρo)

,

whereζo > ζ∗ be the optimal solution of problem (19). Thismeans thatwlRDl(p

oD,ρo)

P total

Dl(po

D,ρo)

> ζ∗, ∀l ∈ L, which implies that

wlRDl(poD,ρo) − ζ∗P total

Dl (poD,ρo) > 0, ∀l ∈ L. Therefore,

we have

maxpD,ρ∈D

minl∈L


Dl (pD,ρ)]

≥minl∈L


Dl (pD,ρ)]

>0,

(65)

which contradicts with the assumption in (21). Therefore,ζ∗ =

minl∈L

wlRDl(p∗D,ρ∗)

P total

Dl(p∗

D,ρ∗)

is the optimal solution of problem (12).

APPENDIX CPROOF OFPROPOSITION2

Since the dual decomposition algorithm proposed to solveproblem (20) for givenζ always converges if we choose thestep sizesθ andκ appropriately [35], and the bisection methodto updateζ always converges, the iterative loops of Algorithm2 always converge. Hence, Algorithm 2 returns a feasiblesolution of problem (19).

From the dual decomposition procedure, we have

(p∗D,ρ∗) = argmax

pD∈X ,ρ∈C

∑

l∈L

µ∗l

[

wlRDl(pD,ρ)− ζ∗P total

Dl

]

+∑

l∈L

λ∗l (P

max

D −∑

k∈N

ρkl pkDl). (66)


We denoteXf = {pD ∈ X |∑

k∈Nρkl p

kDl ≤ Pmax

Dl ,ρ ∈ C} as

the set of feasible power allocation solutions of problem (19);hence,Xf ⊂ X . Then, we have

maxpD∈Xf ,ρ∈C

∑

l∈L

µ∗l

[


Dl (pD,ρ)]

(67)

(a)≤ max

pD∈Xf ,ρ∈C

∑

l∈L

µ∗l

[


Dl (pD,ρ)]

+∑

l∈L

λ∗l (P

max

D −∑

k∈N

ρkl pkDl) (68)

(b)≤ max

pD∈X ,ρ∈C

∑

l∈L

µ∗l

[


Dl (pD,ρ)]

+∑

l∈L

λ∗l (P

max

D −∑

k∈N

ρkl pkDl) (69)

(c)= 0. (70)

Inequality(a) holds because of∑

l∈Lλ∗l (P

max

D − ∑

k∈Nρkl p

kDl) ≥

0; inequality (b) is the result ofXf ⊂ X ; and equality(c) isdue to the assumption of Proposition 2. Therefore, we have

maxpD∈Xf ,ρ∈C

∑

l∈L

µ∗l

[


Dl (pD,ρ)]

= 0. (71)

Since we haveµ∗l > 0, ∀l ∈ L, the following holds

maxpD∈Xf ,ρ∈C

minl∈L

[


Dl (pD,ρ)]

= 0. (72)

Therefore, by using the results of Theorem 1,ζ∗ is the optimalsolution of problem (19).

APPENDIX DPROOF OFPROPOSITION3

If (z∗,p∗D,ρ∗) is the optimal solution of problem (48), we

can expresss∗D assk∗

Dl = ρk∗

l pk∗

Dl, ∀l ∈ L, ∀k ∈ N . Therefore,(z∗, s∗D,ρ∗) is a feasible solution of problem (49). On theother hand, if(z∗, s∗D,ρ∗) is the optimal solution of problem(49), p∗

D is given as

pk∗

Dl =

0, if ρk∗

l = 0

sk∗

Dl

ρk∗

l

, otherwise.(73)

Consequently,(z∗,p∗D,ρ∗) is a feasible solution of problem

(48). Hence,z∗ is the optimal objective value of problem (48)iff it is the optimal objective value of problem (49), whichmeans that problems (48) and (49) are equivalent.

In the following, we prove that (48) is convex. It can beseen that in problem (48), the objective is a linear functionof variable z, and (49c)-(49e) are the linear constraints.Therefore, we only need to prove thatgl(sDl,ρ) are concavefunctions of(sDl,ρ), ∀l ∈ L where

gl(sDl,ρ) , wlRDl(sDl,ρ)− ζP total

Dl (sDl,ρ). (74)

Moreover, ingl(sDl,ρ), P total

Dl (sDl,ρ) = 2P l0+αl

∑

k∈KskDl is a

linear combination ofsDl. Therefore, the remaining task is toprove thatRDl(sDl,ρ) are concave functions of(sDl,ρ), ∀l ∈L.

