smoothed lp-minimization for green cloud-ran with user...

1022 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 4, APRIL 2016

Smoothed L p-Minimization for Green Cloud-RANWith User Admission Control

Yuanming Shi, Member, IEEE, Jinkun Cheng, Jun Zhang, Senior Member, IEEE, Bo Bai, Member, IEEE,Wei Chen, Senior Member, IEEE, and Khaled B. Letaief, Fellow, IEEE

Abstract—The cloud radio access network (Cloud-RAN) hasrecently been proposed as one of the cost-effective and energy-efficient techniques for 5G wireless networks. By moving the signalprocessing functionality to a single baseband unit (BBU) pool,centralized signal processing and resource allocation are enabledin cloud-RAN, thereby providing the promise of improving theenergy efficiency via effective network adaptation and interferencemanagement. In this paper, we propose a holistic sparse opti-mization framework to design green cloud-RAN by taking intoconsideration the power consumption of the fronthaul links, mul-ticast services, as well as user admission control. Specifically, wefirst identify the sparsity structures in the solutions of both thenetwork power minimization and user admission control prob-lems, which call for adaptive remote radio head (RRH) selectionand user admission. However, finding the optimal sparsity struc-tures turns out to be NP-hard, with the coupled challenges of the�0-norm-based objective functions and the nonconvex quadraticQoS constraints due to multicast beamforming. In contrast tothe previous works on convex but nonsmooth sparsity inducingapproaches, e.g., the group sparse beamforming algorithm basedon the mixed �1/�2-norm relaxation, we adopt the nonconvex butsmoothed � p-minimization (0 < p ≤ 1) approach to promote spar-sity in the multicast setting, thereby enabling efficient algorithmdesign based on the principle of the majorization–minimization(MM) algorithm and the semidefinite relaxation (SDR) technique.In particular, an iterative reweighted-�2 algorithm is developed,which will converge to a Karush–Kuhn–Tucker (KKT) point of therelaxed smoothed � p-minimization problem from the SDR tech-nique. We illustrate the effectiveness of the proposed algorithms

Manuscript received April 15, 2015; revised September 12, 2015; acceptedDecember 11, 2015. Date of publication March 24, 2016; date of currentversion May 11, 2016. This work is supported in part by the Hong KongResearch Grant Council under Grant 16200214, in part by the National BasicResearch Program of China (973 Program) under Grant 2013CB336600, inpart by the NSFC Excellent Young Investigator Award no. 61322111, in partby the Opening Research Funding of the State Key Laboratory of Networkingand Switching Technology, National Nature Science Foundation of China(NSFC) under Grant 61401249, and in part by the Specialized ResearchFund for the Doctoral Program of Higher Education (SRFDP) under Grant20130002120001.

Y. Shi is with the School of Information Science and Technology,ShanghaiTech University, Shanghai, China (e-mail: [email protected]).

J. Zhang is with the Department of Electronic and Computer Engineering,Hong Kong University of Science and Technology, Hong Kong (e-mail:[email protected]).

K. B. Letaief is with Hamad bin Khalifa University, Ar-Rayyan, Qatar, andalso with the Hong Kong University of Science and Technology, Hong Kong(e-mail: [email protected]; [email protected]).

J. Cheng, B. Bai, and W. Chen are with the Department of ElectronicEngineering, Tsinghua University, Beijing, China (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2016.2544578

with extensive simulations for network power minimization anduser admission control in multicast cloud-RAN.

Index Terms—5G networks, green communications, Cloud-RAN, multicast beamforming, sparse optimization, semidefi-nite relaxation, smoothed � p-minimization, and user admissioncontrol.

I. INTRODUCTION

T HE GREAT success of wireless industry is driving theproposal of new services and innovative applications,

such as Internet of Things (IoT) and mobile Cyber-Physicalapplications, which yield an exponential growth of wirelesstraffic with billions of connected devices. To handle the enor-mous mobile data traffic, network densification and hetero-geneity supported by various radio access technologies (e.g.,massive MIMO [2] and millimeter-wave communications [3])have become an irreversible trend in 5G wireless networks [4].However, this will have a profound impact and bring formidablechallenges to the design of 5G wireless communication systemsin terms of energy efficiency, capital expenditure (CAPEX),operating expenditure (OPEX), and interference management[5]. In particular, the energy consumption will become pro-hibitively high in such dense wireless networks in the era ofmobile data deluge. Therefore, to accommodate the upcomingdiversified and high-volume data services in a cost-effective andenergy-efficient way, a paradigm shift is required in the designof 5G wireless networks.

By leveraging the cloud computing technology [6], the cloudradio access network (Cloud-RAN) [7], [8] is a disruptive tech-nology to address the key challenges of energy efficiency in5G wireless networks. Specifically, by moving the basebandunits (BBUs) into a single BBU pool (i.e., a cloud data cen-ter) with shared computation resources, scalable and parallelsignal processing, coordinated resource allocation and coop-erative interference management algorithms [9], [10] can beenabled among a large number of radio access points, therebysignificantly improving the energy efficiency [1], [11] and spec-tral efficiency [12]. As the conventional compact base stationsare replaced by low-cost and low-power remote radio heads(RRHs), which are connected to the BBU pool through high-capacity and low-latency fronthaul links, Cloud-RAN providesa cost-effective and energy-efficient way to densify the radioaccess networks [5].

While Cloud-RAN has a great potential to reduce the energyconsumption of each RRH, with additional fronthaul link com-ponents and dense deployment of RRHs, new challenges arisefor designing green Cloud-RAN. In particular, instead of only

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SHI et al.: SMOOTHED L P-MINIMIZATION FOR GREEN CLOUD-RAN 1023

minimizing the total transmit power consumption via coordi-nated beamforming [13], the network power consumption con-sisting of the fronthaul link power consumption and the RRHpower consumption should be adopted as the performance met-ric for designing green Cloud-RAN [1], [11], [14]. To minimizethe network power consumption, a group sparse beamformingframework was proposed in [1] to adaptively select the activeRRHs and the corresponding fronthaul links via controlling thegroup sparsity structures of the beamforming vectors. Such anidea of exploiting sparsity structures in the solutions has alsodemonstrated its effectiveness in solving other mixed combi-natorial optimization problems in Cloud-RAN, e.g., the dataassignment problem [15] and the joint uplink and downlinknetwork power minimization problem [11].

Although network adaption by selecting the active RRHsprovides a promising way to minimize the network power con-sumption, it is critical to maximize the user capacity (i.e., thenumber of admitted users) when the network power minimiza-tion problem is infeasible [16]–[18]. This infeasibility issuemay often occur in the scenarios with a large number of mobiledevices requesting high data rates or some users with unfavor-able channel conditions. Furthermore, exploiting the benefits ofintegrating diversified unicast and multicast services [19] hasbeen well recognized as a promising way to improve the energyefficiency and user capacity, and thus multicast beamform-ing should be incorporated in Cloud-RAN. From the systemdesign perspective, to design a green Cloud-RAN with mul-ticast transmission, a holistic approach is needed to enablenetwork adaptation for RRH selection and user admission ina unified way.

Unfortunately, such design problems fall into the categoryof highly complicated mixed combinatorial optimization prob-lems. The key observation to address this challenge is that thenetwork power minimization and user admission control can beachieved by adaptively selecting the active RRHs and admittingthe mobile users (MUs) via controlling the sparsity structures inthe corresponding solutions. Specifically, for the network powerminimization problem, selecting active RRHs is equivalent tocontrolling the group sparsity structure in the aggregative mul-ticast beamforming vector [1]. That is, all the beamformingcoefficients of a particular RRH that is switched off need tobe set to zeros simultaneously. For the user admission controlproblem that is needed when the network power minimizationproblem is infeasible, maximizing the number of admitted usersis equivalent to minimizing the number of violated QoS con-straints [20]. Mathematically, this is the same as minimizing thesparsity of the auxiliary vector indicating the violations of theQoS constraints. Based on these observations, we will thus for-mulate both design problems as sparse optimization problemsbased on the �0-norm minimization in a unified framework,based on which efficient algorithms will be developed.

A. Contributions

Based on the above discussions, we propose a sparse opti-mization framework to design a multicast green Cloud-RANas shown in Fig. 2, thereby enabling adaptive RRH selec-tion and user admission. However, in contrast to the previous

works on the multicast beamforming problem [21] with con-vex objectives but nonconvex QoS constraints and the groupsparse beamforming problem [1] with convex QoS constraintsbut nonconvex objective functions in unicast Cloud-RAN, todesign efficient algorithms for the resulting sparse optimiza-tion problems in multicast Cloud-RAN, we need to addressthe following coupled challenges in a unified way: Nonconvexquadratic QoS constraints due to multicast transmission;

Combinatorial objective functions for RRH selection anduser admission.

