1

Joint Network Optimization and Downlink

Beamforming for CoMP Transmissions using Mixed

Integer Conic ProgrammingYong Cheng, Student Member, IEEE, Marius Pesavento, Member, IEEE, and Anne Philipp

Abstract—Coordinated multipoint (CoMP) transmission is apromising technique to mitigate intercell interference and toincrease system throughput in single frequency reuse networks.Despite the remarkable benefits, the associated operational costsfor exchanging user data and control information between multi-ple cooperating base stations (BSs) limit practical applications ofCoMP processing. To facilitate wide usage of CoMP transmission,we consider in this paper the problem of joint network opti-mization and downlink beamforming (JNOB), with the objectiveto minimize the overall BS power consumption (including theoperational costs of CoMP transmission) while guaranteeingthe quality-of-service (QoS) requirements of the mobile stations(MSs). We address this problem using a mixed integer second-order cone program (MI-SOCP) framework and develop anextended MI-SOCP formulation that admits tighter continuousrelaxations, which is essential for reducing the computationalcomplexity of the branch-and-cut (BnC) method. Analytic studiesof the MI-SOCP formulations are carried out. Based on theanalyses, we introduce efficient customizing strategies to furtherspeed up the BnC algorithm through generating tight lowerbounds of the minimum total BS power consumptions. Forpractical applications, we develop polynomial-time inflation- anddeflation procedures to compute high-quality solutions of theJNOB problem. Numerical results show that the inflation- anddeflation procedures yield total BS power consumptions that areclose to the lower bounds, e.g., exceeding the lower bounds byabout 12.9% and 9.0%, respectively, for a network with 13 BSsand 25 MSs. Simulation results also show that minimizing thetotal BS power consumption results in sparse network topologiesand reduced operational overhead in CoMP transmission, andthat some of the BSs are switched off when possible.

Index Terms—Coordinated Multipoint Transmission, NetworkOptimization, Downlink Beamforming, Mixed Integer ConicProgramming, Low-complexity Heuristic Algorithms

I. INTRODUCTION

Coordinated multipoint (CoMP) processing is widely rec-

ognized as an effective mechanism for managing intercell

Manuscript received Oct. 17, 2012; revised Mar. 2, 2013 and Apr. 19, 2013;accepted Apr. 21, 2013. The associate editor coordinating the review of thispaper and approving it for publication was Prof. Anthony So. This work wassupported by the European Research Council (ERC) Advanced InvestigatorGrants Program under Grant 227477-ROSE, and the LOEWE Priority ProgramCocoon (www.cocoon.tu-darmstadt.de). Preliminary results of this work werepresented at the conferences WSA’12 [1] and ICASSP’12 [2].

Copyright (c) 2012 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Y. Cheng and M. Pesavento are with the Communication Systems Group,and A. Philipp is with the Dept. of Math., Technische Universitat Darmstadt,64283 Darmstadt, Germany. Emails: pesavento, cheng@nt.tu-darmstadt.de,aphilipp@mathematik.tu-darmstadt.de.

interference (ICI) and improving system throughput in cel-

lular networks with universal frequency reuse [3]–[20]. The

potential of CoMP transmission has been validated in both

theoretic studies [3]–[5] and field trials [3], [6], [7], and

CoMP processing has therefore already been included in

the emerging wireless communication standards, e.g., LTE-

Advanced [8]. While CoMP operation with full cooperation

between BSs that jointly serve users offers significant increases

in network capacity and cell-edge throughput, it induces also

considerable operational overhead, such as power expended

in collecting and exchanging channel state information (CSI)

among multiple BSs and MSs, signaling beamforming weights

and forwarding user data to multiple cooperating BSs [3], [20].

To balance the benefits and the operational costs, CoMP

processing shall be carried out among a limited number of

cooperating BSs, resulting in the so-called partial BS coop-

eration designs. Several partial BS cooperation schemes have

been proposed in the literature, see, e.g., [3], [9]–[20]. Those

existing contributions can roughly be categorized into two

classes, namely coordinated beamforming [3], [9], [10] and

clustered BS cooperation [3], [11]–[20]. In the coordinated

downlink beamforming designs, the beamforming weights of

the MSs are jointly designed across the network, but each MS

is served by a single BS and therefore there is no need to

route payload data or control information, e.g., beamforming

weights, corresponding to one MS over the backhaul network

to multiple BSs [3], [9], [10]. In the clustered BS cooperation

frameworks, CoMP processing is implemented within clusters

of BSs, with full BS cooperation inside each cluster and no

cooperation between clusters [3], [11]–[20]. Since the CoMP

operation is restricted to a small number of BSs in each cluster,

the communication overhead of CoMP processing is bounded

by the size of the BS clusters [3], [11]–[20].

While the existing approaches [3], [9]–[20] can alleviate

the additional expenses in CoMP transmission to certain

extent, several important issues remain open. For instance, in

coordinated beamforming [3], [9], [10], the performance of

cell-edge MSs may still suffer from ICI and large pathloss, as

in conventional cellular systems. Even though cell-edge MSs

can enjoy the performance gain from CoMP processing in the

clustered BS cooperation frameworks [3], [11]–[20], the MSs

located at the cluster edges still suffer from ICI and large

pathloss. In addition, determining the optimal size of the BS

clusters is a challenging open problem [3], [11]–[20]. More

recently, mechanisms to optimize BS selection and multicell

beamforming are proposed in [18]–[20] to reduce the overhead

2

of CoMP transmission, in which the BS selection is carried out

based on the solution of an optimization problem that gives

preference to sparse beamforming vectors [18]–[20]. However,

the sparsity patterns of the beamformers are more appropriate

for antenna selection, rather than for BS selection or network

topology optimization.

In contrast to the existing contributions [3]–[20], we propose

in this paper a systematic approach to find the optimal tradeoff

between the gain and the overhead of CoMP transmission.

Specifically, we consider the problem of joint network topol-

ogy optimization and downlink beamforming (JNOB), with

the objective to minimize the overall BS power consumption

(including the overhead of CoMP operation) while ensur-

ing the quality-of-service (QoS) requirements of the MSs.

The JNOB problem under consideration includes coordinated

beamforming [3]–[20], and full BS cooperation [3]–[5] as

special cases. In other words, in our systematic approach,

the number of cooperating BSs that transmit to each MS

is optimally determined on-the-fly according to the system

parameters and the channel conditions. In addition, we also

consider the possibility of switching off the power amplifiers

(PAs) of the BSs in the JNOB problem formulation to further

reduce BS power dissipations, which is not considered in the

previous works [3]–[20]. The major contributions of this paper

can be summarized as follows.

• In our JNOB approach we explicitly take into account the

operational overhead of CoMP transmission and consider

switching off the PAs of the BSs when minimizing the

total BS power consumption.

• We address the JNOB problem using a MI-SOCP ap-

proach [21] proposing a standard big-M MI-SOCP for-

mulation that supports the convex continuous relaxation

based BnC method [21]–[23].

• Based on the big-M formulation, we introduce auxiliary

variables and develop an extended MI-SOCP formula-

tion [23], also known as perspective formulation [23],

[24], which exhibits several appealing properties that are

exploited in the numerical algorithms.

• We conduct analytic studies to show that the extended

MI-SOCP formulation admits tighter continuous relax-

ations [21]–[23] than that of the big-M MI-SOCP for-

mulation and thus yields significantly reduced compu-

tational complexity when applying the customized BnC

procedure.

• The insights of the analyses allow us to introduce several

customizing techniques to further speed up the BnC

algorithm [21]–[23] by generating tight lower bounds of

the minimum total BS power consumptions. The tight

lower bounds also serve as the benchmarks for evaluating

the low-complexity heuristic algorithms.

• We propose polynomial-time inflation- and deflation pro-

cedures that yield with very low computational complex-

ity high-quality solutions of the JNOB problem, which

are suitable for practical applications.

Extensive simulations are carried out to evaluate the de-

veloped algorithms and confirm the analytic studies. The

commercial solver CPLEX [25] is used in our numerical

experiments for benchmarking purpose. The simulation re-

sults show that the proposed polynomial-time inflation- and

deflation procedures achieve total BS power consumptions

that are very close to the lower bounds, e.g., exceeding the

lower bounds by less than 12.9% and 9.0%, respectively,

for a large-scale network with 13 BSs and 25 MSs. The

proposed heuristic algorithms outperform the BS clustering

schemes of [15], [18]–[20] in terms of the achieved total

BS power consumption. The reduction in the computational

complexity of the extended formulation over the standard big-

M formulation when applying the BnC method is confirmed in

the simulations. Numerical results also show that minimizing

the total BS power consumptions results in sparse network

topologies rather than full BS cooperation. We observe that the

network topologies become sparser as the power consumption

overhead associated with CoMP transmission is increased, and

that some of the BSs are switched off when possible to reduce

the overall BS power consumptions.

The rest of this paper is organized as follows. In Section II,

we introduce the network model and formulate the JNOB

problem as a MI-SOCP. We discuss the continuous relaxation

of the JNOB problem and provide a brief overview of the BnC

algorithm in Section III. We then introduce auxiliary variables

to develop an extended MI-SOCP formulation and conduct

analytic comparisons of the two MI-SOCP formulations in

Section IV. In Section V, we present several techniques to cus-

tomize the BnC algorithm implemented in the solver CPLEX

to solve the JNOB problem. Polynomial-time inflation- and

deflation procedures for computing high-quality solutions of

the JNOB problem are considered in Section VI. Numerical

results along with discussions are presented in Section VII.

Finally, we conclude the paper in Section VIII.

Notations: Throughout this paper, R and C denote the sets

of real and complex numbers, respectively. The transpose

and Hermitian of the vector q are denoted by qT and qH ,

respectively. Re· and Im· represent respectively the real-

and imaginary parts of a complex variable.

II. SYSTEM MODEL AND PROBLEM STATEMENT

A. Network Model

Considering a cellular network consisting of L multiple-

antenna BSs and K single-antenna MSs, where the lth BS

is equipped with Ml ≥ 1 transmit antennas, ∀l ∈ L ,

1, 2, · · · , L. Similar to [3], [11]–[13], it is assumed that

the BSs are mutually connected over a BS network interface

(e.g., the X2-type interface in LTE systems [8]), and therefore

the data of all MSs can be made available at each BS with

associated backhauling costs [3]. The L BSs are assumed to

be synchronized so that CoMP processing can be employed

for downlink data transmissions [3], [11]–[13].

Let hk,l ∈ CMl×1 denote the frequency-flat quasi-

static channel vector between the lth BS and the kth MS,

∀l ∈ L, k ∈ K , 1, 2, · · · ,K, and define hk ,[hTk,1, hT

k,2, · · · , hTk,L

]T∈ CM×1 as the aggregate channel

vector of the kth MS, ∀k ∈ K, with M ,∑L

l=1 Ml.