We also haveRDl(sDl,ρ) =∑

k∈KRk

Dl(skDl, ρ

kl ) where

RkDl(s

kDl, ρ

kl ) =

ρkl log2(

1 +skDl

aklρkl+bkls

kDl

)

,

if ρkl undetermined

log2(

1 +skDl

akl+bklskDl

)

, if ρkl = 1

0, if ρkl = 0.

(75)

In the following, we will prove thatRkDl(s

kDl, ρ

kl ) are concave

functions for all possible cases ofρkl . First, if ρkl = 1, then

RkDl(s

kDl, ρ

kl ) = log2

(

1 +skDl

akl+bklskDl

)

and the second partial

derivative ofRkDl(s

kDl, ρ

kl ) can be expressed as

∂2RkDl(s

kDl, ρ

kl )

∂skDl

2 =

− aklbkl[akl + (bkl + 1)skDl] + akl(bkl + 1)(akl + bklskDl)

(akl + bklskDl)

2[akl + (bkl + 1)skDl]2

.

(76)

Since akl > 0 and bkl ≥ 0, ∀k ∈ K, l ∈ L, we have∂2Rk

Dl(skDl,ρ

kl )

∂skDl

2 ≤ 0 for all non-negative values ofskDl. As a

result, RkDl(s

kDl, ρ

kl ) is a concave function ofskDl for given

ρkl = 1.We now consider the case whereρkl is an undetermined

variable. Sincegkl (skDl) = log2

(

1 +skDl

akl+bklskDl

)

is a concave

function of variableskDl, the related functiongkl (skDl, ρ

kl ) =

ρkl gkl (

skDl

ρkl

) = ρkl log2(

1 +skDl

aklρkl+bkls

kDl

)

is also a concave

function of (skDl, ρkl ), ∀ρkl > 0. In addition, for the givenskDl

limρkl→0+

ρkl log2

(

1 +skDl

aklρkl + bkls

kDl

)

= limρkl→0+

ρkl log2

1 +1

aklρkl

skDl

+ bkl

= 0. (77)

Hence,gkl (skDl, ρ

kl ) is a continuous function ofρkl . According

to [37], the concavity ofgkl (skDl, ρ

kl ) is preserved in the

boundary of its domain. Therefore,gkl (skDl, ρ

kl ) is a concave

function of (skDl, ρkl ). As a result,Rk

Dl(skDl, ρ

kl ) is a concave

function of(skDl, ρkl ). Finally, sinceRk

Dl(skDl, ρ

kl ) is a concave

function for all the cases ofρkl , gl(sDl,ρ) is a concave functionfor all l ∈ L, which means that problem (48) is a convexoptimization problem.

APPENDIX EPROOF OFPROPOSITION4

Because problems (48) and (49) are equivalent for givennode m and Qm, from the solution of problem (49) wecan obtain the solution of problem (48). Assume that(z(t−1),p(t−1)

D ,ρ(t−1)) and (z(t),p(t)D ,ρ(t)) are the solution

of problem (48) in iterationst − 1 and t respectively. In


addition, let us defineζt = minl∈L

wlRDl(p(t)D

,ρ(t))

P(totalDl

(p(t)D

,ρ(t)), and ζt−1 =

minl∈L

wlRDl(p(t−1)D

,ρ(t−1))

P(totalDl

(p(t−1)D

,ρ(t−1)). Moreover, we have

max(pD,ρ)∈D

minl∈L

wlRDl(pD,ρ)− ζt−1Ptotal

Dl (pD,ρ)

=minl∈L

wlRDl(p(t)D ,ρ(t))− ζtP

total

Dl (p(t)D ,ρ(t))

≥minl∈L

wlRDl(p(t−1)D ,ρ(t−1))− ζt−1P

total

Dl (p(t−1)D ,ρ(t−1))

= 0(78)

whereD is the set of feasible solutions of problem (48). There-fore, min

l∈LwlRDl(p

(t)D ,ρ(t)) − ζtP

total

Dl (p(t)D ,ρ(t)) ≥ 0, which

means thatζt = minl∈L

wlRDl(p(t)D

,ρ(t))

P total

Dl(p(t)

D,ρ(t))

≥ ζt−1. This implies

that Algorithm 3 creates a sequence of feasible solutions ofproblem (48) whose objective values monotonically increaseover iterations; therefore, the algorithm converges. Assumethat at convergence,ζt−1 = ζt = ζ∗ = min

l∈L

wlRDl(p∗D,ρ∗)

P total

Dl(p∗

D,ρ∗)

.