In particular, we summarize the major contributions as fol-lows:

1) Sparse optimization framework based on the �0-normminimization is proposed to design a multicast greenCloud-RAN by adaptive RRH selection and user admis-sion via controlling the sparsity structures of the solu-tions.

2) To address the combinatorial challenges in the objectivefunctions, we propose a nonconvex but smoothed �p-minimization approach to induce the sparsity structuresin the solutions. The main advantage of this method isthat it helps develop the group sparse inducing penaltywith quadratic forms in the multicast beamforming vec-tors. Therefore, the objective function in the resultinggroup sparse inducing optimization problem is compat-ible with the nonconvex quadratic QoS constraints. TheSDR technique can then be adopted to solve the resultingnonconvex quadratic group sparse inducing optimizationproblem.

3) To address the challenges of the nonconvex smoothedobjective functions and the nonconvex quadratic QoSconstraints, we propose an iterative reweighted-�2 algo-rithm to solve the resulting nonconvex smoothed �p-minimization problems based on the principle of theMM algorithm and the SDR technique. This algorithmis proven to converge to a KKT point of the relaxedsmoothed �p-minimization problems over the convexconstraint set using the SDR technique.

4) Simulation results will demonstrate the effectiveness ofthe proposed algorithms to minimize the network powerminimization and maximize the user capacity, and theirperformance gains compared with the existing convexapproximation approaches. In particular, the proposedalgorithms can achieve near-optimal performance in thesimulated settings.

B. Related Works

Sparse optimization by exploiting sparsity structures of thesolutions has been proven to be very powerful to solve varioushard optimization problems in machine learning, compressivesensing and high-dimensional statistics [22]. This approachhas recently received enormous attentions in designing wire-less networks, i.e., the group sparse beamforming approachfor network adaption [1], [11], [23], and data assignment inwireless backhaul networks [15], [24]. In particular, the con-vex relaxation approaches, e.g., the �1-minimization [17], themixed �1/�2-norm [1] and the mixed �1/�∞-norm [25], have


become popular due to the computational efficiency as well asperformance guarantees in some scenarios [26].

To further improve the performance, there has been a greatinterest in applying nonconvex approaches in sparse optimiza-tion [27]–[30] by enhancing sparsity. In particular, it is observedthat the nonconvex �p-minimization approach performs betterthan the traditional convex �1-minimization especially when theunderlying model is very sparse [28]. Motivated by this result,we adopt the �p-minimization approach to closely approxi-mate the resulting �0-norm based sparse optimization problemsfor multicast green Cloud-RAN. Furthermore, to deal with theunique challenges with the coupled nonconvex constraints andcombinatorial objectives, thereby enabling efficient algorithmdesign, we will use a smoothed version of the �p-minimizationapproach to induce the sparsity structures in the solutions. Notethat the existing work on the group sparse beamforming [1]can only handle problems with a combinatorial objective andconvex seconder-order cone QoS constraints in unicast Cloud-RAN, and thus cannot be directly applied in the setting ofmulticast Cloud-RAN with nonconvex QoS constraints.

C. Organization

The remainder of the paper is organized as follows.Section II presents the system model and problem formula-tions. Section III presents an algorithmic framework for net-work power minimization and user admission control basedon the smoothed �p-minimization. The iterative reweighted-�2 algorithm is developed in Section IV. Simulation resultswill be demonstrated in Section V. Finally, conclusions anddiscussions are presented in Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we first introduce the system model of themulticast Cloud-RAN. Then, the network power minimizationproblem is formulated. For the scenario when it is not feasi-ble to serve all the MUs, the user admission control problem isformulated. It will be revealed that both problems are sparseoptimization problems, for which unique challenges will beidentified.

A. System Model

We consider a multicast Cloud-RAN with L multi-antennaRRHs and K single-antenna MUs, where the l-th RRH isequipped with Nl antennas, as shown in Fig. 1. Let L ={1, . . . , L} and N = {1, . . . , K } denote the sets of all the RRHsand all the MUs, respectively. We assume that the K MUs formM non-overlapping and non-empty multicast groups, which aredenoted as {G1,G2, . . . ,GM } with ∪iGi = N and G j ∩ G j = ∅with Gm as the set of MUs in the m-th multicast group. LetM = {1, . . . , M} be the set of all the multicast groups. Weconsider the downlink transmission, and the centralized signalprocessing is performed at the BBU pool [1], [7].

The propagation channel from the l-th RRH to the k-th MUis denoted as hkl ∈ C

Nl ,∀k, l. Let vlm ∈ CNl be the transmit

Fig. 1. The architecture of multicast Cloud-RAN, in which, all the RRHs areconnected to a BBU pool through high-capacity and low-latency optical fron-thaul links. All the MUs in the same dashed circle form a multicast group andrequest the same message. To enable full cooperation among all the RRHs, it isassumed that all the user data and channel state information (CSI) are availableat the BBU pool.

beamforming vector from the l-th RRH to MUs in multicastgroup Gm . Then the transmit signal at the l-th RRH is given by

xl =M∑

m=1

vlmsm,∀l ∈ L, (1)

where sm ∈ C is the encoded information symbol for the multi-cast group m with E[|sm |2] = 1. Assume that all the RRHs havetheir own transmit power constraints, which form the followingfeasible set of the beamforming vectors vlm’s,

V ={

vlm ∈ CNl :

M∑m=1

‖vlm‖22 ≤ Pl ,∀l ∈ L

}, (2)

where Pl > 0 is the maximum transmit power of the l-th RRH.The received signal ykm ∈ C at MU k in the m-th multicast

group is given by

ykm =L∑

l=1

hHklvlmsm +

∑i =m

L∑l=1

hHklvli si + nk,∀k ∈ Gm, (3)

where nk ∼ CN(0, σ 2k ) is the additive Gaussian noise at MU

k. We assume that sm’s and nk’s are mutually independent andall the users apply single user detection. Therefore, the signal-to-interference-plus-noise ratio (SINR) of MU k in multicastgroup m is given by

�k,m(v) = vHm�kvm∑

i =m vHi �kvi + σ 2

k

,∀k ∈ Gm, (4)

where �k = hkhHk ∈ C

N×N with N =∑Ll=1 Nl and hk =

[hHk1, hH

k2, . . . , hHkL ]H ∈ C

N as the channel vector consisting ofthe channel coefficients from all the RRHs to MU k, vm =[vH

1m, vH2m, . . . , vH

Lm]H ∈ CN is the beamforming vector con-

sisting of the beamforming coefficients from all the RRHsto the m-th multicast group, and v = [vl ]L

l=1 ∈ CM N is the

aggregative beamforming vector with vl = [vlm]Mm=1 ∈ C

M Nl

as the beamforming vector consisting of all the beamformingcoefficients from the l-th RRH to all the multicast groups.


B. Network Power Minimization for Green Cloud-RAN

With densely deployed RRHs, it is critical to enable energy-efficient transmission via centralized signal processing at theBBU pool. Coordinated beamforming among RRHs will helpreduce the transmit power. Due to the mobile data traffic varia-tions in temporal and spatial domains, it is also critical to enablenetwork adaptation to switch off some RRHs and the corre-sponding fronthaul links to save power [1], [31]. Thus, we needto consider the network power consumption when designinggreen Cloud-RAN, which is defined as the following combi-natorial composite function parameterized by the beamformingcoefficients [1],

p(v) = p1(v)+ p2(v), (5)

with the total transmit power consumption, denoted as p1(v),and the total relative fronthaul links power consumption,denoted as p2(v), given as

p1(v) =L∑

l=1

M∑m=1

1

ηl‖vlm‖22, (6)

and

p2(v) =L∑

l=1

Pcl I (Supp(v) ∩ Vl = ∅), (7)

respectively. Here, I (Supp(v) ∩ Vl = ∅) is an indicator func-tion that takes value zero if Supp(v) ∩ Vl = ∅ (i.e., all thebeamforming coefficients at the l-th RRH are zeros, indi-cating that the corresponding RRH is switched off) andone otherwise, where Vl is defined as Vl := {M ∑L−1

l=1 Nl +1, . . . , M

∑Ll=1 Nl}, and Supp(v) is the support of the vec-

tor v. In (6), ηl > 0 is the drain inefficiency coefficient of theradio frequency power amplifier with the typical value as 25%,and Pc

l ≥ 0 in (7) is the relative fronthaul link power con-sumption [1], which is the static power saving when both theRRH and the corresponding fronthaul link are switched off.For the passive optical fronthaul network [32], Pc

l is given by(P rrh

a,l + P fna,l)− (P rrh

s,l + P fns,l) with P rrh

a,l (P rrhs,l ) and P fn

a,l (P fns,l) as

the power consumptions for the l-th RRH and the l-th fronthaullink in the active (sleep) mode, respectively. The typical valuesare P rrh

a,l = 6.8W , P rrhs,l = 4.3W , P fn

a,l = 3.85W , P fns,l = 0.75W

and Pcl = 5.6W [1], [32]. Note that the energy consumption

of the optical fronthaul links should depend on the receiv-ing periods and data transmission [32], which is a functionof the beamforming vectors. Given the target SINR require-ments (γ1, γ2, . . . , γK ) for all the MUs, to design a multicastgreen Cloud-RAN, we propose to minimize the network powerconsumption subject to the QoS constraints and the RRH trans-mit power constraints. Specifically, we have the following QoSconstraints

