Research ArticleA Novel Access Control and Energy-Saving Resource AllocationScheme for D2D Communication in 5G Networks

Ning Du 12 Kaishi Sun1 Changqing Zhou3 and Xiyuan Ma4

1College of Electrical Engineering and Automation Shandong University of Science and Technology Qingdao 266590 China2Department of Mathematics and Information Engineering Dongchang College of Liaocheng UniversityLiaocheng 252000 China3Shandong Institute of Space Electronic Technology Yantai 264000 China4Department of Computer Science and Engineering Korea University Seoul 02841 Republic of Korea

Correspondence should be addressed to Ning Du lczhlydn126com

Received 13 July 2019 Revised 29 November 2019 Accepted 12 December 2019 Published 8 January 2020

Academic Editor Eulalia Martınez

Copyright copy 2020NingDu et alis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper investigates access link control and resource allocation for the device-to-device (D2D) communication in the fthgeneration (5G) cellular networks e optimization objective of this problem is to maximize the number of admitted D2D linksand minimize the total power consumption of D2D links under the condition of meeting the minimum transmission raterequirements of D2D links and common cellular links is problem is a two-stage nondeterministic polynomial (NP) problemthe solving process of which is very complex So we transform it into a one-stage optimization problem According to themonotonicity of objective function and constraint conditions a monotone optimization problem is established which is solved byreverse polyblock approximation algorithm In order to reduce the complexity of this algorithm a solution algorithm based oniterative convex optimization is proposed Simulation results show that both algorithms can maximize the number of admittedD2D links and minimize the total power consumption of D2D links e proposed two algorithms are better than the energyeciency optimization algorithm

1 Introduction

As one of the key technologies for the fth generation wirelesscommunication networks D2D communication technologyhas attracted wide attention in academia and industry [1]Cellular user equipment in close proximity can communicatewith each other directly which can improve spectral e-ciency reduce transmission delay and ooad trac from thebase station (BS) [2] e implementation of D2D commu-nication can be divided into two categories [3] One is out-of-band D2D communication which occurs on an unauthorizedfrequency band such as bluetooth andWiFi Directe otheris in-band D2D communication For in-band D2D com-munication D2D users adopt the authorized frequency bandand benet from reasonable resource planning and inter-ference management Based on whether or not D2D usersshare resources with cellular users in-band D2D

communication is divided into underlay mode and overlaymode Overlay D2D communication means that sharing thesame frequency bands with cellular users is prohibited [4]Although overlay D2D communication is simple it cannotmake full use of the advantages of D2D communication toimprove spectrum eciency [5] Underlay cellular D2Dcommunication can improve the eciency of local businessbut may be subject to interference from cellular and D2Dusers [6] Appropriate resource allocation can avoid seriousinterference which keep interference below a reasonable levelus resource allocation is one of the most critical issues forunderlay D2D cellular networks [6 7]

A kind of D2D resource allocation scheme based onenergy eciency was proposed by Xu et al [8] which aimedto maximize the energy eciency of D2D links while en-suring the minimum throughput of cellular users In thisscheme D2D links reused the uplink resources of cellular

HindawiComplexityVolume 2020 Article ID 3696015 11 pageshttpsdoiorg10115520203696015

users and multiple D2D links could reuse all cellular usersrsquoresources at the same time It decomposed the originalnonconvex optimization problem into two subproblemsand iterative algorithm was used to solve the problem but itdid not consider access control of D2D links e method ofgraph theory was introduced into D2D resource allocation[9] which took cellular link D2D link and spectrumchannel as vertices in the graph e superedge in the graphcorresponded to each channel allocation scheme Based onthis the cyclic iteration algorithm and branch and boundmethod were designed to optimize themaximumweight rateof the system However in this scheme each cellular userand each D2D user could occupy one channel at most thuslimiting the spectrum utilization Yang et al [10] studied theresource allocation and power control of D2D and cellularusers in a single cellular D2D network Multiple pairs ofD2D users could share the same resources with cellularusers the goal of which was to maximize the total rate ofD2D users under the condition that cellular usersrsquo rate re-quirements were met Zhu et al [11] proposed a channelallocation and power control algorithm for multiple D2Dusers and cellular users so as to make good use of uplinkresources in cellular systems while ensuring the commu-nication quality of common cellular users Ban and Jung [12]proposed a centralized link access control algorithm toensure maximum transmission rate for overlay D2D cellularusers However D2D users occupied special spectrum re-sources and the spectrum utilization rate was not high butthis scheme was not suitable for underlay D2D communi-cation Qian et al [13] proposed a joint channel selection andpower control algorithm based on convex optimization tomaximize the total rate of D2D users without consideringthe access control problem of D2D users A kind of inter-ference-aware resource optimization for D2D communi-cations in 5G networks was put forward by Hao et al [14]but it did not take access control into account e NP-hardjoint resource allocation problem was formulated as a one-to-one matching problem which also did not consideraccess control problem of D2D links [15] A signal to in-terference plus noise ratio (SINR) aware mode selectionscheduling and resource allocation scheme for D2D com-munications was put forward by Bithas et al [16] Howeverscheduling and resource allocation were considered indi-vidually By exploiting D2D communication for enablinguser collaboration and reducing the edge serverrsquos load theD2D-assisted and nonorthogonal multiple access basedmobile edge computing system was investigated by Diaoet al [17] which just considered power channel allocationand computing resources without taking D2D access controlproblem into account e resource allocation problem foruplink multicarrier nonorthogonal multiple access in D2Dunderlay cellular networks was investigated by Zheng et al[18] which did not consider D2D access control and energyminimization A kind of power allocation scheme was putforward by Wang et al [19] to maximize the energy effi-ciency of the relay-aided D2D link while satisfying theminimum data transmission rates of the cellular links whichdid not consider the minimum data rate requirements ofD2D links and the number of admitted D2D links

Although some current works [6 7 16 20 21] con-sidered access control and resource allocation access controland resource allocation were optimized individually Firstlythey determined whether a D2D pair can be admitted underthe SINR requirements of both D2D users and cellular usersSecondly they allocated resources to D2D users and cellularusers to maximize the overall throughput or energy effi-ciency In our research access control and resource allo-cation were optimized jointly not individually Whileensuring the minimum transmission rates of cellular linksand D2D links we not only minimize the power consumedby D2D users but also maximize the number of D2D usersBesides each D2D link can reuse all subcarriers of cellularusers Specifically the main contributions of this paper arelisted as follows

(1) e optimization objective is to maximize thenumber of admitted D2D links and minimize thetotal power consumption of D2D links whilemeeting the minimum transmission rate require-ments of D2D users and common cellular users isoptimization problem is transformed into a one-stage problem It is proved that joint access controland resource allocation can not only maximize thenumber of admitted D2D links but also minimize thetotal power consumption of D2D links

(2) In order to solve this problem a continuous functionis used to approximate binary discrete access controlvariable e joint access control and resource al-location problem is transformed into a monotoneoptimization problem which is worked out by re-verse polyblock approximation algorithm Besideswe prove that choosing appropriate parameters canmaximize the number of admitted D2D links Inorder to reduce computation complexity a kind ofiterative convex optimization algorithm is proposed

(3) Via numerical simulation we demonstrate that theproposed algorithms can maximize the number ofadmitted D2D links and minimize the total powerconsumption of D2D links e performance of twoalgorithms is better than that of the energy efficiencyoptimization algorithm

is paper is organized as follows In Section 2 wepresent the system model e access control and energyminimization problem is transformed into a one-stageproblem in Section 3 In Section 4 the solving algorithm ofaccess control and energy minimization is put forwardNumerical results are presented in Section 5 followed by theconclusions in Section 6

2 System Model

In the uplink transmission of a single-cell wireless cellularnetwork as shown in Figure 1 the set K 1 K de-notes K cellular links L K + 1 K + L denotes L

D2D links and K cellular links share N orthogonal sub-carriers in the set N 1 N L D2D links reuse N

orthogonal subcarriers M KcupL denotes the set of all

2 Complexity

links K |K| L |L| M |M| N |N| and |middot| denotescardinality of the set hn

kl denotes channel coefficient fromtransmitting link l to receiving link k on subcarrier n and pn

m

represents transmitting power of link m isinM on subcarriern p [pm] forallm isinM represents transmitting power vectorof all links pm [pn

m] foralln isinN represents power allocationvector of link m isinM on all subcarriers We introduce abinary variable ρn

k for the cellular link k isinK ρnk 1 rep-

resents that subcarrier n is assigned to link k otherwiseρn

k 0 ρ [ρn] foralln isinN represents allocation vectors of allsubcarriers ρn [ρn

k] forallk isinK represents allocation vectorsof all cellular links on subcarrier n

Signal to interference plus noise ratio of cellular link k onsubcarrier n is

cnk(ρ p)

pnk hn

kk

111386811138681113868111386811138681113868111386811138682

σnk + 1113936lisinLpn

l hnkl

111386811138681113868111386811138681113868111386811138682 (1)

where 1113936lisinLpnl |hn

kl|2 represents interference fromD2D link to

cellular link k when reusing subcarrier n and σnk represents

noise power on subcarrier n of cellular link k Signal tointerference plus noise ratio of D2D link l on subcarrier n is

cnl (ρ p)

pnl hn

ll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLlp

nlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinKρnkpn

k hnlk

111386811138681113868111386811138681113868111386811138682 (2)

where 1113936kisinKρnkpn

k|hnlk|2 represents interference from cellular

link to D2D link l when reusing subcarrier n 1113936lprimeisinLlpnlprime |h

nllprime |

2

represents interference from other D2D links except for linkl σn

l represents noise power of D2D link l on subcarrier ne spectrum efficiency of cellular link k isinK and D2D linkl isinL is expressed respectively as

Rk(ρ p) 1113944nisinN

ρnklog2 1 + c

nk(ρ p)( 1113857

Rl(ρ p) 1113944nisinN

log2 1 + cnl (ρ p)( 1113857

(3)

3 Problem Transformation

31 Access Control Problem and Energy MinimizationProblem e first problem is access control of D2D linkswhich maximizes the number of admitted D2D linksthrough subcarrier allocation power allocation and linkaccess control while ensuring the transmission rates ofcellular users and D2D users A binary link access controlvector s [s1 s2 sL]T is introduced to representwhether D2D link l meets the demand of minimum datarate For any l isinL sl 0 means that the correspondingD2D link is scheduled otherwise sl 1 e optimizationgoal is to make as many as possible sl equal to 0 whichmeans the sum of all elements in the vector s is as small aspossible e access control problem is formulated as P1

minρps

1113944lisinL

sl (4)

st sl 0 Rl(ρ p)geRmin

l

1 otherwise1113896 foralll isinL (5)

Rk(ρ p)geRmink forallk isinK (6)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (7)

1113944nisinN

ρnkp

nk lePmax forallk isinK (8)