Therefore, the following must hold

max(pD,ρ)∈D

minl∈L

[wlRDl(pD,ρ)− ζt−1Ptotal

Dl (pD,ρ)]

= minl∈L

[

wlRDl(p(t)D ,ρ(t))− ζt−1P

total

Dl (p(t),ρ(t))]

= minl∈L

[

wlRDl(p(t)D ,ρ(t))− ζtP

total

Dl (p(t),ρ(t))]

= minl∈L

[

wlRDl(p∗D,ρ∗)− ζ∗P total

Dl (p∗,ρ∗)]

= 0.

(79)

Since(ζ∗,p∗D,ρ∗) satisfies the sufficient condition of Theorem

1, it is the optimal solution of problem (47).

REFERENCES

[1] K. Doppler, M. Rinne, C. Wijting, C. B. Ribeiro, and K. Hugl, “Device-to-device communication as an underlay to LTE-advanced networks,”IEEE Commun. Mag.,vol. 47, no. 12, pp. 42–49, Dec. 2009.

[2] M. S. Corson, R. Laroia, J. Li, V. Park, T. Richardson, andG. Tsirtsis,“Toward proximity-aware internetworking,”IEEE Wireless Commun.,vol. 17, no. 6, pp. 26–33, Dec. 2010.

[3] G. Fodor, E. Dahlman, G. Mildh, S. Parkvall, N. Reider, G. Miklos,and Z. Turanyi, “Design aspects of network assisted device-to devicecommunications,”IEEE Commun. Mag.,vol. 50, no.3, pp. 170–177,Mar. 2012.

[4] T. D. Hoang, L. B. Le, and T. Le-Ngoc, “Joint subchannel and powerallocation for D2D communications in cellular networks,” inProc. IEEEWCNC’2014,Apr. 2014.

[5] X. Lin, J. G. Andrews, A. Ghosh, and R. Ratasuk, “An overview on3GPP device-to-device proximity services,”IEEE Commun. Mag.,vol.52, no. 4, pp. 40–48, Apr. 2014.

[6] T. D. Hoang, L. B. Le, and T. Le-Ngoc, “Resource allocation for D2Dcommunications under proportional fairness,” inProc. IEEE GLOBE-COM’2014,Dec. 2014.

[7] E. Hossain, L. B. Le, and D. Niyato,Radio Resource Management inMulti-Tier Cellular Wireless Networks.New York, NY, USA: Wiley,2013.

[8] Z. Hasan, H. Boostanimehr, and V. K. Bhargava, “Green cellular net-works: A survey, some research issues and challenges,”IEEE Commun.Surveys & Tutorials,vol. 13, no. 4, pp. 524–540, Nov. 2011.

[9] G. Y. Li, Z. Xu, C. Xiong, C. Yang, S. Zhang, Y. Chen, and S. Xu,“Energy-efficient wireless communications: Tutorial, survey, and openissues,”IEEE Wireless Commun.,vol. 18, no. 6, pp. 28–35, Dec. 2011.

[10] V. N. Ha and L. B. Le, “Fair resource allocation for OFDMAfemtocellnetworks with macrocell protection,”IEEE Trans. Veh. Technol.,vol. 63,no. 3, pp. 1388–1401.

[11] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-efficientresource allocation in OFDMA networks,”IEEE Trans. Commun.,vol.60, no. 12, pp. 3767-3778, Dec. 2012.

[12] C. Isheden, Z. Chong, E. Jorswieck, and G. Fettweis, “Framework forlink-Level energy efficiency optimization with informed transmitter,”IEEE Trans. Wireless Commun.,vol. 11, no. 8, pp. 2946–2957, Aug.2012.

[13] S. Khakurel, L. Musavian, and T. Le-Ngoc, “Energy-efficient resourceand power allocation for uplink multi-user OFDMA systems,” inProc.IEEE PIMRC’2012,pp. 357–361, Sept. 2012.