�k,m(v) ≥ γk,∀k ∈ Gm, m ∈M, (8)

which can be rewritten as the following quadratic constraints

Fk,m(v) = γk

(∑i =m

vHi �kvi + σ 2

k

)− vH

m�kvm ≤ 0, (9)

for any k ∈ Gm and m ∈M, which are nonconvex. Therefore,the network power minimization problem can be formulated as

minimizev∈V

p1(v)+ p2(v)

subject to Fk,m(v) ≤ 0,∀k ∈ Gm, m ∈M, (10)

which is highly intractable due to the nonconvex combinatorialcomposite objective and the nonconvex quadratic QoS con-straints (9). When there are some RRHs needed to be switchedoff to minimize the network power consumption, the solutionof problem (10) has the group sparsity structure [1]. That is, allthe beamforming coefficients in vl , which forms a group at thel-th RRH, are set to be zeros simultaneously if the l-th RRHneeds to be switched off.

Therefore, inspired by the fact that the solution of prob-lem (10) has the group sparsity structure in the aggregativebeamforming vector v, the weighted mixed �1/�2-norm wasproposed in [1] to relax the combinatorial composite functionas the tightest convex surrogate to induce the group sparsitystructure in the solution v to guide the RRH selection, defined as

J(v) =L∑

l=1

ρl‖vl‖2, (11)

where ρl > 0 is the weight for the beamforming coefficientsgroup vl at RRH l.

To handle the nonconvex QoS constraints in (9), we proposeto lift the problem to higher dimensions with variables Qm =vmvH

m ∈ CN×N ,∀m, which will help to apply the semidefinite

relaxation (SDR) technique. However, we cannot extract thevariables Qm’s from the non-smooth mixed �1/�2-norm (11).Therefore, this convex relaxation approach cannot be directlyapplied to solve the network power minimization problem inmulticast Cloud-RAN. Instead, to leverage the SDR technique,we need to develop a new group sparsity inducing approachwith quadratic forms of the beamforming vectors, which willbe presented in Section III and form one major contribution ofthis paper.

C. User Admission Control

With QoS constraints for potentially a large number of MUsin the serving area, it may happen that the network powerminimization problem (10) is infeasible from time to time. Insuch scenarios, the design problem will be to maximize theuser capacity (i.e., the maximum number of MUs that can besupported) via user admission control [16], [17], while serv-ing these MUs with the minimum transmit power. While thisaspect is ignored in [1], it is critical for practical applications.Mathematically, by adding auxiliary variables xk’s to the right-hand side of the corresponding inequalities in (9), to maximizethe number of admitted MUs is equivalent to minimize the num-ber of non-zero xk’s [20, Section 11.4]. Therefore, the useradmission control problem can be formulated as the followingsparsity minimization problem,

minimizev∈V,x∈RK+

‖x‖0

subject to Fk,m(v) ≤ xk,∀k ∈ Gm, m ∈M, (12)


where RK+ represents the K -dimensional nonnegative real vec-

tor. Once the admitted MUs are determined, coordinated beam-forming can then be applied to minimize the total transmitpower. The solution x = [xk] of problem (12) has the individualsparsity structure, i.e., xk = 0 indicates that the k-th MU can beadmitted. Therefore, the sparsity level will be increased if moreMUs can be admitted.

Observing that both the sparse optimization problems (10)and (12) possess the same structure with nonconvex quadraticconstraints and combinatorial objectives, in this paper, we willpropose a unified way to handle them based on a smoothed �p-minimization approach.

D. Problem Analysis

In this subsection, we analyze the unique challenges ofthe network power minimization problem (10) and the useradmission control problem (12) in the context of multicastCloud-RAN. In particular, the differences from the previousworks [1], [11] will be highlighted.

1) Nonconvex Quadratic Constraints: The physical-layermulticast beamforming problem [21] yields nonconvexquadratic QoS constraints (9), while only unicast services wereconsidered in [1], [11]. The SDR technique [33] proves tobe an effective way to obtain good approximate solutions tothese problems by lifting the quadratic constraints into higherdimensions, which will also be adopted in this paper.

2) Combinatorial Objective Functions: Although the SDRtechnique provides an efficient way to convexify the nonconvexquadratic QoS constraints in problems (10) and (12), the inher-ent combinatorial objective functions still make the resultingproblems highly intractable. While the �1-norm can be adoptedto relax the �0-norm in problem (12) after SDR, which is alsoknown as the sum-of-infeasibilities relaxation in optimizationtheory [17], [20], the convex relaxation approach based on thenon-smooth mixed �1/�2-norm [1] cannot be applied to prob-lem (10), as we cannot extract the variables Qm’s after SDR.

Therefore, in this paper, we propose a new powerfulapproach to induce the sparsity structures in the solutions forboth problems (10) and (12), which is based on a smoothed �p-norm [27]. The main advantage of this method is that it canhelp develop group sparsity inducing penalties with quadraticforms in the beamforming vectors, thereby leveraging the SDRtechnique to relax the nonconvex quadratic QoS constraintsfor problem (10). Furthermore, by adjusting the parameter p,this approach has the potential to yield a better approximationfor the original �0-norm based objectives, thereby providingimproved solutions for problems (10) and (12). The smoothed�p-minimization framework will be presented in Section III,while the iterative reweighted-�2 algorithm will be developedin Section IV to solve the smoothed �p-minimization problem.

III. A SMOOTHED �p-MINIMIZATION FRAMEWORK FOR

NETWORK POWER MINIMIZATION WITH USER

ADMISSION CONTROL

In this section, we first present the smoothed �p-minimization method as a unified way to induce sparsity

structures in the solutions of problems (10) and (12), therebyproviding guidelines for RRH selection and user admission.After obtaining the active RRHs and admitted MUs by perform-ing the corresponding selection procedure, we will minimizethe total transmit power consumption for the size-reduced net-work. The algorithmic advantages and performance improve-ment of the proposed smoothed �p-minimization based frame-work will be revealed in the sequel.

A. Smoothed �p-Minimization for Sparsity Inducing

To promote sparse solutions, instead of applying the con-vex �1-minimization approach, we adopt a nonconvex �p-minimization (0 < p ≤ 1) approach to seek a tighter approx-imation of the �0-norm in the objective functions in problems(10) and (12), [27]. This is motivated by the fact that the �0-norm ‖z‖0 is the limit as p→ 0 of ‖z‖p

p in the sense of‖z‖0 = limp→0 ‖z‖p

p = limp→0∑ |zi |p. We thus adopt ‖x‖p

pas the optimization objective function to seek sparser solutions.Furthermore, to enable efficient algorithm design as well asinduce the quadratic forms in the resulting approximation prob-lems, we instead adopt the following smoothed version of ‖z‖p

pto induce sparsity:

f p(z; ε) :=m∑

i=1

(z2i + ε2)p/2, (13)

for z ∈ Rm and some small fixed regularizing parameter ε > 0.

Based on the smoothed �p-norm (13), we will present the algo-rithmic advantages of the smoothed �p-minimization approachin Section IV.

1) Smoothed �p-Minimization for Group Sparsity Inducing:For network power minimization, to seek quadratic forms ofbeamforming vectors in the objective functions to leverage theSDR technique for the non-convex quadratic QoS constraints,we adopt the smoothed �p-norm f p(z; ε) (13) to induce groupsparsity in the aggregative beamforming vector v for problem(10), resulting the following optimization problem:

minimizev∈V

L∑l=1

ρl(‖vl‖22 + ε2)p/2

subject to Fk,m(v) ≤ 0,∀k ∈ Gm, m ∈M. (14)

The induced (approximated) group sparse beamformers willguide the RRH selection. The resulting problem (14) thusbecomes a quadratic optimization problem and enjoys thealgorithmic advantages.

2) Smoothed �p-Minimization for User Admission Control:For user admission control, we adopt the smoothed �p-norm(13) to approximate the objective function in problem (12),yielding the following optimization problem:

minimizev∈V,x∈RK+

K∑k=1

(x2k + ε2)p/2

subject to Fk,m(v) ≤ xk,∀k ∈ Gm, m ∈M. (15)

This will help to induce individual sparsity in the auxiliaryvariables x, thereby guiding the user admission.