1113944kisinK

ρnk le 1 foralln isinN (9)

ρnk isin 0 1 forallk isinKforalln isinN (10)

where Rmink and Rmin

l represent the minimum rate re-quirement of cellular user k isinK and D2D user l isinL re-spectively Pmax denotes the maximum power of each linkand ρn

k isin 0 1 and 1113936kisinKρnk le 1 mean that each subcarrier

can only be assigned to one cellular link Problem P1 can givethe set of admitted D2D links Llowast l | slowastl 01113864 1113865

e second problem is to minimize total power con-sumption of the admitted D2D linksLlowast l | slowastl 01113864 1113865 whichcan be expressed as P2

minρp

1113944lisinLlowast

1113944nisinN

pnl (11)

st Rl(ρ p)geRminl foralll isinLlowast (12)

Rk(ρ p)geRmink forallk isinK (13)

(9)ndash(11) (14)

1113944nisinN

pnl lePmax foralll isinL

lowast (15)

pnl 0 foralll notinLlowastforalln isinN (16)

CU

CU

D2D

D2D

Communication linkInterference link

Figure 1 System model

Complexity 3

(12) and (15) ensure that each admitted D2D link meetsthe minimum transmission rate and maximum power re-quirement respectively (13) and (14) can ensure that eachcellular link meets the minimum transmission rate andmaximum power requirement respectively

32 One-Stage Problem Both P1 and P2 are NP problems[22 23] which makes us can no longer find their globaloptimum in polynomial time We have to use a high qualityapproximation method to solve these two problems inpolynomial time erefore effective suboptimal approxi-mation is carried out to convert this two-stage problem intoa one-stage problem P3

minρps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (17)

st Rl(ρ p) + δminus 1l sl geRmin

l foralll isinL (18)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (19)

Rk(ρ p)geRmink forallk isinK (20)

1113944nisinN

ρnkp

nk lePmax forallk isinK (21)

1113944kisinK

ρnk le 1 foralln isinN (22)

ρnk isin 0 1 forallk isinKforalln isinN (23)

sl isin 0 1 foralll isinL (24)

Proposition 1 By selecting appropriate α (αge LPmax) andδminus 1

l geRminl (foralll isinL) P3 can not only maximize the number of

admitted D2D links but also minimize the total power con-sumed by D2D links lte proof process is as follows

Proof Suppose that (ρlowastplowast slowast) is a feasible solution toproblem P3 which satisfies constraint (18)ndash(24) e linkvector slowast that can be scheduled belongs to the setLlowast l | slowastl 01113864 1113865 For l notinLlowast foralln isinN pnlowast

l 0 for ρnlowastk 0

pnlowastk 0 for ρnlowast

k 1 pnlowastk gt 0 So (ρlowastplowast slowast) is also a feasible

solution to problem P1 Similarly suppose that (ρprime pprime sprime) isa feasible solution to problem P1 which satisfies constraint(5)ndash(10) and the link vector sprime that can be scheduled belongsto the set Lrsquo l | sl

prime 01113864 1113865 For l notinLprime foralln isinN pnprimel 0 the

corresponding link rate Rl(ρprimepprime) 0 for ρnprimek 0 pnprime

k 0ρnprime

k 1 pnprimek gt 0 As long as it does satisfy δminus 1

l geRminl

(ρprimepprime sprime) is also a feasible solution to problem P3 Soproblem P1 and problem P3 have the same feasible set

It is assumed that (ρlowast plowast slowast) is the optimal solution ofproblem P3 but it cannot maximize the number of admittedD2D links However there is another solution (ρprime pprime sprime)that makes 1113936lisinLslowastl ge1113936lisinLsl

prime + 1 so that the following in-equality is obtained

α 1113944lisinL

slowastl + 1113944

lisinL1113944

nisinNp

nlowastl ge α 1113944

lisinLslowastl ge α 1113944

lisinLslprime + 1⎛⎝ ⎞⎠ge α 1113944

lisinLslprime

+ LPmax ge α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel

(25)

It can be seen from inequality (25) that the new feasiblesolution (ρprime pprime sprime) can obtain smaller objective functionvalue than the optimal solution which is inconsistent withthe fact that (ρlowastplowast slowast) is the optimal solution of problemP3 so problem P3 can maximize the number of admittedD2D linkse next step is to prove that P3 canminimize thepower consumption of D2D links Suppose that there is afeasible solution (ρprime pprime sprime) which can maximize the numberof admitted D2D links and can get lower D2D powerconsumption than (ρlowast plowast slowast) so that the following in-equality can be obtained

α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel le α 1113944

lisinLslowastl + 1113944

lisinL1113944

nisinNp

nlowastl (26)

Inequality (26) is inconsistent with the fact (ρlowastplowast slowast) isthe optimal solution of P3 so P3 can maximize the numberof admitted D2D links and minimize the power consumedby D2D links Proposition 1 is proved completely

In order to force cellular links to use orthogonal spec-trum the interference effects of other cellular links areconsidered which is implemented by an introduced verylarge channel gain hv between cellular links [24] so that thesignal to interference plus noise ratio of cellular link k onsubcarrier n is reformulated as

cnk(p)

pnk hn

kk

111386811138681113868111386811138681113868111386811138682

σnk + 1113936kprimeisinKkpn

kprimehv + 1113936lisinLpnl hn

kl

111386811138681113868111386811138681113868111386811138682 (27)

where 1113936kprimeisinKkpnkprimehv represents interference from other

cellular links on subcarrier n Similarly the signal to in-terference plus noise ratio of D2D link l on subcarrier n is

cnl (p)

pnl hn

ll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLlp

nlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinKpnk hn

lk

111386811138681113868111386811138681113868111386811138682 (28)

e spectral efficiency of cellular link k isinK and D2Dlink l isinL is expressed respectively as

Rk(p) 1113944nisinN

log2 1 + cnk(p)1113872 1113873

Rl(p) 1113944nisinN

log2 1 + cnl (p)1113872 1113873

(29)

So problem P3 can be converted into P4

minps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (30)

st Rl(p) + δminus 1l sl geRmin

l foralll isinL (31)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (32)

4 Complexity

Rk(p)geRmink forallk isinK (33)

1113944nisinN

pnk lePmax forallk isinK (34)

sl isin 0 1 foralll isinL (35)

Suppose that (ρlowast plowast slowast) is the optimal solution of P3(plowast slowast) is a feasible solution of problem P4 If (plowast slowast) is anoptimal solution of P4 the subcarrier allocation satisfies

ρnlowastk

1 pnlowastk gt 0

0 otherwise1113896 (36)

Each subcarrier is allocated to one cellular link at mostso (ρlowast plowast slowast) is an optimal solution of P3 erefore theoptimal solution of problem P4 is also the optimal solutionof problem P3

4 Problem Solution

41 Joint Access Control and Power Allocation Based onMonotone Optimization sl isin 0 1 makes the solution ofproblem P4 very difficult In order to solve this problem acontinuous function q(sl) [0 1]⟶ [0 1] is used to ap-proximate binary discrete variable sl as shown in the fol-lowing equation

q sl( 1113857 log 1 + slQ( 1113857( 1113857

log(1 +(1Q)) (37)

where Q is a small enough constant larger than 0 and thisapproximate function satisfies monotone increasing prop-erty for sl 0 q(sl) 0 for sl 1 q(sl) 1 Problem P4 isconverted into problem M1

minps

α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (38)

st (31) (32) (33) (34) (39)

sl isin [0 1] foralll isinL (40)

Problem M1 is a nonconvex optimization problem andthe solving process is very complicated but it has impliedmonotonicity After appropriate transformation thisproblem is transformed into a monotone optimizationproblem which can be solved by reverse polyblock ap-proximation method [25] e regular monotone optimi-zation has the following form

max f(x) | x isin GcapH1113864 1113865 (41)

where f(x) Rn+⟶ R is a monotone increasing function

G sub [0 b] sub Rn+ is a nonempty normal set and H is the

inverse normal set belonging to [0 b]If g(x) Rn

+⟶ R and h(x) Rn+⟶ R are both in-

creasing functions G and H satisfying (42) are normal setand inverse normal set respectively

G x isin Rn+ | g(x)le 01113864 1113865

H x isin Rn+ | h(x) ge 01113864 1113865

(42)

e objective function (38) is an increasing function Inorder to transform M1 into a monotone optimizationproblem all constraints in M1 need to be converted into theform of (42)Rl(p) andRk(p) are nonincreasing functions ofp So a new vector z [zn

m]forallmisinMforallnisinN is definede variablezn

m cnm(p) represents the signal to interference plus noise

ratio of the link m isinM on the channel n isinNP p | 1113936nisinNpn

m lePmax m isinM1113864 1113865 represents the maximumpower constraint of the link m isinM and x (z s) representsthe optimization vector with dimension of D L +

(K + L)N M1 is converted into the following form

minx

f(x) α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (43)

st sl ge 0 foralll isinL (44)

1113944nisinN

log2 1 + znl( 1113857 + δminus 1

l sl minus Rminl ge 0 foralll isinL (45)

1113944nisinN

log2 1 + znk( 1113857 minus R

mink ge 0 forallk isinK (46)

znm ge 0 forallm isinMforalln isinN (47)

znm le cn

m(p) forallm isinMforalln isinNforallp isin P (48)

1113944nisinN

pnl minus Pmax + slPmax le 0 foralll isinL (49)

sl le 1 foralll isinL (50)

is problem needs to minimize a monotone increasingfunction G denotes the normal set which satisfies theconstraints (48)ndash(50) and H denotes the reverse normal setwhich satisfies the constraints (44)ndash(47) e optimal so-lution of problem of M1 is located on the boundary ofX GcapH so we can take advantage of the reverse poly-block approximation method to solve problemM1 as shownin Algorithm 1 where ed is a vector the elements of whichare all zeros except that the d-th element is one and ⊙represents the Hadamard product

After Algorithm 1 is completed binary access controlvector s is obtained by carrying out rounding operationaccording to

sl 0 slowastl le ε

1 otherwise1113896 (51)

According to obtained zlowast we can work out (pnm)lowast using

znm( 1113857lowast

pn

l( 1113857lowast

hnll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLl pn

lprime1113872 1113873lowast

hnllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinK pnk1113872 1113873lowast

hnlk

111386811138681113868111386811138681113868111386811138682

(52)

Complexity 5

In order to judge whether b + λ(x(i) minus b) isin H is true inAlgorithm 2 it needs to judge whether b + λ(x(i) minus b) meetsthe constraints

1113944nisinN

log2 1 +Pmax hn

ll

111386811138681113868111386811138681113868111386811138682

σnl

+ λ znl( 1113857

(i)minus

Pmax hnll

111386811138681113868111386811138681113868111386811138682

σnl

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

+ δminus 1l 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873 minus R

minl ge 0 foralll isinL

1113944nisinN

log2 1 +Pmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

minus Rmink ge 0 forallk isinK

(53)

where (znl )(i) and (sl)