[14] M. Ismail, A. Gamage, W. Zhuang X. Shen, “Energy efficient uplinkresource allocation in a heterogeneous wireless medium,” inProc. IEEEICC’2014, pp. 5275–5280, Jun. 2014.

[15] C. H. Yu, O. Tirkkonen, K. Doppler, and C. Ribeiro, “On the perfor-mance of device-to-device underlay communication with simple powercontrol,” in Proc. IEEE VTC’2009,pp. 1–5, Apr. 2009.

[16] C. H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resourcesharing optimization for device-to-device communication underlayingcellular networks,”IEEE Trans. Wireless Commun.,vol. 10, no. 8, pp.2752–2763, Aug. 2011.

[17] J. Wang, D. Zhu, C. Zhao, G. Y. Li, and M. Lei, “Resource sharingof underlaying device-to-device and uplink cellular communications,”IEEE Commun. Lett.,vol. 17, no. 6, pp. 1148–1151, Jun. 2013.

[18] D. Feng, L. Lu, Y. W. Yi, G. Y. Li, G. Geng, and S. Li, “Device-to-devicecommunications underlaying cellular networks,”IEEE Trans. Commun.,vol. 61, no. 8, pp. 3541–3551, Aug. 2013.

[19] L. B. Le, “Fair resource allocation for device-to-device communicationsin wireless cellular networks,” inProc. IEEE Globecom’2012, pp. 5451–5456, Dec. 2012.

[20] Z. Zhou, M. Dong, K. Ota, J. Wu, and T. Sato, “Energy efficiency andspectral efficiency tradeoff in device-to-device (D2D) communications,”IEEE Commun. Lett.,vol. 3, no. 5, pp. 485–488, Oct. 2014.

[21] F. Wang, C. Xu, L. Song, Z. Han, and B. Zhang, “Energy-efficientradio resource and power allocation for device-to-device communicationunderlaying cellular networks,” inProc. IEEE WCSP’2012,pp. 1–6, Oct.2012.

[22] D. Wu, J. Wang, R. Q. Hu, Y. Cai, and L. Zhou, “Energy-efficientresource sharing for mobile device-to-device multimedia communica-tions,” IEEE Trans. Veh. Technol.,vol. 63, no. 5, pp. 2093–2103, Jun.2014.

[23] T. D. Hoang, L. B. Le, and T. Le-Ngoc, “Dual decompositionmethod forenergy-efficient resource allocation in D2D communications underlyingcellular networks,” inProc. IEEE GLOBECOM’2015,Dec. 2015.

[24] T. D. Hoang, L. B. Le, and T. Le-Ngoc, “Radio resource managementfor optimizing energy efficiency of D2D communications in cellularnetworks,” inProc. IEEE PIMRC’2015,Sep. 2015.

[25] W. O. Oduola, X. Li, L. Qian, and Z. Han, “Power control for device-to-device communications as an underlay to cellular system,” in Proc.IEEE ICC’2014, pp. 5257–5262, Jun. 2014.

[26] C. Gao, X. Sheng, J. Tang, W. Zhang, S. Zou, and G. Mohsen,“Jointmode selection, channel allocation and power assignment for greendevice-to-device communications,” inProc. IEEE ICC’2014, pp. 178–183, Jun. 2014.

[27] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy-efficientresource allocation in OFDMA networks,”IEEE Trans. Wireless Com-mun.,vol. 60, no. 12, pp. 3767–3778, Dec. 2012.

[28] T. D. Hoang, L. B. Le, and T. Le-Ngoc, “Energy-efficient resourceallocation for D2D communications in cellular networks,” inProc. IEEEICC’2015, Jun. 2015.

[29] G. Yu, L. Xu, D. Feng, R. Yin, G. Y. Li, and Y. Jiang, “Jointmodeselection and resource allocation for device-to-device communications,”IEEE Trans. Commun.,vol. 63, no. 11, pp. 3814–3824, Nov. 2014.

[30] J. P. G. Crouzeix and J. A. Ferland, “Algorithms for generalizedfractional programming,”Mathematical Programming,vol. 52, pp. 191–207, 1991.

[31] W. Dinkelbach, “On nonlinear fractional programming,”ManagementScience,vol. 13, pp. 492-498, Mar. 1967.