Although the resulting optimization problems (14) and (15)are still nonconvex, they can readily be solved by the SDRtechnique and the MM algorithm. Specifically, we will demon-strate that the nonconvex quadratic QoS constraints can beconvexified by the SDR technique in the next subsection. Theresulting convex constrained smoothed �p-minimization prob-lem will be solved by the MM algorithm, yielding an iterativereweighted-�2 algorithm, as will be presented in Section IV.

B. SDR for Nonconvex Quadratic Constraints

In this part, we will demonstrate how to apply theSDR technique to resolve the challenge of the noncon-vex quadratic QoS constraints in both problems (14) and(15). Specifically, let Qm = vH

mvm ∈ CN×N with rank(Qm) =

1,∀m ∈M. Therefore, the QoS constraint (9) can be rewrittenas

Lk,m(Q) ≤ 0,∀k ∈ Gm, m ∈M, (16)

with Lk,m(Q) given by

Lk,m(Q) = γk

⎛⎝∑

i =m

Tr(�kQi )+ σ 2k

⎞⎠− Tr(�kQm), (17)

where Q = [Qm]Mm=1 and all Qm’s are rank-one constrained.

The per-RRH transmit power constraints (2) can be rewritten as

Q ={

Qm ∈ CN :

M∑m=1

Tr(ClmQm) ≤ Pl ,∀l ∈ L

}, (18)

where Clm ∈ Rn×n is a block diagonal matrix with the identity

matrix INl as the l-th main diagonal block square matrix andzeros elsewhere.

Therefore, by dropping the rank-one constraints for all thematrices Qm’s based on the principle of the SDR technique,problem (14) can be relaxed as

P : minimizeQ∈Q

L∑l=1

ρl

(M∑

m=1

Tr(ClmQm)+ ε2

)p/2

subject to Lk,m(Q) ≤ 0,∀k ∈ Gm,

Qm � 0,∀m ∈M. (19)

Similarly, by dropping the rank-one constraints of all thematrices Qm’s, problem (15) can be relaxed as

D : minimizeQ∈Q,x∈RK+

K∑k=1

(x2k + ε2)p/2

subject to Lk,m(Q) ≤ xk,∀k ∈ Gm,

Qm � 0,∀m ∈M. (20)

Although problems P and D are still nonconvex due tothe nonconvex objective functions, the resulting smoothed �p-minimization problems preserve the algorithmic advantages,as will be presented in Section IV. In particular, an iterativereweighted-�2 algorithm will be developed in Section IV basedon the principle of the MM algorithm to find a stationarypoint to the non-convex smoothed �p-minimization problemsP and D .

Fig. 2. Sparse optimization for network power minimization and user admis-sion control in multicast Cloud-RAN.

C. A Sparse Optimization Framework for Network PowerMinimization With User Admission Control

Denote the solutions of problems P and D as Q and x,respectively. Based on these induced (approximated) sparsesolutions, we propose a sparse optimization framework fornetwork power minimization and user admission control inmulticast Cloud-RAN. The main idea is illustrated in Fig. 2and details will be explained in the following. In particular,Algorithm 3 will be developed in Section IV, which yields aKKT point for problems P and D .

1) Network Power Minimization: If problem P is feasible,once obtaining its solution Q, we can extract the group sparsitystructure information in the beamforming vector v based on the

relation: ‖vl‖2 =√∑M

m=1 Tr(ClmQm), which will be zero if allthe beamforming coefficients in vl are zeros simultaneously. Byfurther incorporating system parameters to improve the perfor-mance [1], we adopt the following RRH ordering criteria todetermine the priorities of the RRHs that should be switchedoff to minimize the network power consumption,

θl =√

ηlκl

Pcl

(∑M

m=1Tr(ClmQ

m)

)1/2

,∀l ∈ L, (21)

where κl =∑Kk=1 ‖hkl‖22 is the channel gain from the l-th RRH

to all the MUs. The RRHs with a smaller parameter θl will havea higher priority to be switched off. Intuitively, the RRH with alower channel power gain κl , lower drain inefficiency efficiencyηl , lower beamforming gain ‖vl‖2, and higher relative fronthaullink power consumption Pc

l , should have a higher priority to beswitched off.

In this paper, we adopt a simple RRH selection procedure,i.e., the bi-section method, to switch off RRHs. This methodwas shown to provide good performance in [1]. Specifically,based on the RRH ordering rule in (21), we sort the coefficientsin the ascending order: θπ1 ≤ θπ2 ≤ · · · ≤ θπL to determine theactive RRHs. Let J0 be the maximum number of RRHs that canbe switched off such that the remaining RRHs can support theQoS requirements for all the MUs. To find J0, in each bi-sectionsearch iteration, we need to solve the following size-reducedconvex feasibility problems based on the SDR technique,

F (A[i]) : find Q[i]1 , . . . , Q[i]

M

subject to Lk,m({Q[i]m }m∈M) ≤ 0,∀k ∈ Gm,

Q[i]m � 0, Q[i]

m ∈ Q[i],∀m ∈M, (22)


where Q[i]m ∈ C

(∑

l∈A[i] Nl )×(∑

l∈A[i] Nl ) with A[i] ={πi+1, . . . , πL} as the active RRH set, Q[i] represents theper-RRH transmit power constraints for the active RRHs inA[i] and the QoS constraints are obtained after omitting thechannel coefficients corresponding to the left-out RRHs. Ifproblem F (A[i]) is feasible, it implies that a feasible solutionexists to F (A[J ]) for all J < i . Likewise, if problem F (A[i])

is infeasible, it implies that no feasible solution exists forany J > i . Therefore, determining the largest J = J0 thatresults in a feasible solution to problem F (A[J ]) can beaccomplished by solving no more than (1+ �log(1+ L)�)such feasibility problems (22) via bi-section search [20].Specifically, the set {0, 1, . . . , L} is guaranteed to contain J0,i.e., J0 ∈ {0, 1, . . . , L} at each step. In each iteration, the set isdivided in two sets, i.e., bisected, so the length of the set afterk iterations is 2−k(L + 1) with (L + 1) as the length of theinitial set. It follows that exactly (1+ �log2(1+ L)�) iterationsare required before the bi-section algorithm terminates. Thisprocedure mainly reduces the relative fronthaul link powerconsumption by switching off RRHs and the correspondingfronthaul links.

Finally, denote the set of active RRHs as A ={πJ0+1, . . . , πL}. To further reduce the network powerconsumption, we need to solve the following size-reducedtransmit power minimization problem with |A| RRHs and |N|MUs based on the SDR technique,

PTP(A,N) : minimizeQ[J0]∈Q[J0]

∑l∈A

M∑m=1

1

ηlTr(ClmQ[J0]

m )

subject to Lk,m(Q[J0]) ≤ 0,∀k ∈ Gm,

Q[J0]m � 0,∀m ∈M, (23)

which is a semidefinite programming (SDP) problem and canbe solved in polynomial time using the interior-point method.

The algorithm for solving the network power minimizationproblem is presented in Algorithm 1.

2) User Admission Control: When problem P is infeasi-ble, we need to perform user admission control to maximizethe user capacity. Specifically, let x be the solution to the indi-vidual sparsity inducing optimization problem D . Observe thatxk represents the gap between the target SINR and the achiev-able SINR for MU k. We thus propose to admit the MUs withthe smallest xi ’s [17], [18]. We order the coefficients in thedescending order: xπ1 ≥ xπ2 ≥ · · · ≥ xN . The bi-section searchprocedure will be adopted to find the maximum number ofadmitted MUs. Let N0 be the minimum number of MUs to beremoved such that all the RRHs can support the QoS require-ments for all the remaining MUs. To determine the value of N0,a sequence of the following convex sized-reduced feasibilityproblems need to be solved,

F (S[i]) : find {Qm}m∈M[i]

subject to Lk,m({Qm}m∈M[i]) ≤ 0,∀k ∈ Gm,

Qm � 0, Qm ∈ Q[i],∀m ∈M[i], (24)

where S[i] = {πi+1, . . . , πK } denotes the set of admitted MUs,M[i] = {m : Gm ∩ S[i] = ∅} is the set of multicast groups, and

Algorithm 1. Network Power Minimization

Step 0: Solve the group sparse inducing optimization problemP (19) using Algorithm 3 in Section IV.

1) If it is infeasible, go to Algorithm 2 for user admissioncontrol.

2) If it is feasible, obtain the solutions Qm’s, calculate the

ordering criterion (21), and sort them in the ascendingorder: θπ1 ≤ · · · ≤ θπL , go to Step 1.

Step 1: Initialize Jlow = 0, Jup = L , i = 0.Step 2: Repeat

1) Set i ← � Jlow+Jup2 �.

2) Solve problem F (A[i]) (22): if it is infeasible, set Jup = i ;otherwise, set Jlow = i .