(i) respectively represent the values ofzn

l and sl in the i-th iteration In order to determine whetherρH(x(i)) meets constraints (48)ndash(50) the solution of prob-lem M1-1 is as follows

minpisinP

0 (54)

stPmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠le cnm(p)

forallm isinMforalln isinN

(55)

1113944nisinN

pnl minus Pmax + 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873Pmax le 0 foralll isinL

(56)

1 + λ sl( 1113857(i)

minus 11113872 1113873le 1 foralll isinL (57)

If the constraints of problemM1-1 are feasible it returnsthe value 0 Otherwise it returns +infin where the numeratorand denominator of cn

m(p) are linear functions of pcn

m(p) Γnumnm (p)Γdennm (p) forallm isinM

Γnumnm (p) pnm h

nmm

111386811138681113868111386811138681113868111386811138682 forallm isinMforalln isinN

Γdennk (p) σnk + 1113944

kprimeisinKk

pnkprimehv + 1113944

lisinLp

nl h

nkl

111386811138681113868111386811138681113868111386811138682 forallk isinK

Γdennl (p) σnl + 1113944

lprimeisinLl

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nlk

111386811138681113868111386811138681113868111386811138682 foralll isinL

(58)

So (55) can be converted into

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠Γdennm (p)leΓnumnm (p)

(59)

For a given λ M1-1 is transformed into the followinglinear programming problem

Initialization e number of iterations is i 1 Vertex set is V(1) x(1)1113864 1113865 with x(1) (z(1) s(1)) 0 Set CBV0 +infinRepeatStep 1 Calculate x(i) argminxisinV(i) f(x)

Update the lower boundary flow f(x(i))Step 2 Work out ρH(x(i)) according to Algorithm 2 If f(ρH(x(i)))leCBViminus 1 and ρH(x(i)) satisfies (48)ndash(50) judged by executionof Algorithm M1-1 update the current optimal value CBVi f(ρH(x(i))) and the optimal solution x(i)lowast ρH(x(i)) otherwisex(i)lowast x(iminus 1)lowast CBVi CBViminus 1Step 3 Calculate the auxiliary vertex setVi 1113864x(i)

1 x(i)D 1113865 x(i)

d x(i) + (ρH(x(i)) minus x(i))⊙ ed foralld isin 1 D Update the vertexset for the next iteration V(i+1) (V(i) minus x(i)1113864 1113865)cupVi and increase the number of iterations i i + 1

Until CBVi minus flow lt δOutput xlowast (zlowast slowast)

ALGORITHM 1 Joint access control and power allocation based on monotone optimization

Input x(i) H

Output λ argmax λgt 0 | b + λ(x(i) minus b) isin H1113864 1113865

Step 1 Initialize λmin 0 λmax 1 and δ gt 0 represents a small positive numberStep 2 Repeat the following steps

λ (λmin + λmax)2Judge whether λ is feasible which is equivalent to judge whether b + λ(x(i) minus b) isin H is true If it is true λmin λ otherwiseλmax λUntil λmax minus λmin le δ

Step 3 Output λ λmin ρH(x(i)) b + λ(x(i) minus b)

ALGORITHM 2 Calculation process of ρH(x(i))

6 Complexity

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

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Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

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Engineering Mathematics

International Journal of

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom

links K |K| L |L| M |M| N |N| and |middot| denotescardinality of the set hn

kl denotes channel coefficient fromtransmitting link l to receiving link k on subcarrier n and pn

m

represents transmitting power of link m isinM on subcarriern p [pm] forallm isinM represents transmitting power vectorof all links pm [pn

m] foralln isinN represents power allocationvector of link m isinM on all subcarriers We introduce abinary variable ρn

k for the cellular link k isinK ρnk 1 rep-

resents that subcarrier n is assigned to link k otherwiseρn

k 0 ρ [ρn] foralln isinN represents allocation vectors of allsubcarriers ρn [ρn

k] forallk isinK represents allocation vectorsof all cellular links on subcarrier n

Signal to interference plus noise ratio of cellular link k onsubcarrier n is

cnk(ρ p)

pnk hn

kk

111386811138681113868111386811138681113868111386811138682

σnk + 1113936lisinLpn

l hnkl

111386811138681113868111386811138681113868111386811138682 (1)

where 1113936lisinLpnl |hn

kl|2 represents interference fromD2D link to

cellular link k when reusing subcarrier n and σnk represents

noise power on subcarrier n of cellular link k Signal tointerference plus noise ratio of D2D link l on subcarrier n is

cnl (ρ p)

pnl hn

ll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLlp

nlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinKρnkpn

k hnlk

111386811138681113868111386811138681113868111386811138682 (2)

where 1113936kisinKρnkpn

k|hnlk|2 represents interference from cellular

link to D2D link l when reusing subcarrier n 1113936lprimeisinLlpnlprime |h

nllprime |

2

represents interference from other D2D links except for linkl σn

l represents noise power of D2D link l on subcarrier ne spectrum efficiency of cellular link k isinK and D2D linkl isinL is expressed respectively as

Rk(ρ p) 1113944nisinN

ρnklog2 1 + c

nk(ρ p)( 1113857

Rl(ρ p) 1113944nisinN

log2 1 + cnl (ρ p)( 1113857

(3)

3 Problem Transformation

31 Access Control Problem and Energy MinimizationProblem e first problem is access control of D2D linkswhich maximizes the number of admitted D2D linksthrough subcarrier allocation power allocation and linkaccess control while ensuring the transmission rates ofcellular users and D2D users A binary link access controlvector s [s1 s2 sL]T is introduced to representwhether D2D link l meets the demand of minimum datarate For any l isinL sl 0 means that the correspondingD2D link is scheduled otherwise sl 1 e optimizationgoal is to make as many as possible sl equal to 0 whichmeans the sum of all elements in the vector s is as small aspossible e access control problem is formulated as P1

minρps

1113944lisinL

sl (4)

st sl 0 Rl(ρ p)geRmin

l

1 otherwise1113896 foralll isinL (5)

Rk(ρ p)geRmink forallk isinK (6)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (7)

1113944nisinN

ρnkp

nk lePmax forallk isinK (8)

1113944kisinK

ρnk le 1 foralln isinN (9)

ρnk isin 0 1 forallk isinKforalln isinN (10)

where Rmink and Rmin

l represent the minimum rate re-quirement of cellular user k isinK and D2D user l isinL re-spectively Pmax denotes the maximum power of each linkand ρn

k isin 0 1 and 1113936kisinKρnk le 1 mean that each subcarrier

can only be assigned to one cellular link Problem P1 can givethe set of admitted D2D links Llowast l | slowastl 01113864 1113865

e second problem is to minimize total power con-sumption of the admitted D2D linksLlowast l | slowastl 01113864 1113865 whichcan be expressed as P2

minρp

1113944lisinLlowast

1113944nisinN

pnl (11)

st Rl(ρ p)geRminl foralll isinLlowast (12)

Rk(ρ p)geRmink forallk isinK (13)

(9)ndash(11) (14)

1113944nisinN

pnl lePmax foralll isinL

lowast (15)

pnl 0 foralll notinLlowastforalln isinN (16)

CU

CU

D2D

D2D

Communication linkInterference link

Figure 1 System model

Complexity 3

(12) and (15) ensure that each admitted D2D link meetsthe minimum transmission rate and maximum power re-quirement respectively (13) and (14) can ensure that eachcellular link meets the minimum transmission rate andmaximum power requirement respectively

32 One-Stage Problem Both P1 and P2 are NP problems[22 23] which makes us can no longer find their globaloptimum in polynomial time We have to use a high qualityapproximation method to solve these two problems inpolynomial time erefore effective suboptimal approxi-mation is carried out to convert this two-stage problem intoa one-stage problem P3

minρps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (17)

st Rl(ρ p) + δminus 1l sl geRmin

l foralll isinL (18)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (19)

Rk(ρ p)geRmink forallk isinK (20)

1113944nisinN

ρnkp

nk lePmax forallk isinK (21)

1113944kisinK

ρnk le 1 foralln isinN (22)

ρnk isin 0 1 forallk isinKforalln isinN (23)

sl isin 0 1 foralll isinL (24)

Proposition 1 By selecting appropriate α (αge LPmax) andδminus 1

l geRminl (foralll isinL) P3 can not only maximize the number of

admitted D2D links but also minimize the total power con-sumed by D2D links lte proof process is as follows

Proof Suppose that (ρlowastplowast slowast) is a feasible solution toproblem P3 which satisfies constraint (18)ndash(24) e linkvector slowast that can be scheduled belongs to the setLlowast l | slowastl 01113864 1113865 For l notinLlowast foralln isinN pnlowast

l 0 for ρnlowastk 0

pnlowastk 0 for ρnlowast

k 1 pnlowastk gt 0 So (ρlowastplowast slowast) is also a feasible

solution to problem P1 Similarly suppose that (ρprime pprime sprime) isa feasible solution to problem P1 which satisfies constraint(5)ndash(10) and the link vector sprime that can be scheduled belongsto the set Lrsquo l | sl

prime 01113864 1113865 For l notinLprime foralln isinN pnprimel 0 the

corresponding link rate Rl(ρprimepprime) 0 for ρnprimek 0 pnprime

k 0ρnprime

k 1 pnprimek gt 0 As long as it does satisfy δminus 1

l geRminl

(ρprimepprime sprime) is also a feasible solution to problem P3 Soproblem P1 and problem P3 have the same feasible set

It is assumed that (ρlowast plowast slowast) is the optimal solution ofproblem P3 but it cannot maximize the number of admittedD2D links However there is another solution (ρprime pprime sprime)that makes 1113936lisinLslowastl ge1113936lisinLsl

prime + 1 so that the following in-equality is obtained

α 1113944lisinL

slowastl + 1113944

lisinL1113944

nisinNp

nlowastl ge α 1113944

lisinLslowastl ge α 1113944

lisinLslprime + 1⎛⎝ ⎞⎠ge α 1113944

lisinLslprime

+ LPmax ge α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel

(25)

It can be seen from inequality (25) that the new feasiblesolution (ρprime pprime sprime) can obtain smaller objective functionvalue than the optimal solution which is inconsistent withthe fact that (ρlowastplowast slowast) is the optimal solution of problemP3 so problem P3 can maximize the number of admittedD2D linkse next step is to prove that P3 canminimize thepower consumption of D2D links Suppose that there is afeasible solution (ρprime pprime sprime) which can maximize the numberof admitted D2D links and can get lower D2D powerconsumption than (ρlowast plowast slowast) so that the following in-equality can be obtained

α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel le α 1113944

lisinLslowastl + 1113944

lisinL1113944

nisinNp

nlowastl (26)

Inequality (26) is inconsistent with the fact (ρlowastplowast slowast) isthe optimal solution of P3 so P3 can maximize the numberof admitted D2D links and minimize the power consumedby D2D links Proposition 1 is proved completely

