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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.
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