Accordingly, we denote wk,l ∈ CMl×1 as the beamforming

3

vector (i.e., antenna weights) used at the lth BS for trans-

mitting data to the kth MS, ∀l ∈ L, k ∈ K, and define

wk ,[wT

k,1, wTk,2, · · · , wT

k,L

]T∈ CM×1 as the collection

of all beamforming weights corresponding to the kth MS,

∀k ∈ K. When all BSs share the same frequency bands

and CoMP processing is employed in the downlink data

transmissions, the received signal yk ∈ C at the kth MS can

be written as (see, e.g., [3], [11]–[13])

yk = hHk wkxk +

K∑

j=1,j 6=k

hHk wjxj + zk, ∀k ∈ K (1)

where xk ∈ C denotes the normalized data symbol designated

for the kth MS with unit-power, i.e., E|xk|

2= 1, and zk ∈

C stands for the additive white Gaussian noise (AWGN) at the

kth MS, with mean zero and variance σ2k, ∀k ∈ K.

Similar to the existing works [3]–[20], it is assumed that

the data symbols for different MSs are mutually statistically

independent and also independent from the noise, and single

user detection is adopted at the MSs, i.e., the co-channel

interference in (1) is treated as noise at the MSs. When

the channel vectors hk, ∀k ∈ K are quasi-static and the

beamformers wk, ∀k ∈ K are adaptive only to the instan-

taneous channel vectors, the received signal-to-interference-

plus-noise-ratio (SINR) at the kth MS, denoted by SINRk,

can be expressed as (see, e.g., [3], [11]–[13])

SINRk ,

∣∣hHk wk

∣∣2∑K

j=1,j 6=k

∣∣hHk wj

∣∣2 + σ2k

, ∀k ∈ K. (2)

We remark that when the lth BS does not participate in

transmitting data to the kth MS in CoMP transmission, i.e.,

the lth BS is not assigned to the kth MS, for some l ∈ L and

k ∈ K, then the equality wk,l = 0 shall hold.

Throughout this paper, it is assumed that there exists a

central processing node (CPN), which has knowledge of

the instantaneous channel vectors hk, ∀k ∈ K. This is a

common assumption made in the existing contributions, see,

e.g., [3]–[20]. The CPN dynamically designs the optimal

network topology by assigning a single or multiple BSs to each

MS, and computes the corresponding optimal beamforming

vectors wk, ∀k ∈ K.

B. BS Power Consumption Model

According to [3], [26]–[30], the power consumption of a

cellular BS can be categorized into non-transmission related

power dissipations (e.g., battery backup costs) and transmis-

sion related power consumptions (e.g., signal processing over-

head and power amplifier costs). The non-transmission related

power consumption, i.e., the offset power, can be treated as a

constant [3], [26]–[30], while the transmission related power

consumption of a BS depends on the activities of the power

amplifier (PA). The PA (and also the RF chain) of a BS may

be in one of the three states, namely (i) powered off (OFF),

(ii) powered on but not transmitting, i.e., idle (IDL), and (iii)

powered on and transmitting. We introduce the binary variable

bl ∈ 0, 1 to indicate that the PA of the lth BS is switched

on with bl = 1, and bl = 0 otherwise, ∀l ∈ L. Further, we

adopt the binary indicatorsak,l ∈ 0, 1, ∀k ∈ K, ∀l ∈ L

to represent BS assignments, with ak,l = 1 meaning that the

lth BS is assigned to the kth MS, and ak,l = 0 otherwise.

In case that ak,l = 0, the equalities wk,l = 0 shall hold.

Clearly, if the PA of the lth BS is powered off, the lth BS

cannot be assigned to any MSs, i.e., ak,l = 0, ∀k ∈ K. Hence,

the case of bl = 0 implies that ak,l = 0,wk,l = 0, ∀k ∈ K.The aforementioned properties regarding the binary variables

ak,l, bl, ∀k ∈ K, ∀l ∈ L can be summarized into the

following conventions:

wk,l = ak,lwk,l, ∀k ∈ K, ∀l ∈ L (3)

bl

K∑

k=1

‖wk,l‖22 =

K∑

k=1

‖wk,l‖22, ∀l ∈ L (4)

bl

K∑

k=1

ak,lP(CMP)k,l =

K∑

k=1

ak,lP(CMP)k,l , ∀l ∈ L (5)

where the user-specific constant P(CMP)k,l represents the fixed

power consumption for forwarding data and the beamforming

weights wk,l of the kth MS to the lth BS, i.e., the constantsP

(CMP)k,l , ∀k ∈ K, ∀l ∈ L

model the operational overhead

associated with CoMP transmission.

Let the constants P(OFT)l , P

(IDL)l , and P

(TPA)l denote the

offset power, the idle-state PA power consumption, and the

power required to turn off and on the PA, respectively, of the

lth BS, ∀l ∈ L. We consider here the scenarios that P(TPA)l <

P(IDL)l , ∀l ∈ L, so that powering off an idle-state PA can

indeed save power [26]–[28], [31]. With the constant 1/Λl

denoting the PA efficiency, the total power consumption of

the lth BS, denoted by P(TOT)l , can then be expressed as

(see, e.g., [3], [26]–[30])

P(TOT)l , P

(OFT)l + bl

(P

(IDL)l + Λl

K∑

k=1

‖wk,l‖22

)+

(1 − bl)P(TPA)l + bl

K∑

k=1

ak,lP(CMP)k,l

= P(OFT)l + blP

(IDL)l +

Λl

K∑

k=1

‖wk,l‖22 +

K∑

k=1

ak,lP(CMP)k,l , ∀l ∈ L (6)

where Eqs. (4) and (5) are used in the development of

Eq. (6), with the new constants P(OFT)l , P

(OFT)l + P

(TPA)l

and P(IDL)l , P

(IDL)l − P

(TBA)l > 0. Since the constants

P(OFT)l , ∀l ∈ L

are immaterial to the network optimization

problem, for ease of elaboration, it is assumed that P(OFT)l =

0, ∀l ∈ L, and we define the total BS power consumption

function f(ak,l, bl, wk,l

)as

f (ak,l, bl, wk,l) ,

L∑

l=1

blP(IDL)l +

L∑

l=1

(Λl

K∑

k=1

‖wk,l‖22 +

K∑

k=1

ak,lP(CMP)k,l

). (7)

4

C. Joint Network Optimization and Downlink Beamforming

In order to limit the overall power dissipations, the cellular

network shall be operated in a power-efficient way. Towards

this end, we consider here the network optimization problem

with the objective to minimize the overall power consumptions

of the L BSs while ensuring the minimum QoS requirements

of the K MSs. Similar to [3], [10], [20], [32], we adopt the

following QoS constraints for the K MSs:

SINRk =

∣∣hHk wk

∣∣2∑K

j=1,j 6=k

∣∣hHk wj

∣∣2 + σ2k

≥ Γ(MIN)k , ∀k ∈ K (8)

where the constant Γ(MIN)k > 0 denotes the minimum SINR

requirement of the kth MS, and SINRk is defined in Eq. (2).

We observe from Eqs. (6) and (8) that the beamformers are

phase-invariant in the sense that if the beamformerswk, ∀k ∈

K

are feasible for the SINR constraints (8), the beamformerswke

θk√−1, ∀k ∈ K

also satisfy the SINR requirements (8),

∀θk ∈ (0, 2π], ∀k ∈ K. Further, the beamformerswk, ∀k ∈

K

andwke

θk√−1, ∀k ∈ K

result in the same total per-

BS power consumption (6). Hence, without loss of generality,

the phase of the beamformer wk can be chosen such that the

term hHk wk is real and non-negative, ∀k ∈ K, and the SINR

constraints defined in (8) can be rewritten as second-order

cone (SOC) constraints (see, e.g., [20], [32])∥∥[hH

k W, σk

]∥∥2≤ γkRehH

k wk, ∀k ∈ K (9a)

ImhHk wk = 0, ∀k ∈ K (9b)

where the matrix W ∈ CM×K and the constant γk > 1 are

respectively defined as

W , [w1, w2, · · · , wK ] (10)

γk ,

√1 + 1/Γ

(MIN)k , ∀k ∈ K. (11)

With the BS power consumption model (6) and the SINR

constraints (9), the JNOB problem can be formulated as the

following MI-SOCP [21]–[23]

Φ(bmi) , minwk,l,ak,l,bl

f(ak,l, bl, wk,l

)(12a)

s.t.∥∥[hH

k W, σk

]∥∥2≤ γkRehH

k wk, ∀k ∈ K (12b)

ImhHk wk = 0, ∀k ∈ K (12c)√√√√

K∑

k=1

‖wk,l‖22 ≤ bl

√P

(MAX)l , ∀l ∈ L (12d)

‖wk,l‖2 ≤ ak,l

√P

(MAX)l , ∀k ∈ K, ∀l ∈ L (12e)

ak,l ∈ 0, 1, bl ∈ 0, 1, ∀k ∈ K, ∀l ∈ L (12f)

ak,l ≤ bl, ∀k ∈ K, ∀l ∈ L (12g)

L∑

l=1

ak,l ≥ 1, ∀k ∈ K (12h)

where the constraints (12d) denote the per-BS sum-power

constraints, with the constant P(MAX)l denoting the maxi-

mum transmission power of the lth BS, and the objective

function f (ak,l, bl, wk,l) is defined in Eq. (7). The

constraints in Eqs. (12g) and (12h) are redundant and can be

removed without loss of generality, i.e., Eqs. (12g) and (12h)

represent problem-specific cuts, which will be discussed in

Section III-B. Note that the on-off constraints in (12e) imple-

ment the well-known big-M method [22], [23] that is used in

problem (12) to ensure that the beamforming vector wk,l = 0

if the indicator ak,l = 0 (see Eq. (3)), and that no additional

constraint is enforced on the vector wk,l in problem (12) when

ak,l = 1. The latter property follows because the per-BS sum-

power budget P(MAX)l represents an upper bound on the term

‖wk,l‖22 according to Eq. (12d). In the following we refer

to problem (12) as the big-M integer (bmi) JNOB problem

formulation.

We remark that the JNOB problem (12) includes as special

cases the coordinated beamforming designs [3], [9], [10],

clustered BS cooperation schemes [3], [11]–[20], and full BS

cooperation scenarios [3]–[5]. Specifically, by introducing the

constraints∑L

l=1 ak,l = 1, ∀k ∈ K

,1 <

∑L

l=1 ak,l <

L, ∀k ∈ K

, and∑L

l=1 ak,l = L, ∀k ∈ K

, the proposed

formulation (12) can be reduced into the problems of coor-

dinated beamforming [3], [9], [10], (dynamically) clustered

BS cooperation [3], [11]–[20], and full BS cooperation [3]–

[5], respectively. Moreover, the proposed MI-SOCP formula-

tion (12) considers switching off the PAs of the BSs to further

save unnecessary power dissipations [28], [31], which up to

now has not been considered in CoMP transmission [3].

III. OPTIMAL SOLUTIONS VIA THE BNC METHOD

The formulated JNOB problem (12), like other MI-SOCPs,

can be solved using the convex continuous relaxation based

BnC method [21]–[23]. In this section, we first discuss the

continuous relaxation of the JNOB problem (12). Based on

that, we present a brief overview of the convex continuous

relaxation based BnC algorithm [21]–[23].