In order to force cellular links to use orthogonal spec-trum the interference effects of other cellular links areconsidered which is implemented by an introduced verylarge channel gain hv between cellular links [24] so that thesignal to interference plus noise ratio of cellular link k onsubcarrier n is reformulated as

cnk(p)

pnk hn

kk

111386811138681113868111386811138681113868111386811138682

σnk + 1113936kprimeisinKkpn

kprimehv + 1113936lisinLpnl hn

kl

111386811138681113868111386811138681113868111386811138682 (27)

where 1113936kprimeisinKkpnkprimehv represents interference from other

cellular links on subcarrier n Similarly the signal to in-terference plus noise ratio of D2D link l on subcarrier n is

cnl (p)

pnl hn

ll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLlp

nlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinKpnk hn

lk

111386811138681113868111386811138681113868111386811138682 (28)

e spectral efficiency of cellular link k isinK and D2Dlink l isinL is expressed respectively as

Rk(p) 1113944nisinN

log2 1 + cnk(p)1113872 1113873

Rl(p) 1113944nisinN

log2 1 + cnl (p)1113872 1113873

(29)

So problem P3 can be converted into P4

minps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (30)

st Rl(p) + δminus 1l sl geRmin

l foralll isinL (31)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (32)

4 Complexity

Rk(p)geRmink forallk isinK (33)

1113944nisinN

pnk lePmax forallk isinK (34)

sl isin 0 1 foralll isinL (35)

Suppose that (ρlowast plowast slowast) is the optimal solution of P3(plowast slowast) is a feasible solution of problem P4 If (plowast slowast) is anoptimal solution of P4 the subcarrier allocation satisfies

ρnlowastk

1 pnlowastk gt 0

0 otherwise1113896 (36)

Each subcarrier is allocated to one cellular link at mostso (ρlowast plowast slowast) is an optimal solution of P3 erefore theoptimal solution of problem P4 is also the optimal solutionof problem P3

4 Problem Solution

41 Joint Access Control and Power Allocation Based onMonotone Optimization sl isin 0 1 makes the solution ofproblem P4 very difficult In order to solve this problem acontinuous function q(sl) [0 1]⟶ [0 1] is used to ap-proximate binary discrete variable sl as shown in the fol-lowing equation

q sl( 1113857 log 1 + slQ( 1113857( 1113857

log(1 +(1Q)) (37)

where Q is a small enough constant larger than 0 and thisapproximate function satisfies monotone increasing prop-erty for sl 0 q(sl) 0 for sl 1 q(sl) 1 Problem P4 isconverted into problem M1

minps

α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (38)

st (31) (32) (33) (34) (39)

sl isin [0 1] foralll isinL (40)

Problem M1 is a nonconvex optimization problem andthe solving process is very complicated but it has impliedmonotonicity After appropriate transformation thisproblem is transformed into a monotone optimizationproblem which can be solved by reverse polyblock ap-proximation method [25] e regular monotone optimi-zation has the following form

max f(x) | x isin GcapH1113864 1113865 (41)

where f(x) Rn+⟶ R is a monotone increasing function

G sub [0 b] sub Rn+ is a nonempty normal set and H is the

inverse normal set belonging to [0 b]If g(x) Rn

+⟶ R and h(x) Rn+⟶ R are both in-

creasing functions G and H satisfying (42) are normal setand inverse normal set respectively

G x isin Rn+ | g(x)le 01113864 1113865

H x isin Rn+ | h(x) ge 01113864 1113865

(42)

e objective function (38) is an increasing function Inorder to transform M1 into a monotone optimizationproblem all constraints in M1 need to be converted into theform of (42)Rl(p) andRk(p) are nonincreasing functions ofp So a new vector z [zn

m]forallmisinMforallnisinN is definede variablezn

m cnm(p) represents the signal to interference plus noise

ratio of the link m isinM on the channel n isinNP p | 1113936nisinNpn

m lePmax m isinM1113864 1113865 represents the maximumpower constraint of the link m isinM and x (z s) representsthe optimization vector with dimension of D L +

(K + L)N M1 is converted into the following form

minx

f(x) α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (43)

st sl ge 0 foralll isinL (44)

1113944nisinN

log2 1 + znl( 1113857 + δminus 1

l sl minus Rminl ge 0 foralll isinL (45)

1113944nisinN

log2 1 + znk( 1113857 minus R

mink ge 0 forallk isinK (46)

znm ge 0 forallm isinMforalln isinN (47)

znm le cn

m(p) forallm isinMforalln isinNforallp isin P (48)

1113944nisinN

pnl minus Pmax + slPmax le 0 foralll isinL (49)

sl le 1 foralll isinL (50)

is problem needs to minimize a monotone increasingfunction G denotes the normal set which satisfies theconstraints (48)ndash(50) and H denotes the reverse normal setwhich satisfies the constraints (44)ndash(47) e optimal so-lution of problem of M1 is located on the boundary ofX GcapH so we can take advantage of the reverse poly-block approximation method to solve problemM1 as shownin Algorithm 1 where ed is a vector the elements of whichare all zeros except that the d-th element is one and ⊙represents the Hadamard product

After Algorithm 1 is completed binary access controlvector s is obtained by carrying out rounding operationaccording to

sl 0 slowastl le ε

1 otherwise1113896 (51)

According to obtained zlowast we can work out (pnm)lowast using

znm( 1113857lowast

pn

l( 1113857lowast

hnll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLl pn

lprime1113872 1113873lowast

hnllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinK pnk1113872 1113873lowast

hnlk

111386811138681113868111386811138681113868111386811138682

(52)

Complexity 5

In order to judge whether b + λ(x(i) minus b) isin H is true inAlgorithm 2 it needs to judge whether b + λ(x(i) minus b) meetsthe constraints

1113944nisinN

log2 1 +Pmax hn

ll

111386811138681113868111386811138681113868111386811138682

σnl

+ λ znl( 1113857

(i)minus

Pmax hnll

111386811138681113868111386811138681113868111386811138682

σnl

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

+ δminus 1l 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873 minus R

minl ge 0 foralll isinL

1113944nisinN

log2 1 +Pmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

minus Rmink ge 0 forallk isinK

(53)

where (znl )(i) and (sl)

(i) respectively represent the values ofzn

l and sl in the i-th iteration In order to determine whetherρH(x(i)) meets constraints (48)ndash(50) the solution of prob-lem M1-1 is as follows

minpisinP

0 (54)

stPmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠le cnm(p)

forallm isinMforalln isinN

(55)

1113944nisinN

pnl minus Pmax + 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873Pmax le 0 foralll isinL

(56)

1 + λ sl( 1113857(i)

minus 11113872 1113873le 1 foralll isinL (57)

If the constraints of problemM1-1 are feasible it returnsthe value 0 Otherwise it returns +infin where the numeratorand denominator of cn

m(p) are linear functions of pcn

m(p) Γnumnm (p)Γdennm (p) forallm isinM

Γnumnm (p) pnm h

nmm

111386811138681113868111386811138681113868111386811138682 forallm isinMforalln isinN

Γdennk (p) σnk + 1113944

kprimeisinKk

pnkprimehv + 1113944

lisinLp

nl h

nkl

111386811138681113868111386811138681113868111386811138682 forallk isinK

Γdennl (p) σnl + 1113944

lprimeisinLl

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nlk

111386811138681113868111386811138681113868111386811138682 foralll isinL

(58)

So (55) can be converted into

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠Γdennm (p)leΓnumnm (p)

(59)

For a given λ M1-1 is transformed into the followinglinear programming problem

Initialization e number of iterations is i 1 Vertex set is V(1) x(1)1113864 1113865 with x(1) (z(1) s(1)) 0 Set CBV0 +infinRepeatStep 1 Calculate x(i) argminxisinV(i) f(x)

Update the lower boundary flow f(x(i))Step 2 Work out ρH(x(i)) according to Algorithm 2 If f(ρH(x(i)))leCBViminus 1 and ρH(x(i)) satisfies (48)ndash(50) judged by executionof Algorithm M1-1 update the current optimal value CBVi f(ρH(x(i))) and the optimal solution x(i)lowast ρH(x(i)) otherwisex(i)lowast x(iminus 1)lowast CBVi CBViminus 1Step 3 Calculate the auxiliary vertex setVi 1113864x(i)

1 x(i)D 1113865 x(i)

d x(i) + (ρH(x(i)) minus x(i))⊙ ed foralld isin 1 D Update the vertexset for the next iteration V(i+1) (V(i) minus x(i)1113864 1113865)cupVi and increase the number of iterations i i + 1

Until CBVi minus flow lt δOutput xlowast (zlowast slowast)

ALGORITHM 1 Joint access control and power allocation based on monotone optimization

Input x(i) H

Output λ argmax λgt 0 | b + λ(x(i) minus b) isin H1113864 1113865

Step 1 Initialize λmin 0 λmax 1 and δ gt 0 represents a small positive numberStep 2 Repeat the following steps

λ (λmin + λmax)2Judge whether λ is feasible which is equivalent to judge whether b + λ(x(i) minus b) isin H is true If it is true λmin λ otherwiseλmax λUntil λmax minus λmin le δ

Step 3 Output λ λmin ρH(x(i)) b + λ(x(i) minus b)

ALGORITHM 2 Calculation process of ρH(x(i))

6 Complexity

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

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(12) and (15) ensure that each admitted D2D link meetsthe minimum transmission rate and maximum power re-quirement respectively (13) and (14) can ensure that eachcellular link meets the minimum transmission rate andmaximum power requirement respectively

32 One-Stage Problem Both P1 and P2 are NP problems[22 23] which makes us can no longer find their globaloptimum in polynomial time We have to use a high qualityapproximation method to solve these two problems inpolynomial time erefore effective suboptimal approxi-mation is carried out to convert this two-stage problem intoa one-stage problem P3

minρps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (17)

st Rl(ρ p) + δminus 1l sl geRmin

l foralll isinL (18)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (19)

Rk(ρ p)geRmink forallk isinK (20)

1113944nisinN

ρnkp

nk lePmax forallk isinK (21)

1113944kisinK

ρnk le 1 foralln isinN (22)

ρnk isin 0 1 forallk isinKforalln isinN (23)

sl isin 0 1 foralll isinL (24)

Proposition 1 By selecting appropriate α (αge LPmax) andδminus 1

l geRminl (foralll isinL) P3 can not only maximize the number of

admitted D2D links but also minimize the total power con-sumed by D2D links lte proof process is as follows