[32] Y. Xu, J. Wang, Q. Wu, A. Anpalagan, and Y.-D. Yao, “Opportunisticspectrum access in cognitive radio networks: Global optimization usinglocal interaction games,”IEEE J. Sel. Topics Signal Process.,vol. 6, no.2, pp. 180–194, Apr. 2012.

[33] Y. Xu, J. Wang, Q. Wu, A. Anpalagan, and Y.-D. Yao, “Opportunisticspectrum access in unknown dynamic environment: A game-theoreticstochastic learning solution,”IEEE Trans. Wireless Commun.,vol. 11,no. 4, pp. 1380 –1391, Apr. 2012.

[34] S. Boyd and L. Vandenberghe,Convex optimization, Cambridge Univer-sity Press, 2009.


[35] D. P. Bertsekas,Nonlinear programming, 2nd ed. Cambridge, MA, USA:MIT Press, 2008.

[36] A. Nemirovski, Interior point polynomial methods in convex program-ming, Lecture notes, Georgia Inst. of Technol., Atlanta, GA, USA, 2004,

Tuong Duc Hoang (S’14) received the B.Eng.(Honor Program) degree from Hanoi University ofTechnology, Vietnam, in 2010, and the M.S. de-gree from Korea Advanced Institute of Science andTechnology (KAIST), Daejeon, Korea, in 2013. Heis currently a Ph.D. student at the Institut Nationalde la Recherche Scientifique-Energy, Materials, andTelecommunications Center (INRS-EMT), Univer-site du Quebec, Montreal, QC, Canada. His re-search interest includes radio resource managementfor wireless communication systems with special

emphasis on heterogeneous networks including D2D communications anddense networks.

Long Bao Le (S’04-M’07-SM’12) received theB.Eng. (with Highest Distinction) degree from HoChi Minh City University of Technology, Vietnam,in 1999, the M.Eng. degree from Asian Instituteof Technology, Pathumthani, Thailand, in 2002, andthe Ph.D. degree from the University of Manitoba,Winnipeg, MB, Canada, in 2007.

He was a postdoctoral researcher at MassachusettsInstitute of Technology (2008-2010) and Universityof Waterloo (2007-2008). Since 2010, he has beenan assistant professor with the Institut National de

la Recherche Scientifique (INRS), Universite du Quebec, Montreal, QC,Canada. His current research interests include smartgrids,cognitive radioand dynamic spectrum sharing, radio resource management, network controland optimization for wireless networks. He is a co-author of the bookRadio Resource Management in Multi-Tier Cellular WirelessNetworks(Wiley,2013). Dr. Le is a member of the editorial board ofIEEE CommunicationsSurveys and Tutorialsand IEEE Wireless Communications Letters. He hasserved as a technical program committee chair/co-chair for several IEEEconferences including IEEE WCNC, IEEE VTC, and IEEE PIMRC.

ch. 10, p. 169.

[37] J.-B. Hiriart-Urruty and C. Lemarechal,Fundamental of Convex Analy-sis, Springer Verlag, Heigelbereg, 2001.

[38] A. H. Land, A. G. Doig, “An automatic method of solving discreteprogramming problems,”Econometrica: Journal of the EconometricSociety,vol. 28, no. 3, pp. 497–520, Jul. 1960.

Tho Le-Ngoc (F’97) obtained his B.Eng. (withDistinction) in Electrical Engineering in 1976, hisM.Eng. in 1978 from McGill University, Montreal,and his Ph.D. in Digital Communications in 1983from the University of Ottawa, Canada. During1977-1982, he was with Spar Aerospace Limited andinvolved in the development and design of satellitecommunications systems. During 1982-1985, he wasan Engineering Manager of the Radio Group in theDepartment of Development Engineering of SRT-elecom Inc., where he developed the new point-to-

multipoint DA-TDMA/TDM Subscriber Radio System SR500. During 1985-2000, he was a Professor at the Department of Electrical and ComputerEngineering of Concordia University. Since 2000, he has been with theDepartment of Electrical and Computer Engineering of McGill University.His research interest is in the area of broadband digital communications. Heis a fellow of the Institute of Electrical and Electronics Engineers (IEEE), theEngineering Institute of Canada (EIC), the Canadian Academyof Engineering(CAE) and the Royal Society of Canada (RSC). He is the recipient of the 2004Canadian Award in Telecommunications Research, and recipient of the IEEECanada Fessenden Award 2005. He holds a Canada Research Chair (Tier I)on Broadband Access Communications.

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