Step 3: Until Jup − Jlow = 1, obtain J0 = Jlow and obtain theoptimal active RRH set A = {πJ0+1, . . . , πL}.Step 4: Solve problem PTP(A,N) (23) to obtain the multicastbeamforming vectors for the active RRHs.End

Algorithm 2. User Admission Control

Step 0: Solve the individual sparse inducing optimization prob-lem D (20) using Algorithm 3 in Section IV. Obtain thesolution x and sort the entries in the descending order: xπ1 ≥· · · ≥ xN , go to Step 1.Step 1: Initialize Nlow = 0, Nup = K , i = 0.Step 2: Repeat

1) Set i ←⌊

Nlow+Nup2

⌋.

2) Solve problem F (S[i]) (24): if it is feasible, set Nup = i ;otherwise, set Nlow = i .

Step 3: Until Nup − Nlow = 1, obtain N0 = Nup and obtain theadmitted MU set S = {πN0+1, . . . , πK }.Step 4: Solve problem PTP(L, S) (23) to obtain the multicastbeamforming vectors for the admitted MUs.End

Q[i] represents the per-RRH transmit power constraints with theserved multicast groups M[i]. In this way, the QoS constraintsof the admitted MUs will be satisfied.

Finally, let S = {πN0+1, . . . , πK } be the admitted MUs. Weneed to solve the same type of size-reduced transmit power min-imization problem (23) with |L| RRHs and |S| MUs to findthe multicast transmit beamforming vectors for all the admittedMUs. We denote this problem as PTP(L, S). The proposeduser admission control algorithm is presented in Algorithm 2.

Remark 1 (Rank-One Approximation After SDR): The solu-tions for the SDR based optimization problems P , D , F (A[i]),F (S[i]) and PTP(A, S) may not be rank-one. If the rank-onesolutions are failed to be obtained, the Gaussian randomizationmethod [33] will be employed to obtain the feasible rank-one approximate solution. Specifically, the candidate multicastbeamforming vectors are generated from the solution of theSDR problems, and one is picked yielding a feasible solution tothe original problem with the minimum value of the objectivefunction. The feasibility for the original problem is guaranteed


by solving a sequence of multigroup multicast power con-trol problems with the fixed beamforming directions via linearprogramming [33]. Please refer to [33, Section IV] for moredetails on the Gaussian randomization method and we adoptthis method in our simulations to find approximate rank-onefeasible solutions. While the optimality of this randomiza-tion method for general problems remains unknown, it hasbeen widely applied and shown to provide good performance[21], [25].

3) Complexity Analysis and Discussions: To implementAlgorithm 1 and Algorithm 2, a sequence of SDP optimiza-tion or feasibility problems (e.g., P , D , F (A[i]), F (S[i])

and PTP(A, S)) need to be solved. In particular, to find theactive RRH set A and admitted MU set S, we need tosolve no more than (1+ �log(1+ L)�) and (1+ �log(1+K )�) SDP feasibility problems F (A[i]) and F (S[i]), respec-tively. In addition, to solve the SDP problem PTP(L,N) (23)with M matrix optimization variables of size N × N and(K + L) linear constraints, the interior-point method [20] willtake O(

√M N log(1/ε)) iterations and cost O(M3 N 6 + (K +

L)M N 2) floating point operations to achieve an optimal solu-tion with accuracy ε > 0. Therefore, this makes the proposednetwork power minimization and user admission algorithmsdifficult to scale to large problem sizes with a large number ofRRHs and/or MUs. To further improve the computational effi-ciency of the proposed SDP based algorithms, one promisingapproach is to apply the alternating direction method of multi-pliers (ADMM) algorithm [34] by leveraging parallelism in thecloud computing environment in the BBU pool [12]. This is,however, an on-going research topic, and we will leave it as ourfuture work.

IV. ITERATIVE REWEIGHTED-�2 ALGORITHM FOR

SMOOTHED �p-MINIMIZATION

In this section, we first develop an iterative reweighted-�2 algorithm to solve a general non-convex smoothed �p-minimization problem based on the principle of the MMalgorithm. We then present how to apply this algorithm tosolve the problems P and D to induce sparsity structuresin the solutions, thereby guiding the RRH selection and useradmission.

A. Iterative Reweighted-�2 Algorithm

Consider the following smoothed �p-minimization problem,

Psm(ε) : minimizez∈C

f p(z; ε) :=m∑

i=1

(z2i + ε2)p/2, (25)

where C is an arbitrary convex set, z ∈ Rm and ε > 0 is some

fixed regularizing parameter. In the following, we first provethat the optimal solution of the smoothed �p-minimizationproblem Psm(ε) is also optimal for the original non-smooth�p-minimization problem (i.e., Psm(0)) when ε is small. Wethen demonstrate the algorithmic advantages of the smooth-ness in the procedure of developing the iterative reweighted-�2algorithm.

1) Optimality of Smoothing the �p-Norm: The set of KKTpoints of problem Psm(ε) is given as

�(ε) = {z ∈ C : 0 ∈ ∇z f p(z; ε)+NC(z)}, (26)

where NC(z) is the normal cone of a convex set C at point z con-sisting of the outward normals to all hyperplanes that support Cat z, i.e.,

NC(z) := {s : 〈s, x− z〉 ≤ 0,∀x ∈ C}. (27)

Define the deviation of a given set Z1 from another set Z2as [35],

D(Z1,Z2) = supz1∈Z1

(inf

z2∈Z2

‖z1 − z2‖)

. (28)

We then have the following theorem on the relationshipbetween the smoothed �p-minimization problem Psm(ε) andthe original non-smooth �p-minimization problem Psm(0).

Theorem 1: Let �ε be the set of KKT points of problemPsm(ε). Then, we have

limε↘0

D(�(ε),�(0)) = 0. (29)

Proof: Please refer to Appendix A for details. �This theorem indicates that any limit of the sequence of KKT

points of problem Psm(ε) is a KKT pair of problem Psm(0)

when ε is small enough. That is, at least a local optimal solutioncan be achieved. In the sequel, we will focus on finding a KKTpoint of problem Psm(ε) with a small ε, yielding good approx-imations to the KKT points of the �p-minimization problemPsm(0) to induce sparsity in the solutions.

2) The MM Algorithm for the Smoothed �p-Minimization:With the established asymptotic optimality, we then leveragethe principle of the MM algorithm to solve problem (25).Basically, this algorithm generates the iterates {zn}∞n=1 by suc-cessively minimizing upper bounds Q(z; z[n]) of the objectivefunction f p(z; ε). The quality of the upper bounds will controlthe convergence (rate) and optimality of the resulting algo-rithms. Inspired by the results in the expectation-maximization(EM) algorithm [36], [37], we adopt the upper bounds in thefollowing proposition to approximate the smoothed �p-norm.

Proposition 1: Given the iterate z[n] at the n-th iteration,an upper bound for the objective function of the smoothed�p-norm f p(z; ε) can be constructed as follows,

Q(z;ω[n]) :=m∑

i=1

ω[n]i z2

i , (30)

where

ω[n]i =

p

2

[(z[n]

i

)2 + ε2] p

2−1

,∀i = 1, . . . , m. (31)

From the weights given in (31), it is clear that, by addingthe regularizer parameter ε > 0, we can avoid yielding infinitevalues when some zi ’s become zeros in the iterations.


Proof: Define the approximation error as

f p(z; ε)− Q(z;ω[n]) =m∑

i=1

[κ(z2i )− κ ′((z[n]

i )2)z2i ], (32)

where κ(s) = (s + ε2)p/2 with s ≥ 0. The sound property ofthe Q-function (30) is that the approximation error (32) attainsits maximum at z = z[n]. In particular, we only need to provethat the function g(s) = κ(s)− κ ′(s[n])s with s ≥ 0 attainsthe maximum at s = s[n]. This is true based on the facts thatg′(s[n]) = 0 and κ(s) is strictly concave. �

Let z[n+1] be the minimizer of the upper bound functionQ(z;ω[n]) at the n-th iteration, i.e.,

z[n+1] := arg minz∈C

Q(z;ω[n]). (33)

Based on Proposition 1 and (33), we have

f p(z[n+1]; ε) =Q(z[n+1];ω[n])+ f p(z[n+1]; ε)− Q(z[n+1];ω[n])

≤Q(z[n+1];ω[n])+ f p(z[n]; ε)− Q(z[n];ω[n])

≤Q(z[n];ω[n])+ f p(z[n]; ε)− Q(z[n];ω[n])

= f p(z[n]; ε), (34)

where the first inequality is based on the fact that function( f p(z; ε)− Q(z;ω[n])) attains its maximum at z = z[n], andthe second inequality follows from (33). Therefore, minimiz-ing the upper bound, i.e., the Q-function in (30), can reduce theobjective function f p(z; ε) successively.

Remark 2: In the context of the EM algorithm [38] forcomputing the maximum likelihood estimator of latent vari-able models, the functions − f p(z; ε) and −Q(z;ω[n]) can beregarded as the log-likelihood and comparison functions (i.e.,the lower bound of the log-likelihood), respectively [36].