Proof Suppose that (ρlowastplowast slowast) is a feasible solution toproblem P3 which satisfies constraint (18)ndash(24) e linkvector slowast that can be scheduled belongs to the setLlowast l | slowastl 01113864 1113865 For l notinLlowast foralln isinN pnlowast

l 0 for ρnlowastk 0

pnlowastk 0 for ρnlowast

k 1 pnlowastk gt 0 So (ρlowastplowast slowast) is also a feasible

solution to problem P1 Similarly suppose that (ρprime pprime sprime) isa feasible solution to problem P1 which satisfies constraint(5)ndash(10) and the link vector sprime that can be scheduled belongsto the set Lrsquo l | sl

prime 01113864 1113865 For l notinLprime foralln isinN pnprimel 0 the

corresponding link rate Rl(ρprimepprime) 0 for ρnprimek 0 pnprime

k 0ρnprime

k 1 pnprimek gt 0 As long as it does satisfy δminus 1

l geRminl

(ρprimepprime sprime) is also a feasible solution to problem P3 Soproblem P1 and problem P3 have the same feasible set

It is assumed that (ρlowast plowast slowast) is the optimal solution ofproblem P3 but it cannot maximize the number of admittedD2D links However there is another solution (ρprime pprime sprime)that makes 1113936lisinLslowastl ge1113936lisinLsl

prime + 1 so that the following in-equality is obtained

α 1113944lisinL

slowastl + 1113944

lisinL1113944

nisinNp

nlowastl ge α 1113944

lisinLslowastl ge α 1113944

lisinLslprime + 1⎛⎝ ⎞⎠ge α 1113944

lisinLslprime

+ LPmax ge α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel

(25)

It can be seen from inequality (25) that the new feasiblesolution (ρprime pprime sprime) can obtain smaller objective functionvalue than the optimal solution which is inconsistent withthe fact that (ρlowastplowast slowast) is the optimal solution of problemP3 so problem P3 can maximize the number of admittedD2D linkse next step is to prove that P3 canminimize thepower consumption of D2D links Suppose that there is afeasible solution (ρprime pprime sprime) which can maximize the numberof admitted D2D links and can get lower D2D powerconsumption than (ρlowast plowast slowast) so that the following in-equality can be obtained

α 1113944lisinL

slprime + 1113944

lisinL1113944

nisinNp

nprimel le α 1113944

lisinLslowastl + 1113944

lisinL1113944

nisinNp

nlowastl (26)

Inequality (26) is inconsistent with the fact (ρlowastplowast slowast) isthe optimal solution of P3 so P3 can maximize the numberof admitted D2D links and minimize the power consumedby D2D links Proposition 1 is proved completely

In order to force cellular links to use orthogonal spec-trum the interference effects of other cellular links areconsidered which is implemented by an introduced verylarge channel gain hv between cellular links [24] so that thesignal to interference plus noise ratio of cellular link k onsubcarrier n is reformulated as

cnk(p)

pnk hn

kk

111386811138681113868111386811138681113868111386811138682

σnk + 1113936kprimeisinKkpn

kprimehv + 1113936lisinLpnl hn

kl

111386811138681113868111386811138681113868111386811138682 (27)

where 1113936kprimeisinKkpnkprimehv represents interference from other

cellular links on subcarrier n Similarly the signal to in-terference plus noise ratio of D2D link l on subcarrier n is

cnl (p)

pnl hn

ll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLlp

nlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinKpnk hn

lk

111386811138681113868111386811138681113868111386811138682 (28)

e spectral efficiency of cellular link k isinK and D2Dlink l isinL is expressed respectively as

Rk(p) 1113944nisinN

log2 1 + cnk(p)1113872 1113873

Rl(p) 1113944nisinN

log2 1 + cnl (p)1113872 1113873

(29)

So problem P3 can be converted into P4

minps

α 1113944lisinL

sl + 1113944lisinL

1113944nisinN

pnl (30)

st Rl(p) + δminus 1l sl geRmin

l foralll isinL (31)

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinL (32)

4 Complexity

Rk(p)geRmink forallk isinK (33)

1113944nisinN

pnk lePmax forallk isinK (34)

sl isin 0 1 foralll isinL (35)

Suppose that (ρlowast plowast slowast) is the optimal solution of P3(plowast slowast) is a feasible solution of problem P4 If (plowast slowast) is anoptimal solution of P4 the subcarrier allocation satisfies

ρnlowastk

1 pnlowastk gt 0

0 otherwise1113896 (36)

Each subcarrier is allocated to one cellular link at mostso (ρlowast plowast slowast) is an optimal solution of P3 erefore theoptimal solution of problem P4 is also the optimal solutionof problem P3

4 Problem Solution

41 Joint Access Control and Power Allocation Based onMonotone Optimization sl isin 0 1 makes the solution ofproblem P4 very difficult In order to solve this problem acontinuous function q(sl) [0 1]⟶ [0 1] is used to ap-proximate binary discrete variable sl as shown in the fol-lowing equation

q sl( 1113857 log 1 + slQ( 1113857( 1113857

log(1 +(1Q)) (37)

where Q is a small enough constant larger than 0 and thisapproximate function satisfies monotone increasing prop-erty for sl 0 q(sl) 0 for sl 1 q(sl) 1 Problem P4 isconverted into problem M1

minps

α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (38)

st (31) (32) (33) (34) (39)

sl isin [0 1] foralll isinL (40)

Problem M1 is a nonconvex optimization problem andthe solving process is very complicated but it has impliedmonotonicity After appropriate transformation thisproblem is transformed into a monotone optimizationproblem which can be solved by reverse polyblock ap-proximation method [25] e regular monotone optimi-zation has the following form

max f(x) | x isin GcapH1113864 1113865 (41)

where f(x) Rn+⟶ R is a monotone increasing function

G sub [0 b] sub Rn+ is a nonempty normal set and H is the

inverse normal set belonging to [0 b]If g(x) Rn

+⟶ R and h(x) Rn+⟶ R are both in-

creasing functions G and H satisfying (42) are normal setand inverse normal set respectively

G x isin Rn+ | g(x)le 01113864 1113865

H x isin Rn+ | h(x) ge 01113864 1113865

(42)

e objective function (38) is an increasing function Inorder to transform M1 into a monotone optimizationproblem all constraints in M1 need to be converted into theform of (42)Rl(p) andRk(p) are nonincreasing functions ofp So a new vector z [zn

m]forallmisinMforallnisinN is definede variablezn

m cnm(p) represents the signal to interference plus noise

ratio of the link m isinM on the channel n isinNP p | 1113936nisinNpn

m lePmax m isinM1113864 1113865 represents the maximumpower constraint of the link m isinM and x (z s) representsthe optimization vector with dimension of D L +

(K + L)N M1 is converted into the following form

minx

f(x) α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (43)

st sl ge 0 foralll isinL (44)

1113944nisinN

log2 1 + znl( 1113857 + δminus 1

l sl minus Rminl ge 0 foralll isinL (45)

1113944nisinN

log2 1 + znk( 1113857 minus R

mink ge 0 forallk isinK (46)

znm ge 0 forallm isinMforalln isinN (47)

znm le cn

m(p) forallm isinMforalln isinNforallp isin P (48)

1113944nisinN

pnl minus Pmax + slPmax le 0 foralll isinL (49)

sl le 1 foralll isinL (50)

is problem needs to minimize a monotone increasingfunction G denotes the normal set which satisfies theconstraints (48)ndash(50) and H denotes the reverse normal setwhich satisfies the constraints (44)ndash(47) e optimal so-lution of problem of M1 is located on the boundary ofX GcapH so we can take advantage of the reverse poly-block approximation method to solve problemM1 as shownin Algorithm 1 where ed is a vector the elements of whichare all zeros except that the d-th element is one and ⊙represents the Hadamard product

After Algorithm 1 is completed binary access controlvector s is obtained by carrying out rounding operationaccording to

sl 0 slowastl le ε

1 otherwise1113896 (51)

According to obtained zlowast we can work out (pnm)lowast using

znm( 1113857lowast

pn

l( 1113857lowast

hnll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLl pn

lprime1113872 1113873lowast

hnllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinK pnk1113872 1113873lowast

hnlk

111386811138681113868111386811138681113868111386811138682

(52)

Complexity 5

In order to judge whether b + λ(x(i) minus b) isin H is true inAlgorithm 2 it needs to judge whether b + λ(x(i) minus b) meetsthe constraints

1113944nisinN

log2 1 +Pmax hn

ll

111386811138681113868111386811138681113868111386811138682

σnl

+ λ znl( 1113857

(i)minus

Pmax hnll

111386811138681113868111386811138681113868111386811138682

σnl

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

+ δminus 1l 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873 minus R

minl ge 0 foralll isinL

1113944nisinN

log2 1 +Pmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

minus Rmink ge 0 forallk isinK

(53)

where (znl )(i) and (sl)

(i) respectively represent the values ofzn

l and sl in the i-th iteration In order to determine whetherρH(x(i)) meets constraints (48)ndash(50) the solution of prob-lem M1-1 is as follows

minpisinP

0 (54)

stPmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠le cnm(p)

forallm isinMforalln isinN

(55)

1113944nisinN

pnl minus Pmax + 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873Pmax le 0 foralll isinL

(56)

1 + λ sl( 1113857(i)

minus 11113872 1113873le 1 foralll isinL (57)

If the constraints of problemM1-1 are feasible it returnsthe value 0 Otherwise it returns +infin where the numeratorand denominator of cn

m(p) are linear functions of pcn

m(p) Γnumnm (p)Γdennm (p) forallm isinM

Γnumnm (p) pnm h

nmm

111386811138681113868111386811138681113868111386811138682 forallm isinMforalln isinN

Γdennk (p) σnk + 1113944

kprimeisinKk

pnkprimehv + 1113944

lisinLp

nl h

nkl

111386811138681113868111386811138681113868111386811138682 forallk isinK

Γdennl (p) σnl + 1113944

lprimeisinLl

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nlk

111386811138681113868111386811138681113868111386811138682 foralll isinL

(58)

So (55) can be converted into

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠Γdennm (p)leΓnumnm (p)

(59)

For a given λ M1-1 is transformed into the followinglinear programming problem

Initialization e number of iterations is i 1 Vertex set is V(1) x(1)1113864 1113865 with x(1) (z(1) s(1)) 0 Set CBV0 +infinRepeatStep 1 Calculate x(i) argminxisinV(i) f(x)

Update the lower boundary flow f(x(i))Step 2 Work out ρH(x(i)) according to Algorithm 2 If f(ρH(x(i)))leCBViminus 1 and ρH(x(i)) satisfies (48)ndash(50) judged by executionof Algorithm M1-1 update the current optimal value CBVi f(ρH(x(i))) and the optimal solution x(i)lowast ρH(x(i)) otherwisex(i)lowast x(iminus 1)lowast CBVi CBViminus 1Step 3 Calculate the auxiliary vertex setVi 1113864x(i)

1 x(i)D 1113865 x(i)

d x(i) + (ρH(x(i)) minus x(i))⊙ ed foralld isin 1 D Update the vertexset for the next iteration V(i+1) (V(i) minus x(i)1113864 1113865)cupVi and increase the number of iterations i i + 1