A. The Continuous Relaxation of the Big-M Formulation and

Analytic Studies

The continuous relaxation of a MI-SOCP is the SOCP

obtained by relaxing all the integer constraints [21]–[23].

Hence, the continuous relaxation of the formulated JNOB

problem (12) can be expressed as the following SOCP, which

is named as the big-M continuous relaxation (bmc):

Φ(bmc) , minwk,l,ak,l,bl

f(ak,l, bl, wk,l

)(13a)

s.t. (12b) – (12e), (12g), and (12h) (13b)

0 ≤ ak,l ≤ 1, 0 ≤ bl ≤ 1, ∀k ∈ K, ∀l ∈ L (13c)

where the variables ak,l, bl, ∀k ∈ K, ∀l ∈ L, originally con-

strained to take binary values in (12f), are relaxed to continues

variables confined in the closed interval [0, 1] in (13c).

Assume that the point characterized by the parameter tuplew

(bmc)k,l , a

(bmc)k,l , b

(bmc)l , ∀k ∈ K, ∀l ∈ L

is an optimal

solution (not necessarily unique) of the SOCP (13). Since the

objective function in (13a) is minimized, we can easily prove

5

by contradicting argument that:

L∑

l=1

b(bmc)l ≥ 1 (14)

K∑

k=1

a(bmc)k,l ≥ b

(bmc)l , ∀l ∈ L. (15)

Assume the pointw

(bmi)k,l , a

(bmi)k,l , b

(bmi)l , ∀k ∈ K, ∀l ∈ L

is an optimal solution (unnecessarily unique) of the JNOB

problem (12). We show next that the optimal objective value

of the SOCP continuous relaxation (13) is strictly smaller than

that of the JNOB problem (12) for practical systems with

CoMP transmission. Towards this end, we first present the

necessary conditions for which the JNOB problem (12) and the

associated continuous relaxation (13) achieve the same optimal

objective value, as summarized in the following theorem.

Theorem 1 (Necessary Conditions): If the JNOB problem

(12) and the associated SOCP continuous relaxation (13)

achieve the same optimal objective value, i.e., if Φ(bmi) =Φ(bmc), the following conditions must hold:

K∑

j=1

a(bmi)j,l =

K∑

j=1

a(bmi)j,m = 1, if a

(bmi)k,l = a

(bmi)k,m = 1,

for some k ∈ K, l 6= m, ∀l,m ∈ L. (16)

That is if the lth BS cooperates with the mth BS to serve the

kth MS, then the lth and the mth BSs exclusively serve the

kth MS in the case that Φ(bmi) = Φ(bmc).

Proof: Please refer to Appendix A for the proof.

We know from Theorem 1 that the special case of Φ(bmi) =Φ(bmc) may occur if each of the cooperating BSs (i.e., the BSs

that jointly serve MSs in CoMP transmission) serves only a

single MS. However, in practical systems employing CoMP

transmission the necessary conditions (16) do not hold, since

cooperating BSs usually serve multiple MSs to suppress ICI

and to improve spectral efficiency. As a result, the following

corollary represents a direct application of Theorem 1.

Corollary 1: In cellular networks with multiple MSs served

jointly by cooperating BSs in CoMP transmission, the optimal

objective value of the SOCP (13) is strictly smaller than that

of the JNOB problem (12), i.e.,

Φ(bmc) < Φ(bmi). (17)

We further observe that we can set ak,l = 1 and bl = 1,

∀k ∈ K, ∀l ∈ L, for testing the feasibility of the JNOB prob-

lem (12). That is if problem (12) is feasible, then a fully con-

nected network is a feasible network topology. This suggests

that if the SOCP (13) is feasible, e.g., with a feasible solution

given by the parameter tuplew

(bmc)k,l , a

(bmc)k,l , b

(bmc)

l , ∀k ∈

K, ∀l ∈ L

, then the pointw

(bmc)k,l , ak,l = 1, bl = 1, ∀k ∈

K, ∀l ∈ L

is a feasible solution of problem (12). As a result,

the JNOB problem (12) is feasible if and only if the associated

SOCP continuous relaxation (13) is feasible.

B. Overview of the BnC Algorithm and the Solver CPLEX

Thanks to the vast advancement of parallel computing, the

convex continuous relaxation based BnC algorithm [21]–[23],

[25] is widely adopted for solving MI-SOCPs and is imple-

mented in the commercial solvers, e.g., IBM CPLEX [25].

We present here a brief overview of the continuous relaxation

based BnC method [21]–[23], [25], based on the JNOB prob-

lem in (12) and the associated continuous relaxation in (13).

The BnC algorithm is a combination of the branch-and-

bound (BnB) and the cutting plane (CP) methods [21]–[23],

[25]. As in the BnB procedure, binary search trees consisting

of nodes are constructed in the BnC algorithm, with each

node representing the continuous relaxation, which is a SOCP

as that of the SOCP in (13), of a subproblem resulted from

fixing one or more binary variables in the original MI-

SOCP (12) [21]–[23], [25]. The BnC search tree is initialized

with one node, e.g., the root node that represents the continu-

ous relaxation (13) of the JNOB problem (12). If the solution

of the SOCP represented by a node is not integer-feasible,

the BnC procedure chooses one relaxed binary variable that

is not binary-valued in the solution to perform a branching

step. Hence, parting from the current node, two subproblems

are created by fixing the chosen variable to be one and zero,

respectively, which are represented by two descendant nodes

of the current node. This branching process is carried out

recursively at each node. Considering a minimization problem

such as the JNOB problem (12), a node and its descendants

(i.e., the subtree rooted at that node) can be removed from the

BnC search tree if one of the following pruning conditions is

satisfied [21]–[23], [25]:

(C1) The SOCP continuous relaxation represented by the

node is infeasible (deleting the node).

(C2) The solution of the SOCP continuous relaxation at

the node is integer-feasible (deleting the node and

recording the integer-feasible solution).

(C3) The optimal objective value of the SOCP con-

tinuous relaxation at the node is larger than the

best-known objective value (i.e., the smallest upper

bound) among the recorded integer-feasible solutions

(deleting the node and the associated subtree).

We know from the pruning conditions (C1) – (C3) that the

size of the search tree and computational complexity of the

BnC algorithm depend critically on the formulation of the MI-

SOCP, as well as the tightness of the continuous relaxation

of the sub-MI-SOCP at each node [21]–[23], [25]. In this

paper, the tightness of a continuous relaxation refers to the

gap between the optimal objective value of a MI-SOCP and

that of the associated continuous relaxation. For instance, the

term(Φ(bmi) − Φ(bmc)

)represents the tightness of the SOCP

continuous relaxation in Eq. (13).

The solution of the SOCP continuous relaxation at a leaf

node on the BnC search tree provides a local lower bound

on the optimal objective value of the corresponding sub-MI-

SOCP at that node [21]–[23], [25]. The minimum among the

local lower bounds represents a global lower bound (simply

called lower bound) of the original MI-SOCP [21]–[23], [25],

i.e., the JNOB problem in (12). While the local lower bounds

are important for pruning nodes and reducing the size of the

BnC search tree, the (global) lower bound is important for

computing optimality certificates [21]–[23], [25]. In the BnC

6

procedure, the (global) lower bound on the optimal objective

value of the original MI-SOCP (12) is successively improved

due to the branching on some of the relaxed binary variables.

Hence, the optimality certificate is eventually obtained as the

branching process continues.

During the searching process of the BnC algorithm, cuts

may be generated at each node. Cuts are linear (and/or convex)

constraints added to a MI-SOCP to reduce the size of the

feasible set of the associated continuous relaxations [21]–[23],

[25]. That is, cuts are constraints that are redundant (i.e., not

affecting the feasible set) for the original MI-SOCPs, but they

reduce the size of the feasible sets of the continuous relax-

ations [21]–[23], [25]. For instance, the following constraints,

i.e., the constraints in Eq. (12h):

L∑

l=1

ak,l ≥ 1, ∀k ∈ K (18)

are redundant in the JNOB problem (12), but they are not nec-

essarily satisfied in the associated continuous relaxation (13)

(see Section III-A). Hence, adding the cuts (18) into the contin-

uous relaxation (13) can cut away some non-integer solutions

and tighten the continuous relaxation (13). In addition to such

problem-specific cuts (18), there are also general cuts which

are valid for all MI-SOCPs, like Clique-cuts, and Gomory-

cuts, etc. [22], [23].

The MI-SOCP solver CPLEX implements the parallel BnC

method [21], [25]. CPLEX offers users the full control of the

BnC solution process, such as adding problem-specific cuts,

and stopping the BnC search when needed, etc. [25], which

are subject of various problem reformulations and customizing

techniques discussed later in Section IV and Section V,

respectively. Moreover, CPLEX records the best-known lower

bound computed in the BnC procedure, which can be used to

characterize the quality of the solutions found by CPLEX and

to evaluate the performance of fast heuristic algorithms.

IV. THE EXTENDED MI-SOCP FORMULATION AND

ANALYTIC STUDIES

The standard big-M formulation (12) results in loose contin-

uous relaxations (13) and very large BnC search trees [22]–

[24]. To reduce the computational complexity of the JNOB

problem (12) when applying the BnC method, in this section

we introduce auxiliary optimization variables and develop

an extended MI-SOCP formulation [22], [23], also known

as perspective formulation [23], [24], which admits tighter

continuous relaxations, and we carry out analytic comparisons

of the two MI-SOCP formulations.