The MM algorithm for the smoothed �p-minimization prob-lem is presented in Algorithm 3.

The convergence of the iterates {z[n]}∞n=1 (35) is presented inthe following theorem.

Theorem 2: Let {z[n]}∞n=1 be the sequence generated by theiterative reweighted-�2 algorithm (35). Then, every limit pointz of {z[n]}∞n=1 has the following properties

1) z is a KKT point of problem Psm(ε) (25);2) f p(z[n]; ε) converges monotonically to f p(z; ε) for

some KKT point z.

Proof: Please refer to Appendix B for details. �As noted in [39], without the convexity of f p(z; ε), the KKT

point may be a local minimum or other point (e.g., a saddlepoint). We also refer to these points as stationary points [39,Page 194].

Remark 3: The algorithm consisting of the iterate (35)accompanied with weights (36) is known as the iterativereweighted least squares [29], [38], [40] in the fields of statis-tics, machine learning and compressive sensing. In particular,with a simple constraint C, the iterates often yield closed-formswith better computational efficiency. For instance, for the noise-less compressive sensing problem [29], the iterates have closed-form solutions [29, (1.9)]. Therefore, this method has a higher

Algorithm 3. Iterative Reweighted-�2 Algorithm

input: Initialize ω[0] = (1, . . . , 1); I (the maximum number ofiterations)Repeat

1) Solve problem

z[n+1] := arg minz∈C

m∑i=1

ω[n]i z2

i . (35)

If it is feasible, go to 2); otherwise, stop and returnoutput 2.

2) Update the weights as

ω[n+1]i = p

2

[(z[n+1]

i

)2 + ε2] p

2−1

,∀i = 1, . . . , m. (36)

Until convergence or attain the maximum iterations andreturn output 1.

output 1: z; output 2: Infeasible.

computational efficiency compared with the conventional �1-minimization approach for compressive sensing [26], whereina linear programming problem needs to be solved via algo-rithms such as interior-point or barrier methods. Furthermore,empirically, it was observed that the iterative reweighted leastsquares method can improve the signal recovery capabilityby enhancing the sparsity for compressive sensing over the�1-minimization method [29], [40].

In contrast to the existing works on the iterative reweightedleast squares methods, we provide a new perspective to developthe iterative reweighted-�2 algorithm to solve the smoothed�p-minimization problem with convergence guarantees basedon the principle of the MM algorithm. Furthermore, the mainmotivation and advantages for developing the iterates (35) is toinduce the quadratic forms in the objective function in prob-lem P (19) to make it compliant with the SDR technique,thereby inducing the group sparsity structure in the multicastbeamforming vectors via convex programming.

Remark 4: The advantages of the iterative reweighted-�2algorithm include the capability of enhancing the sparsity,as well as inducing the quadratic forms for the multicastbeamforming vectors by inducing the quadratic formulations(35). Note that the reweighted �1-minimization algorithm in[30] can also induce the quadratic forms in the beamformingvectors vl ’s by rewriting the indicator function (7) as the �0-norm of the squared �2-norm of the vectors vl ’s, i.e., ‖vl‖22.Furthermore, the key ideas of the convergence proof of the iter-ative reweighted-�2 algorithm (i.e., Algorithm 3), by leveragingthe EM theory to establish upper bounds in the iterates of theMM algorithm, should be useful for other iterative algorithms,e.g., [41].

B. Sparsity Inducing for RRH Selection and User Admission

In this subsection, we demonstrate how to apply the devel-oped iterative reweighted-�2 algorithm to solve the nonconvex


sparse optimization problems P and D for RRH selection anduser admission, respectively. In this way, we can find a KKTpoint for the nonconvex smoothed �p-minimization problemsP and D with convex constraints. Specifically, let � := {Q ∈Q : Lk,m(Q) ≤ 0, Qm � 0,∀k ∈ Gm, m ∈M} be the set of con-straints in problem P . The iterative reweighted-�2 algorithmfor problem P generates a sequence {Q[n]}∞n=1 as follows:

P [n]SDP : minimize

Q∈�

L∑l=1

ω[n]l

(M∑

m=1

Tr(ClmQm)

), (37)

with the weights as

ω[n]l =

ρl p

2

[∑M

m=1Tr(ClmQ[n]

m

)+ ε2] p

2−1

,∀l ∈ L. (38)

Applying Algorithm 3 to problem D is straightforward.

V. SIMULATION RESULTS

In this section, we will simulate the proposed algorithmsbased on the iterative reweighted-�2 algorithm (IR2A) fornetwork power minimization and user admission control inmulticast Cloud-RAN. We set the parameters as follows: Pl =1W,∀l; Pc

l = [5.6+ l − 1]W,∀l; ηl = 1/4,∀l; σk = 1,∀k.Denote the channel propagation from the l-th RRH and the k-thMU as hkl = Dklgkl with Dkl as the large-scale fading coeffi-cients and gkl ∈ CN(0, I) as the small-scale fading coefficients.We set ε = 10−3 in the iterative reweighted-�2 algorithm andthe algorithm will terminate if either the number of iterationsexceeds 30 or the difference between the objective values ofconsecutive iterations is less than 10−3. The pre-determinednumber of randomization is set to be 50 in the Gaussianrandomization method.

A. Convergence of the Iterative Reweighted-�2 Algorithm

Consider a network with L = 6 2-antenna RRHs and 2multicast groups with 2 single-antenna MUs in each group.The channels are spatially uncorrelated. For each MU k, weset Dlk = 1,∀l ∈ �1 with |�1| = 2; Dlk = 0.7,∀l ∈ �2 with|�2| = 2; Dlk = 0.5,∀l ∈ �3 with |�3| = 2. All the sets �i ’sare uniformly drawn from the RRH set L = {1, . . . , L} with∪�i = �. Fig. 3 illustrates the convergence of the iterativereweighted-�2 algorithm for the smoothed �p-minimizationproblem P with different initial points and different channelrealizations. Specifically, we set p = 1. The three differentchannel realizations are generated uniformly and indepen-dently. For each channel realization, we simulate Algorithm 3with the fixed initial point as ω[0] = (1, . . . , 1) and ran-domly generated initial point ω[0]. The random initializationinstances for the three channel realizations are given asω

[0]ch1 = [0.1389, 0.2028, 0.1987, 0.6038, 0.2722, 0.1988],

ω[0]ch2 = [0.5657, 0.7165, 0.5113, 0.7764, 0.4893, 0.1859],

ω[0]ch3 = [0.4093, 0.4635, 0.6109, 0.0712, 0.3143, 0.6084],

respectively. In particular, this figure shows that the sequenceof the objective functions converges monotonically, which

Fig. 3. Convergence of the iterative reweighted-�2 algorithm for the smoothed�p-minimization problem P with different channel realizations and initialpoints.

confirms the convergence results in Theorem 2. In addition, itillustrates the robustness of the convergence of the reweighted-�2 algorithm with different initial points and different problemparameters. Furthermore, it also demonstrates the fast conver-gence rate of the proposed algorithm in the simulated setting.Empirically, the iterative reweighted-�2 algorithm converges in20 iterations on average in all the simulated channels in thispaper.

B. Network Power Minimization

Consider a network with L = 12 2-antenna RRHs and 5multicast groups with 2 single-antenna MUs in each group.The channels are spatially uncorrelated. For each MU k,we set Dlk = 1,∀l ∈ �1 with |�1| = 4; Dlk = 0.7,∀l ∈ �2with |�2| = 4; Dlk = 0.5,∀l ∈ �3 with |�3| = 4. All the sets�i ’s are uniformly drawn from the RRH set L = {1, . . . , L}with ∪�i = �. The proposed iterative reweighted-�2 algorithmis compared with the reweighted �1/�∞-norm based algo-rithm [25], in which, the objective function in problem Pis replaced by

∑Ll1=1

∑Ll2 maxm maxnl1

maxnl2|Qm(nl1, nl2)|

with Qm(i, j) as the (i, j)-th entry in Qm . And the RRHswith smaller beamforming beamforming coefficients (measuredby �∞-norm) have higher priorities to be switched off. Fig. 4demonstrates the average network power consumption with dif-ferent target SINRs using different algorithms. Each point ofthe simulation results is averaged over 50 randomly generatedchannel realizations for which problem P is feasible. Fromthis figure, we can see that the proposed iterative reweighted-�2 algorithm can achieve near-optimal performance comparedwith the exhaustive search algorithm (i.e., solving problem (10)with convexified QoS constraints based on the SDR technique).It yields lower network power consumption compared with theexisting �1/�∞-norm based algorithm [25], while the coordi-nated multicast beamforming algorithm [13], [21] with all theRRHs active has the highest network power consumption. Notethat the number of SDP problems F (A[i]) (22) needed to be


Fig. 4. Average network power consumption versus target SINR with differentalgorithms.