Until CBVi minus flow lt δOutput xlowast (zlowast slowast)

ALGORITHM 1 Joint access control and power allocation based on monotone optimization

Input x(i) H

Output λ argmax λgt 0 | b + λ(x(i) minus b) isin H1113864 1113865

Step 1 Initialize λmin 0 λmax 1 and δ gt 0 represents a small positive numberStep 2 Repeat the following steps

λ (λmin + λmax)2Judge whether λ is feasible which is equivalent to judge whether b + λ(x(i) minus b) isin H is true If it is true λmin λ otherwiseλmax λUntil λmax minus λmin le δ

Step 3 Output λ λmin ρH(x(i)) b + λ(x(i) minus b)

ALGORITHM 2 Calculation process of ρH(x(i))

6 Complexity

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

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Submit your manuscripts atwwwhindawicom

Rk(p)geRmink forallk isinK (33)

1113944nisinN

pnk lePmax forallk isinK (34)

sl isin 0 1 foralll isinL (35)

Suppose that (ρlowast plowast slowast) is the optimal solution of P3(plowast slowast) is a feasible solution of problem P4 If (plowast slowast) is anoptimal solution of P4 the subcarrier allocation satisfies

ρnlowastk

1 pnlowastk gt 0

0 otherwise1113896 (36)

Each subcarrier is allocated to one cellular link at mostso (ρlowast plowast slowast) is an optimal solution of P3 erefore theoptimal solution of problem P4 is also the optimal solutionof problem P3

4 Problem Solution

41 Joint Access Control and Power Allocation Based onMonotone Optimization sl isin 0 1 makes the solution ofproblem P4 very difficult In order to solve this problem acontinuous function q(sl) [0 1]⟶ [0 1] is used to ap-proximate binary discrete variable sl as shown in the fol-lowing equation

q sl( 1113857 log 1 + slQ( 1113857( 1113857

log(1 +(1Q)) (37)

where Q is a small enough constant larger than 0 and thisapproximate function satisfies monotone increasing prop-erty for sl 0 q(sl) 0 for sl 1 q(sl) 1 Problem P4 isconverted into problem M1

minps

α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (38)

st (31) (32) (33) (34) (39)

sl isin [0 1] foralll isinL (40)

Problem M1 is a nonconvex optimization problem andthe solving process is very complicated but it has impliedmonotonicity After appropriate transformation thisproblem is transformed into a monotone optimizationproblem which can be solved by reverse polyblock ap-proximation method [25] e regular monotone optimi-zation has the following form

max f(x) | x isin GcapH1113864 1113865 (41)

where f(x) Rn+⟶ R is a monotone increasing function

G sub [0 b] sub Rn+ is a nonempty normal set and H is the

inverse normal set belonging to [0 b]If g(x) Rn

+⟶ R and h(x) Rn+⟶ R are both in-

creasing functions G and H satisfying (42) are normal setand inverse normal set respectively

G x isin Rn+ | g(x)le 01113864 1113865

H x isin Rn+ | h(x) ge 01113864 1113865

(42)

e objective function (38) is an increasing function Inorder to transform M1 into a monotone optimizationproblem all constraints in M1 need to be converted into theform of (42)Rl(p) andRk(p) are nonincreasing functions ofp So a new vector z [zn

m]forallmisinMforallnisinN is definede variablezn

m cnm(p) represents the signal to interference plus noise

ratio of the link m isinM on the channel n isinNP p | 1113936nisinNpn

m lePmax m isinM1113864 1113865 represents the maximumpower constraint of the link m isinM and x (z s) representsthe optimization vector with dimension of D L +

(K + L)N M1 is converted into the following form

minx

f(x) α 1113944lisinL

q sl( 1113857 + 1113944lisinL

1113944nisinN

pnl (43)

st sl ge 0 foralll isinL (44)

1113944nisinN

log2 1 + znl( 1113857 + δminus 1

l sl minus Rminl ge 0 foralll isinL (45)

1113944nisinN

log2 1 + znk( 1113857 minus R

mink ge 0 forallk isinK (46)

znm ge 0 forallm isinMforalln isinN (47)

znm le cn

m(p) forallm isinMforalln isinNforallp isin P (48)

1113944nisinN

pnl minus Pmax + slPmax le 0 foralll isinL (49)

sl le 1 foralll isinL (50)

is problem needs to minimize a monotone increasingfunction G denotes the normal set which satisfies theconstraints (48)ndash(50) and H denotes the reverse normal setwhich satisfies the constraints (44)ndash(47) e optimal so-lution of problem of M1 is located on the boundary ofX GcapH so we can take advantage of the reverse poly-block approximation method to solve problemM1 as shownin Algorithm 1 where ed is a vector the elements of whichare all zeros except that the d-th element is one and ⊙represents the Hadamard product

After Algorithm 1 is completed binary access controlvector s is obtained by carrying out rounding operationaccording to

sl 0 slowastl le ε

1 otherwise1113896 (51)

According to obtained zlowast we can work out (pnm)lowast using

znm( 1113857lowast

pn

l( 1113857lowast

hnll

111386811138681113868111386811138681113868111386811138682

σnl + 1113936lprimeisinLl pn

lprime1113872 1113873lowast

hnllprime

111386811138681113868111386811138681113868111386811138682

+ 1113936kisinK pnk1113872 1113873lowast

hnlk

111386811138681113868111386811138681113868111386811138682

(52)

Complexity 5

In order to judge whether b + λ(x(i) minus b) isin H is true inAlgorithm 2 it needs to judge whether b + λ(x(i) minus b) meetsthe constraints

1113944nisinN

log2 1 +Pmax hn

ll

111386811138681113868111386811138681113868111386811138682

σnl

+ λ znl( 1113857

(i)minus

Pmax hnll

111386811138681113868111386811138681113868111386811138682

σnl

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

+ δminus 1l 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873 minus R

minl ge 0 foralll isinL

1113944nisinN

log2 1 +Pmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

minus Rmink ge 0 forallk isinK

(53)

where (znl )(i) and (sl)

(i) respectively represent the values ofzn

l and sl in the i-th iteration In order to determine whetherρH(x(i)) meets constraints (48)ndash(50) the solution of prob-lem M1-1 is as follows

minpisinP

0 (54)

stPmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠le cnm(p)

forallm isinMforalln isinN

(55)

1113944nisinN

pnl minus Pmax + 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873Pmax le 0 foralll isinL

(56)

1 + λ sl( 1113857(i)

minus 11113872 1113873le 1 foralll isinL (57)

If the constraints of problemM1-1 are feasible it returnsthe value 0 Otherwise it returns +infin where the numeratorand denominator of cn

m(p) are linear functions of pcn

m(p) Γnumnm (p)Γdennm (p) forallm isinM

Γnumnm (p) pnm h

nmm

111386811138681113868111386811138681113868111386811138682 forallm isinMforalln isinN

Γdennk (p) σnk + 1113944

kprimeisinKk

pnkprimehv + 1113944

lisinLp

nl h

nkl

111386811138681113868111386811138681113868111386811138682 forallk isinK

Γdennl (p) σnl + 1113944

lprimeisinLl

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nlk

111386811138681113868111386811138681113868111386811138682 foralll isinL

(58)

So (55) can be converted into

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠Γdennm (p)leΓnumnm (p)

(59)

For a given λ M1-1 is transformed into the followinglinear programming problem

Initialization e number of iterations is i 1 Vertex set is V(1) x(1)1113864 1113865 with x(1) (z(1) s(1)) 0 Set CBV0 +infinRepeatStep 1 Calculate x(i) argminxisinV(i) f(x)

Update the lower boundary flow f(x(i))Step 2 Work out ρH(x(i)) according to Algorithm 2 If f(ρH(x(i)))leCBViminus 1 and ρH(x(i)) satisfies (48)ndash(50) judged by executionof Algorithm M1-1 update the current optimal value CBVi f(ρH(x(i))) and the optimal solution x(i)lowast ρH(x(i)) otherwisex(i)lowast x(iminus 1)lowast CBVi CBViminus 1Step 3 Calculate the auxiliary vertex setVi 1113864x(i)

1 x(i)D 1113865 x(i)

d x(i) + (ρH(x(i)) minus x(i))⊙ ed foralld isin 1 D Update the vertexset for the next iteration V(i+1) (V(i) minus x(i)1113864 1113865)cupVi and increase the number of iterations i i + 1

Until CBVi minus flow lt δOutput xlowast (zlowast slowast)

ALGORITHM 1 Joint access control and power allocation based on monotone optimization

Input x(i) H

Output λ argmax λgt 0 | b + λ(x(i) minus b) isin H1113864 1113865

Step 1 Initialize λmin 0 λmax 1 and δ gt 0 represents a small positive numberStep 2 Repeat the following steps

λ (λmin + λmax)2Judge whether λ is feasible which is equivalent to judge whether b + λ(x(i) minus b) isin H is true If it is true λmin λ otherwiseλmax λUntil λmax minus λmin le δ

Step 3 Output λ λmin ρH(x(i)) b + λ(x(i) minus b)

ALGORITHM 2 Calculation process of ρH(x(i))

6 Complexity

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

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Submit your manuscripts atwwwhindawicom

In order to judge whether b + λ(x(i) minus b) isin H is true inAlgorithm 2 it needs to judge whether b + λ(x(i) minus b) meetsthe constraints

1113944nisinN

log2 1 +Pmax hn

ll

111386811138681113868111386811138681113868111386811138682

σnl

+ λ znl( 1113857

(i)minus

Pmax hnll

111386811138681113868111386811138681113868111386811138682

σnl

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

+ δminus 1l 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873 minus R

minl ge 0 foralll isinL

1113944nisinN

log2 1 +Pmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

minus Rmink ge 0 forallk isinK

(53)

where (znl )(i) and (sl)

(i) respectively represent the values ofzn

l and sl in the i-th iteration In order to determine whetherρH(x(i)) meets constraints (48)ndash(50) the solution of prob-lem M1-1 is as follows

minpisinP

0 (54)

stPmax hn

kk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠le cnm(p)

forallm isinMforalln isinN

(55)

1113944nisinN

pnl minus Pmax + 1 + λ sl( 1113857

(i)minus 11113872 11138731113872 1113873Pmax le 0 foralll isinL

(56)

1 + λ sl( 1113857(i)

minus 11113872 1113873le 1 foralll isinL (57)

If the constraints of problemM1-1 are feasible it returnsthe value 0 Otherwise it returns +infin where the numeratorand denominator of cn

m(p) are linear functions of pcn

m(p) Γnumnm (p)Γdennm (p) forallm isinM

Γnumnm (p) pnm h

nmm

111386811138681113868111386811138681113868111386811138682 forallm isinMforalln isinN