A. The Extended MI-SOCP Formulation

To improve the standard big-M formulation (12), we adopt

a similar approach as in [23], [24] and introduce the auxiliary

variable tk,l ≥ 0 to model the power transmitted from the lthBS to the kth MS (i.e., the term ‖wk,l‖

22), ∀k ∈ K, ∀l ∈ L,

and use tk,l to replace the loose upper bound P(MAX)l used in

Eq. (12e) and rewrite the on-off constraints (12e) as

‖wk,l‖22 ≤ ak,ltk,l, ∀k ∈ K, ∀l ∈ L. (19)

which are equivalent to (see, e.g., [33])∥∥[2wT

k,l, ak,l − tk,l]∥∥

2≤ ak,l + tk,l, ∀k ∈ K, ∀l ∈ L. (20)

The on-off constraints (20) become SOC constraints when the

binary variables ak,l, ∀k ∈ K, ∀l ∈ L are relaxed to be

continuous variables taking values in the closed interval [0, 1].We define accordingly the new total BS power consumption

function g (ak,l, bl, tk,l) as

g(ak,l, bl, tk,l

),

L∑

l=1

blP(IDL)l +

L∑

l=1

(Λl

K∑

k=1

tk,l +

K∑

k=1

ak,lP(CMP)k,l

). (21)

With the auxiliary variables tk,l, ∀k ∈ K, ∀l ∈ L, the new

on-off constraints (20), and the new objective function (21),

we can convert the big-M MI-SOCP formulation (12) of the

JNOB problem into the following extended MI-SOCP, which

is labeled as the extended integer (exi) formulation:

Φ(exi) , minwk,l,ak,l,bl,tk,l

g(ak,l, bl, tk,l

)(22a)

s.t.∥∥[hH

k W, σk

]∥∥2≤ γkRehH

k wk, ∀k ∈ K (22b)

ImhHk wk = 0, ∀k ∈ K (22c)

K∑

k=1

tk,l ≤ blP(MAX)l , ∀l ∈ L (22d)

tk,l ≥ 0, ∀k ∈ K, ∀l ∈ L (22e)∥∥[2wTk,l, ak,l − tk,l

]∥∥2≤ ak,l + tk,l,

∀k ∈ K, ∀l ∈ L (22f)

ak,l ∈ 0, 1, bl ∈ 0, 1, ∀k ∈ K, ∀l ∈ L (22g)

ak,l ≤ bl, ∀k ∈ K, ∀l ∈ L (22h)

tk,l ≤ ak,lP(MAX)l , ∀k ∈ K, ∀l ∈ L (22i)

L∑

l=1

ak,l ≥ 1, ∀k ∈ K (22j)

where the constraints in (22d) denote the per-BS sum-power

constraints. Note that the constraints in (22h), (22i) and (22j)

represent problem-specific cuts added to the extended MI-

SOCP formulation (22) to obtain tighter continuous relax-

ations. Particularly, the constraints in (22j) are the exemplary

problem-specific cuts defined in Eq. (18) in Section IV-B.

Assume that the pointw

(exi)k,l , a

(exi)k,l , b

(exi)l , t

(exi)k,l , ∀k ∈

K, ∀l ∈ L

is an optimal solution (unnecessarily unique)

of the proposed extended JNOB problem formulation (22).

From the equivalence of Eqs. (19) and (22f), and considering

that the objective function in (22a) is minimized, we can

straightforwardly establish by contradicting argument that∥∥w(exi)

k,l

∥∥22= a

(exi)k,l t

(exi)k,l = t

(exi)k,l , ∀k ∈ K, ∀l ∈ L. (23)

We know from Eq. (23) that adding the equality constraints

‖wk,l‖22 = tk,l, ∀k ∈ K, ∀l ∈ L, will not change the optimal

solution set of the extended MI-SOCP formulation (22). How-

ever, substituting the KL equalities ‖wk,l‖22 = tk,l, ∀k ∈

K, ∀l ∈ L into MI-SOCP (22), we obtain exactly the big-

M MI-SOCP formulation (12). As a result, the extended

7

formulation (22) and the big-M formulation (12) are equivalent

in the sense that both yield the same optimal objective value,

i.e., Φ(exi) = Φ(bmi), and from an optimal solution of the

extended formulation (22), an optimal solution of the big-M

formulation (12) can directly be computed, and vice versa [33].

We remark that, although the proposed formulations in (22)

and (12) represent the same JNOB problem, the extended

formulation (22) admits tighter continuous relaxations than

that of the big-M formulation (12), which shall be analyzed in

the next subsection, and the former admits less computational

complexity than the latter when applying the BnC method, as

demonstrated in Section VII.

B. Analytic Comparison of the Two MI-SOCP Formulations

The continuous relaxation associated with the extended

formulation (22) can be expressed as the following SOCP,

referred as the extended continuous relaxation (exc):

Φ(exc) , minwk,l,ak,l,bl,tk,l

g(ak,l, bl, tk,l

)(24a)

s.t. (22b) – (22f), and (22h) – (22j) (24b)

0 ≤ ak,l ≤ 1, 0 ≤ bl ≤ 1, ∀k ∈ K, ∀l ∈ L (24c)

where the problem-specific cuts defined in Eqs. (22h), (22i)

and (22j) are added to the continuous relaxation (24) to reduce

the size of the feasible set of the SOCP (24).

Assume that the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈

K, ∀l ∈ L

is an optimal solution (not necessarily unique) of

the SOCP (24). Similar to the development of Eqs. (14), (15),

and (23), using proof-by-contradiction, the following results

can readily be established:

L∑

l=1

b(exc)l ≥ 1 (25)

K∑

k=1

a(exc)k,l ≥ b

(bmc)l , ∀l ∈ L (26)

∥∥w(exc)k,l

∥∥22= a

(exc)k,l t

(exc)k,l ≤ t

(exc)k,l , ∀k ∈ K, ∀l ∈ L. (27)

In case that there exist indices j ∈ K and m ∈ L such that

a(exc)j,m is non-integer valued, i.e., 0 < a

(exc)j,m < 1, we know

from the equalities in (27) and the constraints in (22d) that

∥∥w(exc)j,m

∥∥22< t

(exc)j,m =

∥∥w(exc)j,m

∥∥22

a(exc)j,m

(28)

K∑

k=1

∥∥w(exc)k,m

∥∥22< b(exc)m P (MAX)

m . (29)

Eq. (28) suggests that for a non-integer variable a(exc)j,m , the

objective value in (24a) is strictly larger than that of (13a)

at the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈ K, ∀l ∈ L

.

Eq. (29) further reveals that the feasible set described by

Eqs. (22d) and (22f) when projected onto the variables

wk,l, ak,l, bl, ∀k ∈ K, ∀l ∈ L is always contained in the

corresponding feasible set defined by Eqs. (12d) and (12e).

We know directly from Eqs. (22h) and (22i) that

a(exc)k,l ≤ b

(exc)l , ∀k ∈ K, ∀l ∈ L (30)

t(exc)k,l ≤ a

(exc)k,l P

(MAX)l , ∀k ∈ K, ∀l ∈ L. (31)

Eqs. (27) and (31) together imply that

∥∥w(exc)k,l

∥∥22≤(a(exc)k,l

)2P

(MAX)l , ∀k ∈ K, ∀l ∈ L (32)

and Eqs. (22d), (27), and (30) together suggest that

K∑

k=1

∥∥w(exc)k,l

∥∥22≤(b(exc)l

)2P

(MAX)l , ∀l ∈ L. (33)

Eqs. (32) and (33), together with the constraints

(22h), and (22j), which are respectively the same as

the constraints (12g) and (12h), suggest that the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , ∀k ∈ K, ∀l ∈ L

, i.e., the projection

of the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈ K, ∀l ∈ L

,

satisfies all the constraints in (13) and therefore it is a feasible

solution of the SOCP (13). Based on this result, we can

compare the tightness of the continuous relaxations in (13)

and (24), as summarized in the following theorem.

Theorem 2 (Tighter Continuous Relaxation): The optimal

objective value of the extended continuous relaxation (exc)

in Eq. (24) is no smaller than that of the big-M continuous

relaxation (bmc) in Eq. (13), i.e., it holds that

Φ(exc) ≥ Φ(bmc). (34)

Proof: Please refer to Appendix B for the proof.

We know from Theorem 2 that the extended continuous

relaxation (24) provides a larger lower bound Φ(exc) on the

optimal objective value Φ(exi) = Φ(bmi) than the correspond-

ing lower bound Φ(bmc) provided by the big-M continuous

relaxation (13). We can further show that the optimal objective

value of the SOCP continuous relaxation (24) is strictly larger

than that of the SOCP continuous relaxation (13) for cellular

networks employing CoMP transmission. To this end, we first

make use of Eq. (27) to identify the necessary conditions

for the special case of Φ(bmc) = Φ(exc) to hold, which is

summarized in the following theorem.

Theorem 3 (Necessary Conditions): If the SOCP continu-

ous relaxations (13) and (24) achieve the same optimal objec-

tive value, i.e., if Φ(bmc) = Φ(exc), then it holds that

a(exc)k,l ∈ 0, 1, b

(exc)l ∈ 0, 1, ∀k ∈ K, ∀l ∈ L (35)

Φ(bmc) = Φ(bmi) = Φ(exi) = Φ(exc) (36)

K∑

j=1

a(exc)j,l =

K∑

j=1

a(exc)j,m = 1, if a

(exc)k,l = a

(exc)k,m = 1,

for some k ∈ K, l 6= m, ∀l,m ∈ L. (37)

That is in the case that Φ(bmc) = Φ(exc), the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈ K, ∀l ∈ L

and the pro-

jected pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , ∀k ∈ K, ∀l ∈ L

are

optimal solutions of problems (22) and (12), respectively.

Further, the special case of Φ(bmc) = Φ(exc) may occur if

each of the cooperating BSs (i.e., the BSs that jointly serve

MSs in CoMP transmission) exclusively serves a single MS.

8

Proof: Please refer to Appendix C for the proof.

It is important to note that in wireless networks employing

CoMP transmission, the cooperating BSs usually serve more

than one MS to mitigate ICI and to improve spectral efficiency,

and therefore the necessary conditions in Eq. (37) do not hold.

As a result, the following corollary can directly be obtained

from Theorem 3.

Corollary 2: In cellular systems with BSs collaboratively

serving multiple MSs in CoMP transmission, the lower bound

of the minimum total BS power consumption Φ(bmi) = Φ(exi)

provided by the SOCP (24) is strictly larger than that given

by the SOCP (13), i.e.,

Φ(bmc) < Φ(exc) ≤ Φ(bmi) = Φ(exi). (38)

The advantages of the extended formulation (22) over the

standard big-M formulation (12) in terms of computational

complexity when applying the BnC method will be further

confirmed with numerical results in Section VII.

V. TECHNIQUES FOR CUSTOMIZING THE BNC

ALGORITHM

We introduce in this section several customizing strategies

to further speed up the parallel BnC algorithm implemented in,

e.g., the solver CPLEX [25], to solve the JNOB problem (22).

The customizing techniques also enable the BnC algorithm to

compute tight lower bounds on the minimum total BS power

consumptions, which can be used to evaluate the performance

of fast heuristic algorithms.

A. Customized Optimality Criterion

Define Ψ and Ψ as the objective value of the best-known

integer-feasible solution of the JNOB problem (22) and the

largest (global) lower bound of the optimal objective value

Φ(exi), respectively, computed in the BnC procedure. By

definition, we have 0 < Ψ ≤ Φ(exi) ≤ Ψ. A commonly used

optimality criterion for MI-SOCPs is the relative mixed integer

program (MIP) gap, defined as [22], [25]: 1−Ψ/Ψ.

Let the constant ǫ > 0 denote the predefined relative

optimality tolerance. According to [22], [25], an integer-

feasible solution computed in the BnC procedure is declared

as an optimal solution of the JNOB problem (22) if

1−Ψ

Ψ≤ ǫ. (39)

We know from Eq. (39) that it is of great interest to find

high quality integer-feasible solutions with small Ψ and to

compute a large (global) lower bound Ψ, which speeds up the

process of computing the optimality certificate [22], [25].