TABLE ITHE AVERAGE NUMBER OF ACTIVE RRHS WITH DIFFERENT

ALGORITHMS

TABLE IITHE AVERAGE RELATIVE FRONTHAUL LINKS POWER CONSUMPTION

WITH DIFFERENT ALGORITHMS

solved grows logarithmically with L for the proposed networkpower minimization algorithm and the previous �1/�∞-normbased algorithm [25]. But the number of SDP problems (22)to be solved for exhaustive search grows exponentially with L .Although the coordinated beamforming algorithm has the low-est computational complexity by only solving the total transmitpower minimization problem PTP(L,N) (23) with all theRRHs active, it yields the highest network power consumption.Our proposed algorithm thus achieves a good trade-off betweencomputational complexity and performance.

Tables I-III show the corresponding average number ofactive RRHs, average fronthaul network power consumptionand average total transmit power consumption, respectively.Specifically, Table I confirms the existence of the group sparsitypattern in the aggregative multicast beamforming vector. Thatis, the switched off RRHs indicate that all the beamformingcoefficients at the RRHs are set to be zeros simultaneously. Italso shows that the proposed iterative reweighted-�2 algorithmhas the capability of enhancing sparsity, thereby switching offmore RRHs compared with the �1/�∞-norm based algorithm

TABLE IIITHE AVERAGE TOTAL TRANSMIT POWER CONSUMPTION WITH

DIFFERENT ALGORITHMS

Fig. 5. Average number of admitted MUs versus target SINR with differentalgorithms.

except for very low SNR. Table II shows that the proposediterative reweighted-�2 algorithm can achieve much lower fron-thaul network power consumption and attain similar values withexhaustive search. With all the RRHs being active, the coor-dinated multicast beamforming algorithm yields the highestfronthaul network power consumption and the lowest trans-mit power consumption as shown in Table II and Table III,respectively. This indicates that it is critical to take the rela-tive fronthaul link power consumption into consideration whendesigning a green Cloud-RAN. Although the exhaustive searchmay yield higher transmit power consumption in some sce-narios as shown in Table III, overall it achieves the lowestnetwork power consumption as much lower relative fronthaulnetwork power consumption can be attained as indicated inTable II. Furthermore, we can see that, in the simulated settings,different values of p in the proposed iterative reweighted-�2algorithm yield similar performance, while all achieve near-optimal performance compared with the exhaustive search.

C. User Admission Control

Consider a network with L = 6 2-antenna RRHs and 4 mul-ticast groups with 2 single-antenna MUs in each group. Thechannel model is the same as Section V-A. The proposediterative reweighted-�2 algorithm based user admission con-trol algorithm is compared with the existing convex relaxationapproach (i.e.. the multicast membership deflation by relaxation(MDR) [16]) and the exhaustive search. The simulation results


are illustrated in Fig. 5 for the average number of admittedMUs. Each point of the simulation results is averaged over 200randomly generated channel realizations for which problem Pis infeasible. From Fig. 5, we can see that the proposed iter-ative reweighted-�2 algorithm outperforms the existing MDRapproach [16]. In particular, the performance of the iterativereweighted-�2 algorithm is almost the same with different val-ues of p and achieves near-optimal performance compared tothe exhaustive search via solving problem (12) with convexifiedQoS constraints based on the SDR technique. Although all thesimulated results demonstrate that the performances are robustto the parameter p, it is very interesting to theoretically identifythe typical scenarios, where smaller values of p will yield muchbetter performances.

VI. CONCLUSIONS AND FUTURE WORKS

This paper developed a sparse optimization framework fornetwork power minimization and user admission in greenCloud-RAN with multicast beamforming. A smoothed �p-minimization method was proposed to induce the sparsitystructures in the solutions, thereby guiding the RRH selectionand user admission. This approach has the advantages in termsof promoting sparsity and assisting algorithmic design by intro-ducing the quadratic forms in the group sparse inducing penalty.In particular, by leveraging the MM algorithm and the SDRtechnique, we developed an iterative reweighted-�2 algorithmwith convergence and optimality guarantees (i.e., KKT points)to solve the resulting smoothed �p-minimization problems Pand D with convex constraints. The effectiveness of the pro-posed algorithms was demonstrated via simulations for networkpower minimization and user admission control.

Several future directions are listed as follows:• Although the proposed methods only need to solve a

sequence of convex optimization problems, the complex-ity of solving the large-scale SDP problem using theinterior-point method is prohibitively high. One promis-ing option is to use the first-order method (e.g., theoperator splitting method [42], [12]) by leveraging theparallel computing platform in the BBU pool [5], [43],which will require further investigation.• It is desirable to establish the optimality and perform

probabilistic analysis for the iterative reweighted-�2 algo-rithm in the context of inducing group sparsity in themulticast beamforming vectors. However, with the com-plicated constraints set, this becomes much more chal-lenging compared with the compressive sensing problemswith simple constraints (e.g., affine constraints) [29]. Itis also interesting to apply this algorithm to solve othermixed combinatorial optimization problems in wirelessnetworks, e.g., wireless caching problems [44].• It is interesting to apply the proposed smoothed �p-

minimization approach with the iterative reweighted-�2algorithm in the scenarios with CSI uncertainty. The onlyrequirement is that the resulting non-convex constraintsdue to the CSI uncertainty can be convexified [45], [46].It is also interesting but challenging to characterize the

performance degradations due to CSI acquisition errorsin multicast Cloud-RANs [47], [48].

APPENDIX APROOF OF THEOREM 1

We first need to prove that

lim supε↘0

�(ε) ⊂ �(0), (39)

where lim supε↘0 �(ε) is defined as [49, Page 152]

lim supε↘0

�(ε) : =⋃

ε[n]↘0

lim supn→∞

�(ε[n])

= { z|∃ε[n] ↘ 0, ∃z[n] → z}, (40)

where z[n] ∈ �(ε[n]). Therefore, for any z ∈ lim supε↘0 �(ε),there exists z[n] ∈ �(ε[n]) such that z[n] → z and ε[n] ↘ 0. Toprove (39), we only need to prove that z ∈ �(0). Specifically,z[n] ∈ �(ε[n]) indicates that

0 ∈ ∇z f p(z[n]; ε[n])+NC(z[n]). (41)

As f p(z; ε) is continuously differentiable in both z and ε, wehave

limn→∞∇z f p(z[n]; ε[n]) = lim

z[n]→ zlim

ε[n]↘0∇z f p(z[n]; ε[n])

=∇z f p( z; 0). (42)

Furthermore, based on results for the limits of normal vectorsin [49, Proposition 6.6], we have

lim supz[n]→ z

NC(z[n]) = NC( z). (43)

Specifically, if z[n] → z, s[n] ∈ NC(z[n]) and s[n] → s, thens ∈ NC( z). That is, the set {(z, s)|s ∈ NC(z)} is closed relativeto C× R

m . Based on (42) and (43) and taking n→∞ in (41),we thus prove (39). Based on [50, Theorem 4] we complete theproof for (29).

APPENDIX BCONVERGENCE OF THE ITERATIVE REWEIGHTED-�2

ALGORITHM

1) We will show that any convergent subsequence {z[nk ]}∞k=1of {z[n]}∞n=1 satisfies the definition of the KKT points ofproblem Psm(ε) (26). Specifically, let z[nk ] → z be onesuch convergent subsequence with

limk→∞ z[nk+1] = lim

k→∞ z[nk ] = z. (44)

As

z[nk+1] := arg minz∈C

Q(z;ω[nk ]), (45)

which is a convex optimization problem, the KKT condi-tion holds at z[nk+1], i.e.,

0 ∈ ∇z Q(z[nk+1];ω[nk ])+NC(z[nk+1]). (46)

Based on [49, Proposition 6.6] and (44), we have


lim supz[nk+1]→ z

NC(z[nk+1]) = NC( z). (47)

Furthermore, based on (44), we also have

limk→∞∇z Q(z[nk+1];ω[nk ]) = lim

k→∞ 2m∑

i=1

ω[nk ]znk+1i

= limk→∞

m∑i=1

pz[nk+1][(z[nk ]

i

)2 + ε2

]1− p2

=∇z f p( z; ε). (48)

Therefore, taking k →∞ in (46), we have

0 ∈ ∇z Q( z; ω)+NC( z), (49)

which indicates that z is a KKT point of problem Psm(ε).We thus complete the proof.

2) As f p(z; ε) is continuous and C is compact, we have thefact that the limit of the sequence f p(z[n]; ε) is finite.Furthermore, we have f p(z[n+1]; ε) ≤ f p(z[n]; ε) accord-ing to (34). Based on the results in 1), we complete theproof. Note that a similar result was presented in [37] byleveraging the results in the EM algorithm theory.

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Yuanming Shi (S’13–M’15) received the B.S. degreein electronic engineering from Tsinghua University,Beijing, China, in 2011, and the Ph.D. degree in elec-tronic and computer engineering from the Hong KongUniversity of Science and Technology (HKUST),Hong Kong, in 2015. He is currently an AssistantProfessor with the School of Information Scienceand Technology, ShanghaiTech University, Shanghai,China. His research interests include large-scale con-vex optimization and analysis, Riemannian optimiza-tion, stochastic optimization, mobile edge computing,

and dense wireless networking.

Jinkun Cheng received the B.S. degree in electronicengineering from Shandong University, Jinan, China,in 2013. She is currently pursuing the Ph.D degreein electronic engineering from Tsinghua University,Beijing, China. Her research interests include cloud-RAN, computation offloading, and convex optimiza-tion theory.

Jun Zhang (S’06–M’10–SM’15) received the B.Eng.degree in electronic engineering from the Universityof Science and Technology of China, Hefei, China,in 2004, the M.Phil. degree in information engineer-ing from the Chinese University of Hong Kong, HongKong, in 2006, and the Ph.D. degree in electrical andcomputer engineering from the University of Texasat Austin, Austin, TX, USA, in 2009. He is currentlya Research Assistant Professor with the Departmentof Electronic and Computer Engineering, Hong KongUniversity of Science and Technology (HKUST),

Hong Kong. He has coauthored the book Fundamentals of LTE (Prentice-Hall,2010). His research interests include wireless communications and networking,green communications, and signal processing. He is an Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS and served as a MACTrack Co-Chair for the IEEE WCNC 2011. He was the recipient of the 2014Best Paper Award for the EURASIP Journal on Advances in Signal Processing,and the PIMRC 2014 Best Paper Award.

Bo Bai (S’09–M’11) received the B.S. degree(with the highest Hons.) in communication engi-neering from Xidian University, Xi’an, China, in2004, and the Ph.D. degree in electronic engineer-ing from Tsinghua University, Beijing China, in2010. He also received the Honor of OutstandingGraduates of Shaanxi Province and the Honor ofYoung Academic Talent of Electronic Engineeringin Tsinghua University. From 2009 to 2012, hewas a Visiting Research Staff (Research Assistant)from April 2009 to September 2010 and Research

Associate from October 2010 to April 2012 at the Department of Electronicand Computer Engineering, Hong Kong University of Science and Technology(HKUST). Currently, he is an Assistant Professor with the Department ofElectronic Engineering, Tsinghua University.

He has also obtained support from Backbone Talents Supporting Projectof Tsinghua University. He has authored more than 50 papers in major IEEEjournals and flagship conferences. His research interests include hot topicsin wireless networks, information theory, random graph, and combinatorialoptimization. He also served on the Wireless Communications TechnicalCommittee (WTC) and Signal Processing and Communications Electronics(SPCE) Technical Committee. He has served as a TPC member for severalIEEE ComSoc conferences such as ICC, Globecom, WCNC, VTC, ICCC, andICCVE. He has also served as a reviewer for a number of major IEEE jour-nals and conferences. He was the recipient of the Student Travel Grant at IEEEGlobecom’09. He was also invited as a Young Scientist Speaker at IEEE TTM2011.

Wei Chen (S’05–M’07–SM’13) received the B.S.degree in operations research and the Ph.D. degree inelectronic engineering (both with the highest Hons.)from Tsinghua University, Beijing, China, in 2002and 2007, respectively. From 2005 to 2007, he wasalso a Visiting Research Staff member at the HongKong University of Science and Technology. SinceJuly 2007, he has been with the Department ofElectronic Engineering, Tsinghua University, wherehe is a Full Professor, a Vice Department Head,and a University Council Member. He visited the

University of Southampton, Southampton, U.K., from June 2010 to September2010, Telecom ParisTech, Paris, France, from June 2014 to September 2014,and Princeton University, Princeton, NJ, USA, from July 2015 to September2015. He has also visited the Hong Kong University of Science and Technology,Hong Kong and the Chinese University of Hong Kong, Hong Kong, in 2008,2009, 2012, and 2014.

He is a National 973 Youth Project Chief Scientist and a National Youngand Middle-Aged Leader for science and technology innovation. Treatedas a special case of early promotion, he was promoted to Full Professor atTsinghua University, in 2012. He is also supported by the NSFC ExcellentYoung Investigator Project, the New Century Talent program of the Ministryof Education, the Beijing Nova Program, and the 100 Fundamental ResearchTalents program of Tsinghua University (also known as the 221 talentsProgram).

His research interests include wireless communications, information theory,and applied mathematics. He serves as an Editor for the IEEE TRANSACTIONS

ON EDUCATION, IEEE WIRELESS COMMUNICATIONS LETTERS, and ChinaCommunications, a member of the 2013 education working group of IEEEComsoc, a Vice Director of the youth committee of China Institute ofCommunications, and a Vice Director of youth board of education committeeof China Institute of Electronics. He served as a Tutorial Co-Chair of the IEEEICC 2013, a TPC Co-Chair of the IEEE VTC 2011-Spring, and SymposiumCo-Chair for the IEEE ICC, ICCC, CCNC, Chinacom, and WOCC. He alsoholds Guest Professor positions of some universities in China.

Dr. Chen was the recipient of the First Prize of the 14th Henry Fok Ying-Tung Young Faculty Award, the Yi-Sheng Mao Beijing Youth Science andTechnology Award, the 2010 IEEE Comsoc Asia Pacific Board Best YoungResearcher Award, the 2009 IEEE Marconi Prize Paper Award, the 2015 CIEInformation Theory New Star Award, the Best Paper Awards at IEEE ICC 2006,IEEE IWCLD 2007, and IEEE SmartGirdComm 2012, as well as, the FirstPrize in the 7th Young Faculty Teaching Competition in Beijing, China. Hewas also the recipient of the 2011 Tsinghua Raising Academic Star Award andthe 2012 Tsinghua Teaching Excellence Award. He holds the honorary titlesof Beijing Outstanding Teacher and Beijing Outstanding Young Talent. He isthe Champion of the First National Young Faculty Teaching Competition, anda Winner of the National May 1st Medal.


Khaled B. Letaief (S’85–M’86–SM’97–F’03)received the B.S. degree (with distinction) in elec-trical engineering, and the M.S. and Ph.D. degreesin electrical engineering from Purdue University,West Lafayette, IN, USA, in 1984, 1986, and 1990,respectively.

From 1990 to 1993, he was a Faculty Memberwith the University of Melbourne, Parkville Vic.,Australia. He has been with the Hong KongUniversity of Science and Technology, Hong Kong.While at HKUST, he held numerous administrative

positions, including the Head of the Department of Electronic and ComputerEngineering, the Director of the Center for Wireless IC Design, the Directorof Huawei Innovation Laboratory, and the Director of the Hong Kong TelecomInstitute of Information Technology. He has also served as a Chair Professor andthe Dean of HKUST School of Engineering. Under his leadership, the Schoolof Engineering has dazzled in international rankings (ranked #14 in the worldin 2015 according to QS World University Rankings).

In September 2015, he joined HBKU as a Provost to help establish aresearch-intensive university in Qatar in partnership with strategic partners thatinclude Northwestern University, Carnegie Mellon University, Pittsburgh, PA,USA, Cornell University, Ithaca, NY, USA, and Texas A&M, College Station,TX, USA.

He is a world-renowned leader in wireless communications and networks. Inthese areas, he has authored more than 500 journal and conference papers andgiven invited keynote talks as well as courses all over the world. He has made 6major contributions to IEEE Standards along with 13 patents including 11 U.S.patents.

He served as a consultant for different organizations and is the found-ing Editor-in-Chief of the prestigious IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS. He has served on the Editorial Board of other pres-tigious journals including the IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS—Wireless Series (as an Editor-in-Chief). He has beeninvolved in organizing a number of major international conferences.

Dr. Letaief has been a dedicated educator committed to excellence inteaching and scholarship. He had the privilege to serve on various profes-sional and government organizations and is currently serving as an IEEEComSoc Vice-President for Technical Activities, a member of the IEEE TABPeriodicals Committee, and the IEEE Fellow Committee. He is a Fellow ofHKIE. He is also recognized by Thomson Reuters as an ISI Highly CitedResearcher. He was the recipient of the Mangoon Teaching Award from PurdueUniversity in 1990; the HKUST Engineering Teaching Excellence Award(4 times); and the Michael Gale Medal for Distinguished Teaching (high-est university-wide teaching award at HKUST). He was also the recipient ofmany other distinguished awards including the 2007 IEEE Joseph LoCiceroPublications Exemplary Award; the 2009 IEEE Marconi Prize Award inWireless Communications; the 2010 Purdue University Outstanding Electricaland Computer Engineer Award; the 2011 IEEE Harold Sobol Award; the 2011IEEE Wireless Communications Technical Committee Recognition Award; and11 IEEE Best Paper Awards.

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