Γdennk (p) σnk + 1113944

kprimeisinKk

pnkprimehv + 1113944

lisinLp

nl h

nkl

111386811138681113868111386811138681113868111386811138682 forallk isinK

Γdennl (p) σnl + 1113944

lprimeisinLl

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nlk

111386811138681113868111386811138681113868111386811138682 foralll isinL

(58)

So (55) can be converted into

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

+ λ znk( 1113857

(i)minus

Pmax hnkk

111386811138681113868111386811138681113868111386811138682

σnk

⎛⎝ ⎞⎠⎛⎝ ⎞⎠Γdennm (p)leΓnumnm (p)

(59)

For a given λ M1-1 is transformed into the followinglinear programming problem

Initialization e number of iterations is i 1 Vertex set is V(1) x(1)1113864 1113865 with x(1) (z(1) s(1)) 0 Set CBV0 +infinRepeatStep 1 Calculate x(i) argminxisinV(i) f(x)

Update the lower boundary flow f(x(i))Step 2 Work out ρH(x(i)) according to Algorithm 2 If f(ρH(x(i)))leCBViminus 1 and ρH(x(i)) satisfies (48)ndash(50) judged by executionof Algorithm M1-1 update the current optimal value CBVi f(ρH(x(i))) and the optimal solution x(i)lowast ρH(x(i)) otherwisex(i)lowast x(iminus 1)lowast CBVi CBViminus 1Step 3 Calculate the auxiliary vertex setVi 1113864x(i)

1 x(i)D 1113865 x(i)

d x(i) + (ρH(x(i)) minus x(i))⊙ ed foralld isin 1 D Update the vertexset for the next iteration V(i+1) (V(i) minus x(i)1113864 1113865)cupVi and increase the number of iterations i i + 1

Until CBVi minus flow lt δOutput xlowast (zlowast slowast)

ALGORITHM 1 Joint access control and power allocation based on monotone optimization

Input x(i) H

Output λ argmax λgt 0 | b + λ(x(i) minus b) isin H1113864 1113865

Step 1 Initialize λmin 0 λmax 1 and δ gt 0 represents a small positive numberStep 2 Repeat the following steps

λ (λmin + λmax)2Judge whether λ is feasible which is equivalent to judge whether b + λ(x(i) minus b) isin H is true If it is true λmin λ otherwiseλmax λUntil λmax minus λmin le δ

Step 3 Output λ λmin ρH(x(i)) b + λ(x(i) minus b)

ALGORITHM 2 Calculation process of ρH(x(i))

6 Complexity

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

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Submit your manuscripts atwwwhindawicom

minpisinP

0

st(59)forallm isinMforalln isinN(56) (57)

(60)

e above linear programming problem can be solved bythe simplex method or interior point method

Proposition 2 In problem M1 slowastl could be a fractionalnumber which is clearly not the optimal solution to problemP4 lten Algorithm 1 can maximize the number of admittedD2D links by choosing appropriate ε satisfying(

1 + 1QLminus 1L

radicminus 1)Qlt εlt 1 lte proof process is as follows

Suppose that (zlowast slowast) is the optimal solution of problemM1 and plowast is the corresponding optimal power vector if slowastl isan integer the proposition is proved If slowastl is a fractionalnumber suppose that s0 and p0 are the optimal accesscontrol vector and power allocation vector of problem P4respectively slowast ne s0 z0 [(zn

m)0]forallmisinM foralln isinN(zn

m)0 cnm(p0) (z0 s0) is a feasible solution of the problem

M1 and we can obtain

α 1113944lisinL

q slowastl( 1113857 + 1113944

lisinL1113944

nisinNp

nl( 1113857lowast lt α 1113944

lisinLq s

0l1113872 1113873 + 1113944

lisinL1113944

nisinNp

nl( 1113857

0

(61)

where αge LPmax (pnl )lowast and (pn

l )0 are bounded variables andwe can obtain

1113944lisinL

q slowastl( 1113857le 1113944

lisinLq s

0l1113872 1113873 (62)

where s [sl]foralllisinL represents the binary access control so-lution after rounding according to (62) e admitted D2Dlink should meet the following equation

sl 0 slowastl le ε

1 otherwise1113896 (63)

en inequality (63) is established

1113944lisinL

q slowastl( 1113857ge 1113944

lisinLq sl( 1113857 + L

log(1 +(εQ))

log(1 +(1Q))minus L (64)

Since (1+1QLminus 1L

radicminus 1)Qltεlt1 minus 1ltL((log(1+ (εQ)))

(log(1+1Q))) minus Llt0 holds and we can obtain1113944lisinL

q sl( 1113857le 1113944lisinL

q s0l1113872 1113873 (65)

erefore Algorithm 1 can maximize the number ofadmitted D2D links

I represents the total number of iterations in Algo-rithm 1 e computational complexity of calculating thelower boundary of step 1 in each iteration is O(D) where D

represents the dimension of optimization vector ecomputational complexity of step 2 is O(M35N35) whichadopts interior point method to calculate linear program-ming e computational complexity of Algorithm 1 isO(I(D + M35N35)) in polynomial time

42 Access Control and Resource Allocation Algorithm Basedon Iterative Convex Optimization As discussed in the lastparagraph of the previous section the algorithm based onmonotone optimization can achieve the asymptoticallyoptimal solution but the computational complexity is highSo we propose an iterative convex optimization approxi-mation algorithm with low complexity sl in problem P4 isrelaxed the value of which belongs to [0 1] In the constraintcondition Rm(p) m isinKcupL is a nonconvex functionwhich can be expressed as Rm(p) fm(p) minus gm(p)

For m isinK

fm(p) 1113944nisinN

log2 σnm + p

nm h

nmm

111386811138681113868111386811138681113868111386811138682

+ 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

kprimeisinKmm

pnkprimehv + 1113944

lisinLp

nl h

nml

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(66)

For m isinL

fm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinL

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

gm(p) 1113944nisinN

log2 σnm + 1113944

lprimeisinLm

pnlprime h

nllprime

111386811138681113868111386811138681113868111386811138682

+ 1113944kisinK

pnk h

nmk

111386811138681113868111386811138681113868111386811138682⎛⎝ ⎞⎠

(67)

where fm(p) and gm(p) are concave functions Rm(p) hasthe difference form of concave functions [26] and gm(p)

satisfies the inequality

gm(p)legm p(k)1113872 1113873 + nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873 (68)

e dimension of the vector nablagTm(p) is (K + L)N and

nablagTm(p(k)) represents the gradient vector of function gm(p)

at p p(k) According to this approximation the lowerboundary of the rate for link m is Rm(p)leRm(p)

Rm(p) fm(p) minus gm p(k)1113872 1113873 minus nablagT

m p(k)1113872 1113873 p minus p(k)

1113872 1113873

forallm isinKcupL

(69)

According to the given power pk problem P4 is con-verted into the following problem CP4

minps

α 1113944lisinLi

sl + 1113944lisinLi

1113944nisinN

pnl

st Rl(p) + δminus 1l sl geRmin

l foralll isinLi

1113944nisinN

pnl le 1 minus sl( 1113857Pmax foralll isinLi

Rk(p)geRmink forallk isinK

1113944nisinN

pnk lePmax forallk isinK

sl isin [0 1] foralll isinLi

(70)

It is easy to verify that it is a convex optimization problemwhich can be solved by standard convex optimization

Complexity 7

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

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Operations ResearchAdvances in

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

techniques such as the interior point method e solvingprocess of problem P4 is described in Algorithm 3

e complexity of iterative computation in this algo-rithm is O(L) the complexity of solving convex optimiza-tion by using interior point method is O(N3M35) and thetotal computational complexity of solving problem P4 isO(LN3M35) in polynomial time

5 Numerical Simulation

In order to test the performance of proposed algorithmswe perform numerical simulation based on MATLABplatform In the wireless cellular network that supportsD2D communication the coverage radius of the basestation is 500m the number of cellular links is K 4 thenumber of D2D links is L 26 and the number of sub-carriers is N 5e maximum transmission power of theuser is 23 dBm the distance between D2D transmittingendpoint and receiving endpoint is randomly distributedbetween 10m and 50m and the cellular users are evenlydistributed in the cell e numerical simulation pa-rameters are shown in Table 1 e minimum rate re-quirement of each cellular link is Rmin

k 2 bpsHz and theminimum rate requirement of each D2D link isRmin

l 5 bpsHz All numerical results are obtained byaveraging 1000 randomly implemented channel gains Inthe numerical simulation process reverse polyblock ap-proximation algorithm is used to solve monotone opti-mization problem low complexity algorithm representsthe iterative convex optimization algorithm with lowcomplexity and maximizing energy efficiency algorithmrepresents the method which can maximize energy effi-ciency [27] e energy efficiency is defined as the ratio oftotal sum rate to overall consumed power of all D2D links[27] e comparison of access ratio of different algo-rithms is shown in Figure 2 e reverse polyblock ap-proximation algorithm has the highest access ratio theaccess ratio of the iterative convex optimization algorithmwith low complexity decreases about 5 on averagecompared with reverse polyblock approximation algo-rithm and the maximizing energy efficiency algorithm has

the lowest access ratio and is reduced by about 26 onaverage compared with reverse polyblock approximationalgorithm

e total power consumption comparison of differentalgorithms is shown in Figure 3 e power consumption ofmaximizing energy efficiency algorithm is greater than it-erative convex optimization algorithm and reverse polyblockapproximation algorithm Iterative convex optimizationalgorithm consumes about 10more power on average thanreverse polyblock approximation algorithm e powerconsumption of the maximizing energy efficiency algorithmis increased by about 30 times as much as that of reversepolyblock approximation algorithm Figure 4 presents theobjective function value of different algorithms It can beseen from this figure that reverse polyblock approximationalgorithm has the smallest objective function value followedby the iterative convex optimization algorithm and themaximum energy efficiency algorithm has the largest ob-jective function value

e relationship between objective function value andD2D bit rate requirement is shown in Table 2 As the bitrate requirement of D2D links increases the objectivefunction value of reverse polyblock approximation al-gorithm increases from 151827 to 342001 the objectivefunction value of iterative convex optimization algorithmincreases from 229407 to 388148 and the objectivefunction value of maximizing energy efficiency algorithmincreases from 1349136 to 1925249 e average objec-tive function value of maximizing energy efficiency al-gorithm is about 5 times that of iterative convexoptimization algorithm on averagee objective functionvalue of reverse polyblock approximation algorithm isreduced by about 20 on average compared with iterativeconvex optimization algorithm

In order to test the access ratio and power con-sumption of D2D links under different number of cellularusers we perform another experiment e number ofcellular links is varied from 4 to 10 and the number ofsubcarriers is 10 e access ratio and power consumptionunder different number of cellular users are shown inFigures 5 and 6 respectively As the number of cellularlinks increases the access ratio of D2D links decreases and