B. Customized Node Selection and Branching Rules

The computational complexity of solving the JNOB prob-

lem (22) with the BnC method depends on the total number of

nodes on the BnC search tree that are visited. We can reduce

the number of nodes that need to be processed by customizing

the BnC algorithm according to the specific characteristics of

the JNOB problem (22). Several customizing strategies can

be applied to control the execution of the BnC search pro-

cess, e.g., defining the branching priorities. The customizing

strategies are supported by the solver CPLEX [25].

When the SOCP at a node of the BnC search tree is solved, a

decision needs to be taken on which of the non-integer valued

variable among the relaxed binary variables in the solution

to branch, i.e., which variable to fix to integer value in the

next step of the BnC algorithm. Branching variable selection

at a node is carried out according to the branching priorities

of the (relaxed) binary variables. At each branching step, the

variable that has the largest branching priority among all the

non-integer valued relaxed binary variables is selected.

Recall that the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈

K, ∀l ∈ L

represents an optimal solution of the SOCP con-

tinuous relaxation (24) and therefore the vectorsw

(exc)k,l ∀k ∈

K, ∀l ∈ L

can be treated as the virtual beamformers

under a fully connected network. Due to the specific scaler

ambiguity of the variablesa(exc)k,l , ∀k ∈ K, ∀l ∈ L

and

t(exc)k,l , ∀k ∈ K, ∀l ∈ L

expressed in the left equality of

Eq. (27), it is generally not useful to choose a variable to

branch based solely on the values ofa(exc)k,l , ∀k ∈ K, ∀l ∈ L

.

Hence, to determine proper branching priorities of the non-

integer valued relaxed binary variables, we define in this paper

the incentive measure, denoted by Υk,l, of assigning the lthBS to serve the kth MS (i.e., setting ak,l = 1) as:

Υk,l ,

∑K

j=1

∣∣hHj,lw

(exc)k,l

∣∣2

Λlt(exc)k,l + P

(CMP)k,l

, ∀k ∈ K, ∀l ∈ L. (40)

The numerator of Eq. (40) represents the total power received

at the K MSs from the beamformer w(exc)k,l , and the denom-

inator of Eq. (40) can be interpreted as the power expended

to obtain this total received power. As a result, the incentive

measure in Eq. (40) can be interpreted as the normalized

system utility obtained from assigning the lth BS to the kth

MS. In other words, the incentive measure Υk,l represents the

normalized importance of the link between the lth BS and the

kth MS to the entire network and to the JNOB problem (22).

Similarly, we define the incentive measure Ωl of switching

on the PA of the lth BS (i.e., setting bl = 1) as:

Ωl ,

∑K

k=1

∑K

j=1

∣∣hHj,lw

(exc)k,l

∣∣2

Λl

∑K

k=1 t(exc)k,l + P

(IDL)l

, ∀l ∈ L. (41)

The numerator of Eq. (41) represents the total power received

at the K MSs when the lth BS is switched on and transmitting,

and the denominator of Eq. (41) represents the total power

expended at the lth BS when it is transmitting. Hence, the

incentive measure Ωl given in Eq. (41) can be interpreted as

the normalized system utility that can be potentially gained

from powering on the lth BS. In other words, the incentive

measure Ωl represents the normalized importance of the lthBS to the whole network and to the JNOB problem (22).

Intuitively, the relaxed binary variables that have large

impacts (i.e., large incentive measures) on the JNOB prob-

lem (22) shall be processed first. We propose here to carry out

variable selection in the BnC procedure based on the proposed

incentive measures defined in Eqs. (40) and (41). Specifically,

9

we define the branching priority, denoted as Υk,l, associated

with the (relaxed) binary variable ak,l as

Υk,l ,

K∑

j=1

L∑

m=1

I (Υj,m ≤ Υk,l) , ∀k ∈ K, ∀l ∈ L (42)

where the indicator function I (Υj,m ≤ Υk,l) is defined as

I (Υj,m ≤ Υk,l) =

1, if Υj,m ≤ Υk,l

0, otherwise.(43)

Accordingly, we define the branching priority Ωl of the

(relaxed) binary variable bl as

Ωl , maxj∈K,m∈L

Υj,m +

L∑

m=1

I (Ωm ≤ Ωl) , ∀l ∈ L (44)

where the term maxj∈K,m∈L Υj,m enforces larger branching

priorities of the variables bl, ∀l ∈ L than that of the variables

ak,l, ∀k ∈ K, ∀l ∈ L, so that the PA of a BS is powered on

(off) before assigning (unassigning) the BS to any MSs.

We remark that the proposed branching prioritizing prin-

ciples in (42) and (44) take into account not only the

CSI hk,l, ∀k ∈ K, ∀l ∈ L, but also the system parametersΛl, P

(IDL)l , P

(CMP)k,l , ∀k ∈ K, ∀l ∈ L

. In addition, the de-

pendence of the branching priorities (42) and (44) on the SINR

requirementsΓ(MIN)k , ∀k ∈ K

is implicitly incorporated

through the virtual beamformersw

(exc)k,l , ∀k ∈ K, ∀l ∈ L

,

which are obtained from solving the SOCP in Eq. (24).

C. Integer-Feasible Initializations of the BnC Algorithm

According to the pruning conditions (C3) specified in

Section III-B, high-quality integer-feasible solutions can also

reduce the number of visited nodes in the BnC method

and therefore reduce the computational complexity of the

BnC algorithm. Good integer-feasible initializations can be

obtained through low-complexity heuristic algorithms, which

are considered in the next section.

VI. THE POLYNOMIAL-TIME HEURISTIC ALGORITHMS

Despite the significant enhancements in the improved for-

mulation (22) as compared to the original problem formula-

tion (12), the computational complexity of solving the JNOB

problem (22) may still be prohibitive for large networks in

practice. Moreover, it is often encountered that even if the

optimal solution is found by the BnC procedure, the optimality

certificate can generally not be reached in reasonable time.

This motivates the development of polynomial-time algorithms

that yield close-to-optimal solutions of the JNOB problem

for practical applications in large-scale networks. Further, the

solutions found by the low-complexity algorithms can be

utilized to initialize the BnC algorithm to reduce the computa-

tional complexity. We propose in this section polynomial-time

inflation- and deflation procedures [34]–[36].

In the polynomial-time algorithms, the SOCP (24) is solved

at first to obtain the solutionw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈

K, ∀l ∈ L

. If the SOCP (24) is infeasible, the JNOB

problem (22) is also infeasible. For such cases, certain MS

admission mechanisms (see, e.g., [34]) can be employed to

select a subset of the K MSs to serve, which however is out

of the scope of this paper. We consider in this paper the cases

that the SOCP (24) and the JNOB problem (22) are feasible

and leave MS admission control for future work.

A. The SOCP based Inflation Procedure

We propose here a fast inflation procedure to compute high-

quality integer-feasible solutions of the JNOB problem (12).

Let the pointa(n)k,l , b

(n)l , ∀k ∈ K, ∀l ∈ L

denote the

solution of the relaxed binary variables obtained in the nth

iteration. The inflation procedure initializes with none of

the BSs assigned to the MSs, i.e., a(0)k,l = 0, b

(0)l = 0,

∀k ∈ K, ∀l ∈ L, and a sufficiently large objective value Φ(0),

e.g., set Φ(0) ,∑L

l=1

(P

(IDL)l +ΛlP

(MAX)l +

∑K

k=1 P(CMP)k,l

).

The BSs are gradually assigned to the MSs by fixing one of

the zero-valued variables ina(n−1)k,l , ∀k ∈ K, ∀l ∈ L

to one

in the nth (n ≥ 1) iteration of the inflation procedure.

Apparently, it is a critical decision how to choose and fix to

one a particular zero-valued variables in the seta(n−1)k,l , ∀k ∈

K, ∀l ∈ L

in the nth iteration. Intuitively, we shall consider

the variables that have large impacts on the JNOB prob-

lem (22). Hence, we propose here to select variables according

to the associated incentive measures (40). That is, in the nth it-

eration, the variable that has the largest incentive measure (40)

among the zero-valued variables ina(n−1)k,l , ∀k ∈ K, ∀l ∈ L

is chosen and set to one. If two or more zero-valued variables

in the seta(n−1)k,l , ∀k ∈ K, ∀l ∈ L

have the same largest

incentive measure, we randomly pick one of them. Note that

according to Eqs. (5) and (22h), we need to set b(n)l = 1 if we

fix a(n)k,l = 1, for the chosen k ∈ K, l ∈ L.

After obtaining the binary variablesa(n)k,l , b

(n)l , ∀k ∈

K, ∀l ∈ L

in the nth iteration of the inflation procedure,

we then try to solve the following SOCP, which represents a

subproblem of the JNOB problem in (12) with all the binary

variables ak,l, bl, ∀k ∈ K, ∀l ∈ L fixed:

Φ(n) , minwk,l

f(

a(n)k,l

,b(n)l

,wk,l

)(45a)

s.t.∥∥[hH

k W, σk

]∥∥2≤ γkRehH

k wk, ∀k ∈ K (45b)

ImhHk wk = 0, ∀k ∈ K (45c)

K∑

k=1

‖wk,l‖22 ≤ P

(MAX)l , if b

(n)l = 1, ∀l ∈ L (45d)

wk,l = 0, if a(n)k,l = 0, ∀k ∈ K, ∀l ∈ L (45e)

where the total BS power consumption function

f(ak,l, bl, wk,l) is defined in Eq. (7).

If the SOCP (45) is infeasible, we set Φ(n) = Φ(0)

and proceed to the next iteration. Otherwise, after solving

problem (45), we compare the objective value Φ(n) with that of

Φ(n−1). If Φ(n) ≤ Φ(n−1), we proceed to the next iteration. If

Φ(n) > Φ(n−1), i.e., a worse solution is reached, we stop with

one-step backtracking, i.e., stop and return the objective value

Φ(n−1) and the solutionw

(n−1)k,l , a

(n−1)k,l , b

(n−1)l , ∀k ∈ K, ∀l ∈

L

. The simple necessary conditions that:∑L

l=1 a(n)k,l ≥ 1,

10

∀k ∈ K, can be verified before solving the SOCP (45) to

reduce the computational efforts. The low-complexity inflation

procedure is summarized in Alg. 1.

Initialization: Initialize a sufficiently large Φ(0),

a(0)k,l = 0, b

(0)l = 0, ∀k ∈ K, ∀l ∈ L, and n = 1.

Repeat:

Step 1: Compute: (k∗, l∗) = argmax(k,l)∈P(n)

Υk,l, with the

set P(n) ,(k, l)

∣∣k ∈ K, l ∈ L, a(n−1)k,l = 0

.

Step 2: If no indices (k∗, l∗) can be found, the

algorithm stops and returns the results of the (n− 1)th

iteration. Otherwise, update the indicators a(n)j,l = a

(n−1)j,l ,

b(n)l = b

(n−1)l , ∀j ∈ K, ∀l ∈ L, and set a

(n)k∗,l∗ = b

(n)l∗ = 1.

Step 3: Check the necessary conditions:∑L

l=1 a(n)j,l ≥ 1, ∀j ∈ K. If they are not satisfied, update

the iteration number n← n+ 1 and go back to Step 1.

Step 4: Try to solve problem (45) with the obtained

indicatorsa(n)k,l , b

(n)l , ∀k ∈ K, ∀l ∈ L

.

Step 5: If problem (45) is feasible but Φ(n) > Φ(n−1),

stop and return the results of the (n− 1)th iteration.

Step 6: Update the iteration number n← n+ 1.

Alg. 1: The proposed low-complexity inflation procedure

Since there are KL binary indicatorsak,l, ∀k ∈ K, ∀l ∈

L

, the worst-case computational complexity of the inflation

procedure in Alg. 1 mainly consists in solving K(L − 1)times the SOCP (45) and hence the inflation procedure is

a polynomial-time algorithm and it converges in finite iter-

ations [33]. We will show via numerical examples in Sec-

tion VII that Alg. 1 yields high-quality solutions of prob-

lem (22) with very low computational complexity.

B. The SOCP based Deflation Procedure

Similar to the inflation procedure, we develop here an effi-

cient deflation procedure to compute near-optimal solutions of

the considered JNOB problem (22). In contrast to the inflation

procedure in Alg. 1, the deflation procedure starts with a

fully connected network topology, i.e., a(0)k,l = 1, b

(0)l = 1,

∀k ∈ K, ∀l ∈ L, which is inspired by the fact that if the JNOB

problem (22) is feasible, then a fully-connected configuration

yields a feasible solution. The sparsity of the network topology

is then gradually increased via fixing one of the one-valued

variables in the seta(n−1)k,l , ∀k ∈ K, ∀l ∈ L

to zero in the

nth (n ≥ 1) iteration of the deflation procedure.

Similar to the inflation procedure, the performance of the

deflation procedure depends highly on the rules defining how

a particular one-valued variables ina(n−1)k,l , ∀k ∈ K, ∀l ∈ L

is chosen and set to zero in the nth iteration. Similar as in

Alg. 1, we propose here to select variables according to the

associated incentive measures defined in Eq. (40). Specifically,

in the nth iteration, the variable that has the smallest incentive

measure (40) among the one-valued variables ina(n−1)k,l , ∀k ∈

K, ∀l ∈ L

is selected and set to zero. If multiple one-valued

variables in the seta(n−1)k,l , ∀k ∈ K, ∀l ∈ L

have the same

smallest incentive measure, we randomly choose one of them.

Note that according to Eqs. (5) and (22h), we need to update

b(n)l = maxj∈K a

(n)j,l after setting a

(n)k,l = 0, for the chosen

k ∈ K, l ∈ L, in the nth iteration.

After updating the binary variablesa(n)k,l , b

(n)l , ∀k ∈ K, ∀l ∈

L

in the nth iteration, we then try to solve the SOCP (45).

If problem (45) is feasible and Φ(n) ≤ Φ(n−1), i.e., a better

solution is obtained, we record the results and proceed to the

next iteration. Conversely, if the SOCP (45) is infeasible or

if it is solved with Φ(n) > Φ(n−1), we initiate a one-step

backtracking procedure, i.e., setting a(n)k∗,l∗ = 1, b

(n)l∗ = 1,

Φ(n) = Φ(n−1), and Υk∗,l∗ = +∞ (for preventing loop), with

a(n)k∗,l∗ denoting the variable that is chosen in the nth iteration.

Similar to the inflation procedure, the necessary conditions

that∑L

l=1 a(n)k,l ≥ 1, ∀k ∈ K, can also be used here to quickly

certify the feasibility of problem (45). The low-complexity

deflation procedure is summarized in Alg. 2.

Initialization: Initialize a sufficiently large Φ(0),

a(0)k,l = 0, b

(0)l = 0, ∀k ∈ K, ∀l ∈ L, and n = 1.

Repeat:

Step 1: Compute:

(k∗, l∗) = argmin(k,l)∈Q(n)

Υk,l, s.t.∑L

m=1 ak,m ≥ 2, with the

set Q(n) ,(k, l)

∣∣k ∈ K, l ∈ L, a(n−1)k,l = 1

.

Step 2: If no indices (k∗, l∗) can be found, the

algorithm stops and returns the results of the (n− 1)th

iteration. Otherwise, update the indicators a(n)j,l = a

(n−1)j,l ,

b(n)l = b

(n−1)l , ∀j ∈ K, ∀l ∈ L, and set a

(n)k∗,l∗ = 0 and

b(n)l∗ = maxj∈K a

(n)j,l∗ .

Step 3: Try to solve problem (45) with the obtained

indicatorsa(n)k,l , b

(n)l , ∀k ∈ K, ∀l ∈ L

.

Step 4: If problem (45) is feasible but Φ(n) > Φ(n−1)

or problem (45) is infeasible, then set a(n)k∗,l∗ = 1,

b(n)l∗ = 1, Φ(n) = Φ(n−1), and Υk∗,l∗ = +∞.

Step 5: Update the iteration number n← n+ 1.

Alg. 2: The proposed low-complexity deflation procedure

The computational complexity of the deflation procedure

in Alg. 2 mainly consists in solving K(L − 1) times the

SOCP (45) since there are only KL binary variables ofak,l, ∀k ∈ K, ∀l ∈ L

and therefore the deflation procedure

is a polynomial-time algorithm [33]. In addition, we shall

show via numerical results in Section VII that the deflation

procedure yields close-to-optimal solutions of the JNOB prob-

lem (22) with very low computational complexity.

VII. NUMERICAL RESULTS AND DISCUSSIONS

In the simulations, we consider cellular networks compris-

ing 13 identical hexagonal cells with one BS located at each

cell-center. The layout of the 13 cells in a two-dimensional

coordinate system is depicted in Fig. 1 with a cell-radius of 1kilo meter (km). The MSs are uniformly randomly dropped in

the rectangular coverage area defined by the dashed lines as

shown in Fig. 1. Similar to the existing works [13], [15], [17],

[18], [20], [31], we use the following channel model: (i) the

11

3GPP LTE pathloss (PL) mode: PL = 148.1 + 37.6 log10(d)(in dB), with d (in km) denoting the BS-MS distance, (ii)

Log-norm shadowing with zero mean, 8 dB variance, (iii)

Rayleigh fading with zero mean and unit variance, (iv) transmit

antenna power gain of 9 dB and noise power σ2k = −143 dB,

∀k ∈ K. We adopt homogeneous settings: the BS transmit

power P(MAX)l = 10 dB, the PA efficiency 1/Λl = 25% [30],

the parameters Γ(MIN)k = 6 dB, P

(IDL)l = 10 dB, and

P(CMP)k,l = P (CMP), ∀k ∈ K, ∀l ∈ L, with the values

of P (CMP) listed in the figures and the tables. The relative

optimality tolerance in Eq. (39) is set as ǫ = 1%. The

simulation results presented in Sections VII-A and VII-B are

averaged over 500 Monte Carlo runs (MCRs), and the data

given in Section VII-C are averaged over 300 MCRs.

BS 1 BS 2

BS 3BS 4

BS 5

BS 6 BS 7 BS 8

BS 9

BS 10

1 km(0, 0)

2km

2km

BS 11

BS 12

BS 13

3.46 km3.46 km

Fig. 1: The layout of the 13 cells. The MSs are uniformly

dropped in the rectangular area defined by the dashed lines.

A. Performance of the Low-complexity Algorithms

We first evaluate the performance of the proposed low-

complexity algorithms in Alg. 1 and Alg. 2 in a medium-scale

network with K = 15 MSs. To provide fair comparisons with

the existing schemes [15], [18]–[20] and to further motivate the

proposed incentive measures in Eqs. (40) and (41), we consider

two reference incentive measures, namely (i) channel gain [15]

and (ii) sparsity of the beamformers [18]–[20], in Step 1 of

the inflation- and deflation procedures. The channel gain based

incentive measure [15], denoted by Υk,l, of assigning the lthBS to the kth MS (i.e., setting ak,l = 1) is defined as:

Υk,l , ‖hk,l‖2, ∀k ∈ K, ∀l ∈ L. (46)

In the sparse optimization based approach [18]–[20], the

following regularized convex problem

w

(spa)k

, argmin

wk

K∑

k=1

‖wk‖22 + ξ

K∑

k=1

‖wk‖1 (47a)

s.t. (12b), and (12c) (47b)

K∑

k=1

‖wk,l‖22 ≤ P

(MAX)l , ∀l ∈ L (47c)

is firstly solved if it is feasible to obtain the sparse beam-

formersw

(spa)k , ∀k ∈ K

under full BS cooperation, where

the large constant ξ > 0 denotes the penalty factor on the

l1-norm of the beamformers. We then define accordingly the

sparsity based incentive measure [18]–[20], denoted byΥk,l,

of assigning the lth BS to serve the kth MS as:

Υk,l ,

∥∥w(spa)k,l

∥∥1, ∀k ∈ K, ∀l ∈ L. (48)

We observe in the simulations that the performance of the

inflation- and deflation procedures employing the incentive

measure (48) is not sensitive to the penalty factor ξ, e.g.,

choosing ξ ∈ 102, 103, 104, 105 resulting in the same

performance, and we thus fix ξ = 103 in the simulations.

Fig. 2 and Fig. 3 display the total BS power consumptions

versus the system parameter P (CMP). The curves labeled with

”Lower bound by CPLEX on prob. (22)” correspond to the

largest lower bounds computed by the solver CPLEX applied

to the JNOB problem formulation (22) under the runtime limit

of 300 seconds. The BnC algorithm implemented in CPLEX is

customized according to the techniques discussed in Section V

and it is initialized with the solutions found by the proposed

deflation procedure in Alg. 2 equipped with the proposed

incentive measure in Eq. (40).

0 2 4 6 8 10

150

200

250

300

350

400

Power overhead of CoMP transmission P (CMP)

[dB]

To

tal

BS

po

wer

co

nsu

mp

tio

n [

Wat

ts]

Inflation proc. w/ incentive in Eq. (46)

Inflation proc. w/ incentive in Eq. (48)

Inflation. proc. w/ incentive in Eq. (40)

Feasible soln. by CPLEX on prob. (22)

Lower bound by CPLEX on prob. (22)

Fig. 2: The total BS power consumption vs. the parameter

P (CMP), with different incentive measures used in the inflation

procedure in Alg. 1 and K = 15 MSs.

0 2 4 6 8 10

140

160

180

200

220

240

260

280

300

Power overhead of CoMP transmission P (CMP)

[dB]

To

tal

BS

po

wer

co

nsu

mp

tio

n [

Wat

ts]

Deflation proc. w/ incentive in Eq. (46)

Deflation proc. w/ incentive in Eq. (48)

Deflation. proc. w/ incentive in Eq. (40)

Feasible soln. by CPLEX on prob. (22)

Lower bound by CPLEX on prob. (22)

Fig. 3: The total BS power consumption vs. the parameter

P (CMP), with different incentive measures used in the defla-

tion procedure in Alg. 2 and K = 15 MSs.

We observe from Fig. 2 and Fig. 3 that: (i) the inflation-

and deflation procedures employing the proposed incentive

measure in Eq. (40) outperform in terms of the achieved

total BS power consumptions their counterparts that adopt

the channel gain based incentive measure in Eq. (46) [15]

12

and the sparsity based incentive measure in Eq. (48) [18]–

[20], (ii) the deflation procedure outperforms in terms of the

achieved total BS power consumptions the inflation procedure,

and (iii) the average total BS power consumptions achieved by

the proposed inflation- and deflation procedures are very close

to the lower bounds computed by CPLEX, e.g., exceeding the

lower bounds by less than 11.7% and 7.6%, respectively, for

the considered settings.

Fig. 4 depicts the runtime of the considered schemes versus

the parameter P (CMP). Since almost the same runtime is re-

quired by the inflation procedure employing different incentive

measures, which holds also for the deflation procedure, we

plot in Fig. 4 only the runtime of the inflation- and defla-

tion procedures employing the proposed incentive measure in

Eq. (40). We observe from Fig. 4 that while the proposed

inflation- and deflation procedures yield the total BS power

consumptions that are close to that achieved by the customized

BnC method and close to the lower bounds, the inflation- and

deflation procedures admit much less computational complex-

ity and consume much less runtime, e.g., requiring respectively

less than 0.46% and 21.4% of the runtime required by the

customized BnC method.

0 2 4 6 8 1010

−1

100

101

102

Power overhead of CoMP transmission P (CMP)

[dB]

CP

U t

ime

[sec

on

ds]

Feasible soln. by CPLEX on prob. (22)

Deflation proc. w/ incentive in Eq. (40)

Inflation proc. w/ incentive in Eq. (40)

Fig. 4: The runtime of the considered schemes vs. the param-

eter P (CMP), with K = 15 MSs.

Tab. I lists the number of active BS-MS links versus the

parameter P (CMP). Here, we denote the BS-MS link between

the lth BS and the kth MS as ”active” if ak,l = 1. We

observe from Tab. I that instead of the full BS cooperation with

KL = 195 active links, the average number of active BS-MS

links obtained by applying CPLEX to the JNOB problem (22)

ranges from 21.7 to 15.7 as the power overhead of CoMP

transmission P (CMP) is increased from 0 dB to 10 dB.

This shows that partial BS cooperations and sparse network

topologies are employed in the proposed CoMP transmission

design to minimize the total BS power consumptions, and to

balance the gain and the overhead of CoMP transmission.

Tab. II lists the average number of BSs that are switched

on, i.e., the BSs that are transmitting data to the MSs, in the

proposed design. We see from Tab. II that when taking into

account the idle-state power consumptions of the PAs of the

BSs, some of the BSs are switched off to minimize the total

BS power consumptions, e.g., on average more than 37.7% of

the BSs are switched off in the proposed design.

TABLE I: The average number of active BS-MS links vs. the

parameter P (CMP), with K = 15 MSs.

P(CMP) [dB] 0 2 4 6 8 10

Inflation w/ Eq. (46) 34.7 32.9 31.1 29.8 28.9 28.2

Inflation w/ Eq. (48) 32.5 30.7 29.3 28.1 27.2 26.7

Inflation w/ Eq. (40) 30.6 27.4 24.8 22.6 21.1 19.8

Deflation w/ Eq. (46) 18.9 17.3 16.6 16.1 15.7 15.6

Deflation w/ Eq. (48) 19.8 17.9 16.8 16.1 15.7 15.4

Deflation w/ Eq. (40) 21.0 18.9 17.5 16.8 16.2 15.9

CPLEX w/ prob. (22) 21.7 19.7 18.0 17.0 16.2 15.7

TABLE II: The average number of powered on BSs vs. the

parameter P (CMP), with K = 15 MSs.

P(CMP) [dB] 0 2 4 6 8 10

Inflation w/ Eq. (46) 12.2 12.1 11.9 11.8 11.6 11.5

Inflation w/ Eq. (48) 10.9 10.9 10.8 10.8 10.8 10.8

Inflation w/ Eq. (40) 9.6 9.6 9.6 9.5 9.4 9.4

Deflation w/ Eq. (46) 10.1 9.7 9.6 9.4 9.3 9.2

Deflation w/ Eq. (48) 10.4 10.3 10.1 10.0 10.0 9.9

Deflation w/ Eq. (40) 9.4 9.2 9.1 9.1 9.0 8.9

CPLEX w/ prob. (22) 7.6 7.7 7.8 7.9 8.0 8.1

B. Comparisons of the Two Problem Formulations and the

Effectiveness of the Branching Priorities

In this subsection, we compare the two problem formu-

lations in Eqs. (12) and (22) and the associated continuous

relaxations (13) and (24), and demonstrate the effectiveness

of the proposed branching priorities in Eqs. (42) and (44).

To provide meaningful and fair comparisons, we apply the

solver CPLEX to problems (12) and (22) with (w/) and without

(w/o) issuing the branching priorities in Eqs. (42) and (44),

respectively. CPLEX is initialized with the solutions obtained

from the proposed deflation procedure equipped with the

proposed incentive measure (40), and CPLEX terminates once

a strictly better solution, i.e., a solution with strictly less total

BS power consumption, than the initial solution is reached

within the runtime limitation of 150 seconds.

We first compare the continuous relaxations (13) and (24).

Fig. 5 displays the optimal objective values of the continuous

relaxations in Eqs. (13) and (24) vs. the parameter P (CMP).

The figure clearly shows that the continuous relaxation (24)

associated with the extended formulation (22) provides strictly

larger lower bounds on the minimum total BS power con-

sumptions than that of the continuous relaxation (13) asso-

ciated with the standard big-M formulation (12). The lower

bounds given by the SOCP (24) are almost twice as large as

the lower bounds offered by the SOCP (13).

Fig. 6, Fig. 7, and Fig. 8 depict the percentages of solutions

that are strictly better than the initializations, the normalized

total BS power consumptions achieved by the considered

methods (normalized by the total BS power consumptions

achieved by the proposed deflation procedure), and the algo-

rithm runtime vs. the parameter P (CMP), respectively.

We observe from Figs. 6 – 8 that: (i) applying the BnC algo-

rithm implemented in CPLEX to the extended formulation (22)

yields strictly better solutions, i.e., solutions with strictly less

total BS power consumptions, than that computed by applying

the BnC method to the standard big-M formulation (12),

while the former requires strictly less runtime than the lat-

13

0 2 4 6 8 1050

100

150

200

250

Power overhead of CoMP transmission P (CMP)

[dB]

Op

ti.

ob

jec.

val

ues

of

the

con

t. r

elax

a.

Optimal objective value of the SOCP in Eq. (24)

Optimal objective value of the SOCP in Eq. (13)

Fig. 5: The optimal objective values of the continuous relax-

ations in Eqs. (24) and (13) vs. P (CMP), with K = 15 MSs.

ter, and (ii) employing the proposed branching priorities in

Eqs. (42) and (44) in the BnC method applied to problem (22)

achieves a larger percentage of strictly better solutions than

the initializations with much less runtime than that without

issuing the branching priorities. These observations confirm

that the extended formulation (22) admits less computational

complexity than the standard big-M formulation (12) when

applying the BnC method, and that the proposed branching

priorities (42) and (44) are very effective in reducing the

computational complexity of the BnC method.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Power overhead of CoMP transmission P (CMP)

[dB]

Per

cen

tag

e o

f st

rict

ly b

ette

r so

luti

on

s

CPLEX on prob. (22) w/ branc. priorities

CPLEX on prob. (22) w/o branc. priorities

CPLEX on prob. (12) w/ branc. priorities

CPLEX on prob. (12) w/o branc. priorities

Fig. 6: The percentage of solutions strictly better than the ini-

tial solutions obtained from the deflation procedure employing

the incentive measure (40) vs. P (CMP), with K = 15 MSs.

0 2 4 6 8 10

0.96

0.97

0.98

0.99

1

Power overhead of CoMP transmission P (CMP)

[dB]

No

rmal

ized

to

tal

BS

po

wer

co

nsu

mp

tio

n

CPLEX on prob. (12) w/o branc. priorities

CPLEX on prob. (12) w/ branc. priorities

CPLEX on prob. (22) w/o branc. priorities

CPLEX on prob. (22) w/ branc. priorities

Fig. 7: The normalized total BS power consumption vs.

P (CMP), with K = 15 MSs. Note that CPLEX stops once

a strictly better solution than the initialization is found.

0 2 4 6 8 1040

60

80

100

120

140

160

Power overhead of CoMP transmission P (CMP)

[dB]

CP

U t

ime

[sec

on

ds]

CPLEX on prob. (12) w/o branc. priorities

CPLEX on prob. (12) w/ branc. priorities

CPLEX on prob. (22) w/o branc. priorities

CPLEX on prob. (22) w/ branc. priorities

Fig. 8: The algorithm run-time vs. P (CMP), with K = 15 MSs.

Note that CPLEX terminates once a strictly better solution than

the initialization is found.

C. Performance Evaluation in a Large-Scale Network

In this subsection, we evaluate the performance of the

inflation- and deflation procedures employing the proposed

incentive measures in Eq. (40), and compare the two problems

formulations (12) and (22) in a large-scale network with

K = 25 MSs. Due to the comparably large runtime required

for the solver CPLEX to compute meaningful lower bounds

(LBs) of the total BS power consumptions, which are used

as benchmarks for the heuristic algorithms, with L = 13 BSs

and K = 25 MSs, without loss of generality, we consider two

representative values of the system parameter P (CMP), namely

P (CMP) = 2 dB and P (CMP) = 6 dB. The solver CPLEX

is initialized with the solutions computed by the proposed

deflation procedure equipped with the incentive measures (40),

and the runtime limit of CPLEX is set to 4200 seconds for

computing the lower bounds.

Tab. III lists the total BS power consumptions (Power), the

algorithm runtime (Time), the average number of active links

(AcLks), and the average number of powered on BSs (OnBSs).

We observe from Tab. III that: (i) the proposed inflation- and

deflation procedures yield total BS power consumptions that

are very close to the lower bounds, e.g., exceeding the lower

bounds by less than 12.9% and 9.0%, respectively, while the

inflation- and deflation procedures require much less runtime,

e.g., requiring respectively less than 0.19% and 5.5% of the

runtime required by the BnC method, and (ii) partial BS

cooperation and sparse network topologies are realized in the

proposed design, and about 13.1% of the BSs are switched off

to further reduce the overall BS power consumptions.

We next compare the two problem formulation in Eqs. (12)

and (22). As in Section VII-B, we apply CPLEX to prob-

lems (12) and (22) under a runtime limitation of 800 seconds,

where CPLEX is initialized with the solutions found by the

deflation procedure. CPLEX terminates once a strictly better

solution than the initialization is reached. Tab. IV lists the

percentage of solutions that are strictly better than the initial-

izations (Perct.), the total BS power consumption (Power), and

the algorithm runtime (Time).

We see from Tab. IV that applying the BnC method to the

extended formulation (22) yields much more strictly better

solutions than the initializations with much less runtime than

14

TABLE III: The total BS power consumption (Power) [Watts],

the algorithm runtime (Time) [seconds], the average number

of active links (AcLks), and the average number of powered

on BSs (OnBSs) vs. P (CMP), with K = 25 MSs.

P(CMP) Inflation Deflation CPLEX on (22)

LB[dB] w/ (40) w/ (40) w/ priorities

2

Power 271.2 264.1 253.3 245.0

Time 7.0 206.4 3727.8 –AcLks 57.2 45.3 42.0 –OnBSs 12.0 12.0 10.9 –

6

Power 385.2 368.6 354.0 335.3

Time 6.3 197.8 3967.5 –AcLks 48.4 36.0 35.8 –OnBSs 12.0 11.7 11.3 –

applying the BnC method to the big-M formulation (12). This

confirms that the extended formulation (22) admits less com-

putational complexity when applying the BnC method than

that of the big-M formulation (12) in large-scale networks.

TABLE IV: The percentage of solutions strictly better than the

initial solutions computed by the deflation procedure (Perct.),

the total BS power consumption (Power) [Watts], and the

algorithm runtime (Time) [seconds] vs. P (CMP), with K = 25MSs. Note that CPLEX terminates once a strictly better

solution than the initialization is found.

P(CMP) Deflation CPLEX on (12) CPLEX on (22)[dB] w/ (40) w/ priorities w/ priorities

2

Perct. – 0.0 84.4%

Power 264.1 264.1 260.0

Time 206.4 801.7 504.1

6

Perct. – 0.0 72.0%

Power 368.6 368.6 362.2

Time 197.8 801.8 622.9

VIII. CONCLUSION

We have considered in this paper the JNOB problem aiming

to balance the benefits and operational overhead of CoMP

transmission. The standard big-M MI-SOCP formulation (12)

and the extended MI-SOCP formulation (22) are developed

for the JNOB problem, and the advantages (e.g., admitting

tighter continuous relaxations) of the latter over the former

have been confirmed by analytic studies and numerical results.

Several techniques have been introduced to customize the BnC

algorithm implemented in CPLEX to solve the JNOB problem

and to compute tight lower bounds on the minimum total

BS power consumptions when optimality cannot be reached

due to runtime constraints. We have developed polynomial-

time inflation- and deflation procedures in Alg. 1 and Alg. 2,

respectively, to compute high-quality integer-feasible solutions

of the JNOB problem for practical applications. Simulations

results show that Alg. 1 and Alg. 2 yield with very low compu-

tational complexity the total BS power consumptions that are

close to the lower bounds, e.g., exceeding the lower bounds

by less than 12.9% and 9.0%, respectively, for a network

with L = 13 BSs and K = 25 MSs under the considered

settings. Numerical results have also confirmed the reduction

of computational complexity of the extended formulation (22)

over the big-M formulation (12) and the effectiveness of the

proposed branching priorities when applying the customized

BnC method. Finally, it has been observed in the simulations

that balancing the gain and operational overhead of CoMP

transmission results in partial BS cooperation designs and

sparse network topologies, and BSs are switched off when

possible to reduce the overall BS power consumptions in the

proposed partial BS cooperation design. The proposed MI-

SOCP approach can also be applied to other problems, e.g.,

joint beamforming and discrete rate adaptation [37], sparse

filter design [38], and sparse signal recovery [39], etc.

APPENDIX A

PROOF OF Theorem 1

Recall that the pointw

(bmi)k,l , a

(bmi)k,l , b

(bmi)l , ∀k ∈ K, ∀l ∈

L

represent an optimal solution of the JNOB problem (12).

The necessary conditions in Eqs. (16) can be proved by

contradicting argument.

Assuming that the necessary conditions (16) do not hold,

i.e., assuming that there exist two MSs with indices j, k ∈ Kand two BSs with indices m, l ∈ L such that

a(bmi)

j,l= a

(bmi)

k,l= a

(bmi)

k,m= 1. (49)

That is it is assumed that the lth BS serves the jth and the

kth MSs jointly, and the lth and the mth BSs collaboratively

serve the kth MS. Since∥∥w(bmi)

j,l

∥∥22> 0 when a

(bmc)

j,l= 1,

we know from the per-BS power constraints (12d) that:

∥∥w(bmi)

k,l

∥∥22< P

(MAX)

l= P

(MAX)

l

(a(bmi)

k,l

)2. (50)

We can then define the new variable a(bmi)

k,las: a

(bmi)

k,l,∥∥

w(bmi)

k,l

∥∥2√

P(MAX)

l

, which satisfies

0 < a(bmi)

k,l=

∥∥w(bmi)

k,l

∥∥2√

P(MAX)

l

< a(bmi)

k,l(51)

∥∥w(bmi)

k,l

∥∥22= P

(MAX)

l

(a(bmi)

k,l

)2(52)

∑

l∈L\m,l

a(bmi)

k,l+ a

(bmi)

k,l+ a

(bmi)

k,m> 1. (53)

We can then replace the variable a(bmi)

k,lin the optimal so-

lutionw

(bmi)k,l , a

(bmi)k,l , b

(bmi)l , ∀k ∈ K, ∀l ∈ L

of the JNOB

problem (12) with the variable a(bmi)

k,lto obtain a new feasible

solution of the SOCP (13), which, due to Eq. (51), achieves

a strictly smaller objective value than Φ(bmi). This, however,

contradicts with the fact that Φ(bmc) = Φ(bmi). Hence, the

lth BS cannot serve the jth and the kth MSs jointly when

Φ(bmc) = Φ(bmi). Following a similar contradicting argument,

we can prove that the mth BS must also serve exclusively the

kth MS. As a result, cooperating BSs must serve exclusively

a single MS when Φ(bmc) = Φ(bmi), i.e., the necessary

condition (16) must hold in the case that Φ(bmc) = Φ(bmi).

15

APPENDIX B

PROOF OF Theorem 2

We know from the constraints (22h), and (22j), which

are respectively the same as that of (12g) and (12h), and

Eqs. (32) and (33) that the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , ∀k ∈

K, ∀l ∈ L

, which is obtained from the projection of the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈ K, ∀l ∈ L

, is a feasible

solution of the SOCP in (13). Hence, it holds that

Φ(bmc) ≤ f(

a(exc)k,l

,b(exc)l

,w

(exc)k,l

). (54)

Eq. (27) suggests that f(

a(exc)k,l

,b(exc)l

,w

(exc)k,l

)≤

Φ(exc). Hence, we have Φ(bmc) ≤ Φ(exc).

APPENDIX C

PROOF OF Theorem 3

Recall that the pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈

K, ∀l ∈ L

represent an optimal solution of the SOCP (24).

We first prove Eq. (35). If Φ(bmc) = Φ(exc), i.e., if∥∥w(exc)k,l

∥∥22= t

(exc)k,l , ∀k ∈ K, ∀l ∈ L, we know from Eq. (27)

that the relaxed binary variablesa(exc)k,l , ∀k ∈ K, ∀l ∈ L

take values in the discrete set 0, 1. Due to Eq. (22h), this

is also true for the relaxed binary variablesb(exc)l , ∀l ∈ L

.

Hence, Eq. (35) holds in the case that Φ(bmc) = Φ(exc).

We next prove Eq. (36). We know from Eq. (35) that the

pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , t

(exc)k,l , ∀k ∈ K, ∀l ∈ L

is actu-

ally an optimal solution of the JNOB problem (22) [21]–[23]

and therefore the projected pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , ∀k ∈

K, ∀l ∈ L

is an optimal solution of the JNOB problem (12).

Hence, Eq. (36) holds.

Finally, we know from Eqs (35) and (36) that the projected

pointw

(exc)k,l , a

(exc)k,l , b

(exc)l , ∀k ∈ K, ∀l ∈ L

is an optimal

solution of problem (12) and Φ(bmc) = Φ(bmi) in case that

Φ(bmc) = Φ(exc) holds. As a result, we can directly apply

the results of Theorem 1 to obtain the necessary conditions in

Eq. (37) for the special case of Φ(bmc) = Φ(exc).

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Yong Cheng (S’09) received the B.Eng. (1st honors)and M.Phil. degrees from Zhejiang University (ZJU),Hangzhou, P.R. China, and the Hong Kong Uni-versity of Science and Technology (HKUST), HongKong, in 2006 and 2010, respectively. He is currentlya Ph.D. student at the Communication SystemsGroup, Dept. of Electrical Engineering and Informa-tion Technology, Technische Universitat Darmstadt,Darmstadt, Germany. His current research interestsmainly include mixed integer programming and con-vex optimization in signal processing and wireless

communications, multiple-antenna techniques in LTE/LTE-advanced, as wellas resource allocation and coordinated multipoint processing (CoMP) inheterogeneous networks.

Marius Pesavento (M’00) received the Dipl.-Ing.and M.Eng. degrees from Ruhr-Universitat Bochum,Germany, and McMaster University, Hamilton, ON,Canada, in 1999 and 2000, respectively, and in 2005the Dr.-Ing. degree in Electrical Engineering fromRuhr-Universitat Bochum, Germany. Between 2005and 2007, he was a Research Engineer at FAGIndustrial Services GmbH, Aachen, Germany. From2007 to 2009 he was the Director of the SignalProcessing Section at mimoOn GmbH, Duisburg,Germany. In 2010, he became a Professor for Robust

Signal Processing at the Department of Electrical Engineering and Informa-tion Technology, Darmstadt University of Technology, Darmstadt, Germany,and he is currently the Head of the Communication Systems Group. Hisresearch interests are in the area of robust signal processing and adaptivebeamforming, high-resolution sensor array processing, transceiver designfor cognitive radio systems, cooperative communications in relay networks,MIMO and multiantenna communications, space-time coding, multiuser andmulticarrier wireless communication systems (3+G), convex optimization forsignal processing and communications, statistical signal processing, spectralanalysis, parameter estimation and detection theory. Dr. Pesavento was arecipient of the 2003 ITG/VDE Best Paper Award, the 2005 Young AuthorBest Paper Award of the IEEE Transactions on Signal Processing, and the2010 Best Paper Award of the CROWNCOM conference. He is a member ofthe Editorial board of the EURASIP Signal Processing Journal, an AssociateEditor for the IEEE Transactions on Signal Processing, and a member ofthe Sensor Array and Multichannel (SAM) Technical Committee of the IEEESignal Processing Society (SPS).

Anne Philipp (S’09) received the Diploma degreein mathematics from Technische Universitat Darm-stadt, Darmstadt, Germany in 2011. From 2008 to2009, she studied mathematics at the University ofSaskatchewan, Saskatoon, Canada. She is currently aPh.D. student in the Nonlinear Optimization Group,Department of Mathematics, Technische Univer-sitat Darmstadt, Darmstadt, Germany. Her currentresearch interests include mixed integer nonlinearprogramming and semidefinite programming withapplications in signal processing and wireless com-

munications.