Step 1 Given link L1 L initial power p(0) 0 and iteration times i 0Step 2 Repeat

i i + 1 k 0repeatk k + 1solve problem CP4 to obtain p(k)update nablagT

m(p(k))

until convergencecalculate Rl(p) according to obtained p(k)calculate l argminlisinLi

Rl(p)Rminl if Rl(p)Rmin

l lt 1 Li LilUntil Rl(p)geRmin

l foralll isinLiOutput Li plowast p(k)

ALGORITHM 3 P4

8 Complexity

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

total power consumption increases In this case the in-terference from the cellular link increases resulting in adecrease in the access ratio of the D2D link In order to

Table 1 Numerical simulation parameters

Parameter ValueCell coverage 500mSubcarrier bandwidth 15 kHzNoise power minus 174 dBmHzPath loss index 3Path loss constant 001Maximum transmission power of cellular user 23 dBmMaximum transmission power of D2D user 23 dBmDistance between D2D transmitting endpoint toreceiving endpoint 10mndash50m

Channel fast fading Exponential distribution with mean value of 1

Shadow fading Lognormal distribution with standard deviation of8 dB

5 55 6 65 7 75 8 85 9 95 100

01

02

03

04

05

06

07

08

09

1

Bit rate requirement of each D2D link (bpsHz)

Acce

ss ra

tio

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

Figure 2 Comparison of access ratio of diumlerent algorithms

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 3 Comparison of total power consumption of diumlerentalgorithms

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

5 55 6 65 7 75 8 85 9 95 1010ndash1

100

101

102

103

Bit rate requirement of each D2D link (bpsHz)

Valu

e of o

bjec

tive f

unct

ion

Figure 4 Objective function value of diumlerent algorithms

Table 2 e relationship between objective function value and bitrate requirement

Bit raterequirementof D2D link(bpsHz)

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reversepolyblock

approximationalgorithm

5 1349136 229407 15182755 1416358 251183 1767466 1509760 262547 19125365 1570504 284875 2167247 1624909 290522 22551575 1688731 312595 2507318 1748789 323256 26453685 1796258 346119 2905429 1849805 367865 31543195 1897533 372876 32358410 1925249 388148 342001

Complexity 9

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

meet transmission rate requirements of D2D links moreenergy is required It can be observed that reverse poly-block approximation algorithm and iterative convexoptimization algorithm are superior to maximizing en-ergy eciency algorithm e objective function valueversus the number of cellular users is shown in Figure 7Table 3 presents the numerical results implying the re-lationship between objective function value and thenumber of cellular users It is also validated that reversepolyblock approximation algorithm has the best perfor-mance iterative convex optimization algorithm takes thesecond place and maximizing energy eciency algorithmhas the worst performance

6 Conclusions

In this paper the problem of D2D link access controlsubcarrier allocation and power allocation in the uplinkof single-cell D2D underlay cellular network is studiede purpose is to maximize the number of admitted D2Dlinks and reduce the power consumption of D2D links inthe system while ensuring the minimum data transmissionrate of cellular links and D2D links It is dicult to solvethe problem eumlectively so it is transformed into mono-tone optimization problem en reverse polyblock ap-proximation algorithm is used to solve this monotoneoptimization problem Because the monotone optimiza-tion problem has relatively high complexity this paperproposes an algorithm based on iterative convex opti-mization with low complexity e numerical results showthat reverse polyblock approximation algorithm has thebest performance the low complexity algorithm based oniterative convex optimization has the suboptimal per-formance and the algorithm based on energy eciencymaximization has the lowest access rate and the highestenergy consumption

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Number of cellular users

Acce

ss ra

tio

Figure 5 Access ratio versus the number of cellular users

Maximizing energy efficiency algorithmReverse polyblock approximationLow complexity algorithm

10ndash1

100

101

102

103

4 5 6 7 8 9 10Number of cellular links

Tota

l pow

er co

nsum

ptio

n (m

w)

Figure 6 Total power consumption versus the number of cellularusers

Reverse polyblock approximationLow complexity algorithmMaximizing energy efficiency algorithm

10ndash1

100

101

102

103

5 55 6 65 7 75 8 85 9 95 10Number of cellular links

Valu

e of o

bjec

tive f

unct

ion

Figure 7 Objective function value versus the number of cellular users

Table 3 e relationship between objective function value and thenumber of cellular users

e numberof cellularusers

Maximizingenergy eciency

algorithm

Lowcomplexityalgorithm

Reverse polyblockapproximation

algorithm4 1632548 261520 1969165 1730267 289304 2085206 1846074 309124 2251167 1896626 330678 2443848 2028289 363616 2614789 2138672 406358 31469610 2215871 454877 362362

10 Complexity

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e authors would like to acknowledge the support ofNatural Science Foundation of Shandong Province in China(ZR2015FL028) Project of the 13th Five-Year Planning ofEducation Science in Shandong Province (grant noYC2017081) and Science and Technology Planning Projectof Colleges and Universities in Shandong Province (grantnos J16LN59 and J15LN78)

References

[1] M N Tehrani M Uysal and H Yanikomeroglu ldquoDevice-to-device communication in 5G cellular networks challengessolutions and future directionsrdquo IEEE CommunicationsMagazine vol 52 no 5 pp 86ndash92 2014

[2] L Wei R Hu Y Qian and G Wu ldquoEnable device-to-devicecommunications underlaying cellular networks challengesand research aspectsrdquo IEEE Communications Magazinevol 52 no 6 pp 90ndash96 2014

[3] G Yu L Xu D Feng R Yin G Y Li and Y Jiang ldquoJointmode selection and resource allocation for device-to-devicecommunicationsrdquo IEEE Transactions on Communicationsvol 62 no 11 pp 3814ndash3824 2014

[4] Y Pei and Y-C Liang ldquoResource allocation for device-to-device communications overlaying two-way cellular net-worksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 7 pp 3611ndash3621 2013

[5] W Zhao and S Wang ldquoResource sharing scheme for device-to-device communication underlaying cellular networksrdquoIEEE Transactions on Communications vol 63 no 12pp 4838ndash4848 2015

[6] D Feng L Lu Y Yuan-Wu G Y Li G Feng and S LildquoDevice-to-device communications underlaying cellularnetworksrdquo IEEE Transactions on Communications vol 61no 8 pp 3541ndash3551 2013

[7] Y Gu Y Zhang M Pan and Z Han ldquoMatching and cheatingin device to device communications underlying cellularnetworksrdquo IEEE Journal on Selected Areas in Communica-tions vol 33 no 10 pp 2156ndash2166 2015

[8] H Xu W Xu Z Yang Y Pan J Shi and M Chen ldquoEnergy-efficient resource allocation in D2D underlaid cellular up-linksrdquo IEEE Communications Letters vol 21 no 3pp 560ndash563 2017

[9] T D Hoang L B Le and T Le-Ngoc ldquoResource allocationfor D2D communication underlaid cellular networks usinggraph-based approachrdquo IEEE Transactions on WirelessCommunications vol 15 no 10 pp 7099ndash7113 2016

[10] Z Yang N Huang and H Xu ldquoDownlink resource allocationand power control for device to device communication un-derlaying cellular networksrdquo IEEE Communication Lettersvol 20 no 7 pp 1449ndash1452 2016

[11] D Zhu Y Guo L Wei et al ldquoOptimal and suboptimal resourcesharing schemes for underlaid D2D communicationsrdquo WirelessPersonal Communications vol 98 no 3 pp 2799ndash2817 2018

[12] T-W Ban and B C Jung ldquoOn the link scheduling for cellular-aided device-to-device networksrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 11 pp 9404ndash9409 2016

[13] Y Qian T Zhang and D He ldquoResource allocation formultichannel device-to-device communications underlayingQoS-protected cellular networksrdquo IET Communicationsvol 11 no 4 pp 558ndash565 2017

[14] Y Hao Q Ni H Li S Hou and G Min ldquoInterference-awareresource optimization for device-to-device communicationsin 5G networksrdquo IEEE Access vol 6 pp 78437ndash78452 2018

[15] Z Zhou K Ota M Dong and C Xu ldquoEnergy-Efficientmatching for resource allocation in D2D enabled cellularnetworksrdquo IEEE Transactions on Vehicular Technologyvol 66 no 6 pp 5256ndash5268 2017

[16] P S Bithas K Maliatsos and F Foukalas ldquoAn SINR-awarejoint mode selection scheduling and resource allocationscheme for D2D communicationsrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4949ndash4963 2019

[17] X Diao J Zheng Y Wu and Y Cai ldquoJoint computing re-source power and channel allocations for d2d-assisted andNOMA-based mobile edge computingrdquo IEEE Access vol 7pp 9243ndash9257 2019

[18] H Zheng S Hou H Li Z Song and Y Hao ldquoPower al-location and user clustering for uplink MC-NOMA in D2Dunderlaid cellular networksrdquo IEEE Wireless CommunicationsLetters vol 7 no 6 pp 1030ndash1033 2018

[19] R Wang J Liu G Zhang S Huang and M Yuan ldquoEnergyefficient power allocation for relay-aided D2D communica-tions in 5G networksrdquo China Communications vol 14 no 7pp 54ndash64 2017

[20] Y Li T Jiang M Sheng and Y Zhu ldquoQoS-aware admissioncontrol and resource allocation in underlay device-to-devicespectrum-sharing networksrdquo IEEE Journal on Selected Areasin Communications vol 34 no 11 pp 2874ndash2886 2016

[21] X Li W Zhang H Zhang and W Li ldquoA combining calladmission control and power control scheme for D2Dcommunications underlaying cellular networksrdquo ChinaCommunications vol 13 no 10 pp 137ndash145 2016

[22] Y-F Liu ldquoDynamic spectrum management a completecomplexity characterizationrdquo IEEE Transactions on Infor-mation lteory vol 63 no 1 pp 392ndash403 2017

[23] Y-F Liu andY-HDai ldquoOn the complexity of joint subcarrier andpower allocation for multi-user OFDMA systemsrdquo IEEE Trans-actions on Signal Processing vol 62 no 3 pp 583ndash596 2014

[24] S Hayashi and Z-Q Luo ldquoSpectrum management for in-terference-limited multiuser communication systemsrdquo IEEETransactions on Information lteory vol 55 no 3pp 1153ndash1175 2009

[25] Y J Zhang L Qian and J Huang ldquoMonotonic optimizationin communication and networking systemsrdquo Foundationsand Trends in Networking vol 7 no 1 pp 1ndash75 2012

[26] H H Kha H D Tuan and H H Nguyen ldquoFast globaloptimal power allocation in wireless networks by local DCprogrammingrdquo IEEE Transactions on Wireless Communica-tions vol 11 no 2 pp 510ndash515 2012

[27] J Hu W Heng X Li and J Wu ldquoEnergy-Efficient resourcereuse scheme for D2D communications underlaying cellularnetworksrdquo IEEE Communications Letters vol 21 no 9pp 2097ndash2100 2017

Complexity 11

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom