Group Sparse Beamforming for Multicast Green Cloud ... - IEEE Xplore

IEEE ICC 2015 - Wireless Communications Symposium 1

Group Sparse Beamforming for Multicast Green Cloud-RAN via Parallel Semidefinite Programming Jinkun Cheng∗ , Yuanming Shi† , Bo Bai∗ , Wei Chen∗ , Jun Zhang† , and Khaled B. Letaief† ∗ State Key Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing, China Email: [email protected], {eebobai, wchen}@tsinghua.edu.cn † Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology Hong Kong, E-mail: {yshiac, eejzhang, eekhaled}@ust.hk

Abstract—The Cloud radio access network (Cloud-RAN) has great potentials to improve energy efficiency and increase capacity of wireless networks. In this paper, we investigate multicast beamforming design for network power minimization of CloudRAN, which is shown to be a highly intractable non-convex mixed integer non-linear programming problem. To provide an efficient solution to this highly complicated problem, we propose a three-stage algorithm based on the group-sparsity inducing norm, which minimizes network power by coordinated multicast beamforming and adaptively selecting active remote radio heads (RRHs). In particular, a novel quadratic variational weighted `1 /`2 -norm aided alternating algorithm is proposed to exploit the group-sparsity structure of the beamforming vector, thereby guiding the active RRH set selection. Given the selected RRH set, multicast beamforming is performed to minimize the network power consumption. Furthermore, to enhance the computation efficiency upon utilizing the shared computing resources in the cloud center, we employ the alternating direction method of multipliers (ADMM) algorithm to solve the resulting semidefinite programming problems in parallel. Extensive simulation results will demonstrate the effectiveness of the proposed multicast group sparse beamforming algorithm.

I. I NTRODUCTION Cloud radio access network (Cloud-RAN) [1], [2], is a promising and flexible solution to accommodate the exponential growth of mobile data traffic and diversified multimedia applications with high energy efficiency. In Cloud-RAN, all the baseband signal processing is shifted to the baseband unit (BBU) pool, while conventional base stations (BS) are replaced by geographically distributed low-cost remote radio heads (RRHs), connected to the BBU pool via high-capacity and low-latency optical fronthaul links. As a result, centralized signal processing and scheduling can effectively manage interference and significantly reduce energy consumption. Coordinated beamforming [3] has been widely investigated to achieve high diversity gain and efficient interference suppression. Recently, inter-relay interference cancellation and sub-carrier matching techniques have also been exploited to further improve the spectral efficiency of cooperative relaying network, respectively in [4] and [5]. In Cloud-RAN, the This paper is partially supported by NSFC under grant No. 61322111 and No. 61401249, the National Basic Research Program of China (973 Program) No. 2013CB336600, Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No. 20130002120001, Chuanxin Funding, and Beijing nova program No.Z121101002512051.

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group-sparsity based coordinated beamforming approach [2] is recently proposed for green communications with unicast services via switching off RRHs selectively. Meanwhile, the integrated multigroup service with content-aware applications is becoming increasingly popular, so that the total power consumption of the network can be further reduced. The group sparse beamforming design has also been introduced for the multicast transmission strategy [6] in conventional cellular networks. However, group sparse beamforming with respect to the multicast scenario in the Cloud-RAN network, which can significantly improve energy efficiency and system capacity, has not been investigated in a systematic way. In this paper, we consider the network power minimization problem with multicast beamforming in Cloud-RAN, which is a non-convex mixed integer non-linear programming problem. To solve this highly complicated problem, we propose a three-stage algorithm based on the group-sparsity inducing norm. Firstly, a novel quadratic variational weighted `1 /`2 norm is proposed to induce group-sparsity in the beamforming vector, which is to be compatible with the quadratic QoS constraints. The state-of-art `1 /`2 -norm based group sparse beamforming structure [2] is inapplicable in this scenario because it cannot be lifted to a higher dimension due to its nonsmooth property. With the proposed approach, the non-convex mixed integer non-linear programming can be reformulated into a biconvex problem [7], which can be solved by the alternating optimization algorithm efficiently. Secondly, we perform the active RRH selection with the group-sparsity information attained from the alternating algorithm. Finally, the network power is minimized given the optimal active RRH set, where the transport power consumption is taken into consideration [2]. Simulation results will show that the proposed algorithm with low computation complexity achieves the near-optimal solution compared with exhaustive search. To solve the semidefinite programming (SDP) problems generated in the proposed algorithm, we adopt a first-order method, i.e., the ADMM algorithm [8], via parallel cone projection and parallel subspace projection [9]. With the parallel algorithm, it is capable of taking advantage of the collaborative computation resources gathered at the BBU pool. Moreover, the adopted ADMM algorithm solver [9] obtains the capability of feasibility detection, which plays an important role in guiding RRH selection. Simulation results will demonstrate

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infeasibility detection ability and accuracy of solutions with respect to the ADMM solver. II. S YSTEM M ODEL AND G ROUP S PARSE F ORMULATION In this section, we will present the mathematical description of the system model with respect to the multicast CloudRAN, as well as the network power minimization problem formulation via group sparse beamforming. A. System Model Consider a downlink multicast Cloud-RAN with L RRHs and K single antenna mobile users (MU), where the l-th RRH is equipped with Nl antennas. Denote the set of all the RRHs as L = {1, 2, . . . , L} and the set of all the users as K = {1, 2, . . . , K}. The K MUs are divided into M (1 ≤ M ≤ K) multicast groups according to the traffic, which are denoted as G1 , G2 , . . . , GM with Gi ∩ Gj = ∅ and ∪i Gi = K. Let M = {1, . . . , M } represent the set of multicast groups. Denote the channel vector from the l-th RRH to the k-th MU as hlk ∈ CNl . The transmitted signal for the m-th multicast group is denoted by sm ∈ C, with normalized power, i.e., 2 E[|sm | ] = 1. Denote the beamforming vector from the l-th RRH to the k-th MU in the group m as vlm ∈ CNl . Let A ⊆ L be the active RRH set and Z ⊆ L be the inactive RRH set with A ∪ Z = L. The transmitted signal from the l-th RRH is given by xl =

M X

vli si , ∀l ∈ A.

(1)

i=1

The received signal at MU k can be written as X XX yk = hH hH lk vlm sm + lk vli si + nk , ∀k ∈ Gm , (2) l∈A

i6=m l∈A

2

where nk ∼ CN 0, σk represents the additive white Gaussian noise at the k-th MU, which is assumed to be independent from the transmitted signal sm . For the k-th MU in the multicast group m, the signal-to-interference-plus-noise ratio (SINR) is given by P 2 H l∈A hlk vlm , ∀k ∈ Gm . (3) Γk,m = P P 2 H + σ2 i6=m l∈A hlk vli k We also assume that the l-th RRH has the maximum transmit power constraint with Pl > 0, which can be written as M X

2

kvlm k2 ≤ Pl , ∀l ∈ A.

consumption representing the static power consumption of the RRH and the corresponding transport fronthaul PL link when switched on, and v ∈ CN ×M with N = l=1 Nl is the aggregative beamforming vector given by T T T T T v = v11 , . . . , v1M , . . . , vL1 , . . . , vLM . (6) B. Group Sparse Formulation Based on the power model in Eq. (5), we will focus on the network power minimization problem with the QoS constraints and per-RRH power constraints. Assume that perfect channel state information (CSI) is available at the BBU pool. With target SINRs as γ = (γ1 , γ2 , . . . , γK ), the multicast network power minimization problem can be formulated as follows,

v,A

subject to

M X XX 1 2 kvlm k2 + Plc , η l m=1 l∈A

(5)

2

kvlm k2 ≤ Pl , ∀l ∈ A

(7)

P 2 H l∈A hlk vlm ≥ γk , 2 P P H + σ2 l∈A hlk vli k

i6=m

∀k ∈ Gm , m ∈ M, which is a joint RRH selection and transmit beamforming design problem. However, with the non-convex combinational composite objective function and the non-convex quadratic QoS constraints, it becomes a highly intractable non-convex mixed integer nonlinear programming problem. To facilitate efficient algorithm design, we reformulate the problem in Eq. (7) via exploiting group-sparsity structure of the aggregative beamforming in Eq. (6). DenotT vector Nv×M T T ˜ l = vl1 ed by v , . . . , vlM ∈ C l , the aggregative beamforming vector at RRH l will be set to zero as the corresponding RRH is switched off. Thus, the optimal beamforming vector v? should have the group-sparsity structure. Specifically, we can define the set of total transmit antennas as V P = {1, 2, . . . , N }, and Plthe l-th partition of V as l−1 Sl = {M i=1 Ni + 1, . . . , M i=1 Ni }, l = 1, . . . , L, such ˜ l = [vi ] is indexed by i ∈ Sl . Therefore, the antenna that v index set with non-zero beamforming coefficients is given by T (v) = {i|vi 6= 0} ,

(8)

where v = [vi ] is indexed by i ∈ V. Therefore, we can rewrite the relative transport link power consumption as follow, F (T (v)) =

m=1

P (v; A) =

l∈A

l∈A M X

m=1

(4)

In order to design a green Cloud-RAN network, it is necessary to minimize the network power consumption, which consists of both transmit power consumption and relative fronthaul link power consumption [2], i.e.,

M X XX 1 2 kvlm k2 + Plc η l m=1

minimize

L X

Plc I (T (v) ∩ Sl 6= 0),

(9)

l=1

where I represents the indicator function, which equals 1 when T (v) ∩ Sl 6= 0 and 0 otherwise. Hence, we have successfully connected the relative fronthaul link power consumption with the group sparse structure in v. Moreover, we can rewrite the transmit power consumption as follow based on the groupsparsity structure of the beamforming vector v.

l∈A

where ηl > 0 is the drain efficiency of the radio frequency power amplifier, Plc > 0 is the relative fronthaul link power

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T (v) =

L X M X 1 2 kvlm k2 . η l m=1 l=1

(10)


Therefore, the network power minimization problem in Eq. (7) can be reformulated as the following multicast group sparse beamforming problem, minimizeF (T (v)) + T (v) v

subject to

M X

2

kvlm k2 ≤ Pl , ∀l ∈ L,

m=1

quadratic variational formulation for the mixed `1 /`2 -norm in the following proposition. Proposition 2: [11] Define x = [x1 , . . . , xL ] ∈ RL + and y = [y1 , . . . , yL ] ∈ RL + , and then we have !2 L L L X X X x2i yi2 , s.t. xi yi = inf µi = 1. (13) µi µ∈RL + i=1 i=1 i=1

(11)

The proof is directly from the Cauchy-Schwarz inequality [11]. In particular, when xi yi , (14) µi = PL i=1 xi yi

Although the reformulated problem in Eq. (11) is still combinatorially hard, we will show the algorithmic advantages of this formulation in the next section based on convex relaxation.

the equality is met in (13). Based on proposition 2, we obtain the following squared weighted mixed `1 /`2 -norm Ω(v),

P 2 H l∈L hlk vlm ≥ γk , 2 P P hH vli + σ 2 i6=m

l∈L

lk

k

∀k ∈ Gm , m ∈ M.

Ω2 (v) = P

III. M ULTICAST G ROUP S PARSE B EAMFORMING A LGORITHM BASED ON S EMIDEFINITE P ROGRAMMING In this section, we shall propose a three-stage algorithm for the multicast group sparse beamforming problem in Eq. (11). Specifically, in the first stage, a novel group-sparsity inducing penalty function based on the variational formulation of the weighted `1 /`2 -norm is proposed to relax the nonconvex combinatorial composite objective function of problem in Eq. (11). In the second stage, we will order the RRHs based on the obtained (approximated) group sparse beamforming vector and system parameters to determine which RRHs should be switched off. In the final stage, the transmit power consumption minimization for the determined active RRHs is performed by multicast beamforming design. A. Stage One: Group Sparsity Inducing In this subsection, we will develop a group sparse penalty approach to overcome the combinatorial challenge in problem Eq. (11). Specifically, in [2], we derived a lower bound for the composite objective function in the following proposition. Proposition 1: [2] The tightest convex positively homogeneous lower bound of the objective function in Eq. (11) is given by s L X Plc Ω(v) = 2 k˜ vl k2 , (12) ηl

L l=1

4

µl =1,µl >0

L M X Plc X tr (Clm Qm ), (15) ηl µl m=1 l=1

where Clm denotes the diagonal matrix with Il at the lth main diagonal block and zeros otherwise. Eq. (15) has the same capability of inducing the group-sparsity structure as the nonsmooth one Ω(v) [11]. With the quadratic variables Qm , the per-RRH power constraints are given by M X

tr (Clm Qm ) ≤ Pl , ∀l ∈ L.

(16)

m=1

Furthermore, we can linearize the non-convex quadratic QoS constraints as tr Gm hk hH ≥ γk σk2 , (17) k T where hk = hT1k , hT2k , . . . , hTLk ∈P CN with k ∈ Gm and m ∈ M, as well as Gm = Qm − γk i6=m Qi . Therefore, based on Eqs. (15), (16) and (17), and adopting the SDR technique [10] by dropping the rank-one constraints of Qm , we arrive at the following group-sparsity inducing optimization problem, thereby providing guidelines for the RRH selection as will be shown in Section III-B, ! M L X X Plc tr (Clm Qm + εIN ) minimizie 4 Q,µ ηl µl m=1 l=1

l=1

subject to

which contains the weighted l1 /l2 -norm structure to induce the group-sparsity in the beamforming vector v. Based on the convex relaxation for the objective function, the resultant optimization problem is unfortunately still nonconvex due to the non-convex quadratic QoS constraints. We thus further propose a semidefinite relaxation (SDR) approach [10] to find a good approximate solution to the resultant nonconvex multicast group sparse beamforming problem. Specifically, we lift the problem into higher dimensions by defining H the corresponding optimization variables Qm = vm vm ∈ T T N ×N T T N C , ∀m with vm = v1m , v2m , . . . , vLm ∈ C and rank(Qm ) = 1. To extract the variables Qm , we propose a novel group-sparsity inducing penalty function based on the

inf

M X

tr (Clm Qm ) ≤ Pl , ∀l ∈ L,

(18)

m=1

tr Gm hk hH ≥ γk σk2 , k Qm 0, ∀k ∈ Gm , m ∈ M, where µ = [µl ], l ∈ L, and the perturbed element εINl with ε > 0 is to avoid singularity during the optimization procedure. Observe that the objective function in Eq. (18) is biconvex [7], which means it is convex when fixing either µ or Qm . We shall adopt the alternating optimization algorithm [7] to find we initialize µ[0] = a 1suboptimal solution to it. Specifically, 1 1 [i−1] at the i-th iteration, we L , L , . . . , L . With a fixed µ optimize with respect to Qm , which is an SDP problem and [i] the corresponding optimal solution is denoted as Qm . Then

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the optimal µ[i] can be updated based on proposition 2 with [i] a fixed Qm as follow, r Plc PM [i] tr C Q + εI m lm N m=1 ηl [i] r µl = . (19) PL Plc PM [i] m=1 tr Clm Qm + εIN l=1 ηl With a bounded and non-increasing objective function [7], the alternating algorithm will converge with finite iterations. Simulation results on convergence is shown in Section V. B. Stage Two: RRH Selection

C. Stage Three: Semidefinite Relaxation to the Multicast Beamforming Problem After determining the active RRH set A? , we will minimize the transmit power for the active RRHs. Specifically, by introducing the higher dimension beamforming coefficients Qm and performing the SDP relaxation by dropping the rank-one constraints, we arrive at the following multicast beamforming design problem ! M X 1 X c minimize tr (Clm Qm ) + Pl Qm ηl m=1 ?

Q?m ,

Based on the obtained solution we can determine the active RRH set by extracting the group-sparsity structure information in the beamforming vector v, i.e., 2

k˜ vl k2 =

M X

tr (Clm Qm ), ∀l ∈ L.

Specifically, we adopt the following ordering criterion as in [2], which incorporates both the group-sparsity structure information of the beamforming vector and key system parameters, r M κl ηl X tr(Clm Q?m ), ∀l ∈ L, (21) θl = Plc m=1 PK 2 where κl = k=1 khlk k2 denotes the channel gain between the l-th RRH and the k-th MU. The RRH with a smaller θl will have a higher priority to be switched off. We sort the coefficients θl ’s in the ascending order, i.e., θπ1 ≤ θπ2 ≤ · · · ≤ θπL . The first J smallest coefficients are set to zero, thus the corresponding RRHs are turned off. Denote J0 as the maximum number of RRHs that can be switched off, which can be found by the binary search. Specifically, we need to solve the following feasibility problem, find Q1 , ..., QM M X

tr (Clm Qm ) ≤ Pl , ∀l ∈ A,

m=1

Gm hk hH k

(22)

γk σk2 ,

tr ≥ Qm 0, rank (Qm ) = 1,∀k ∈ Gm , m ∈ M, which is combinatorially hard due to rank-one constraints. Here, we adopt the Phaselift technique [12] to approximate the non-convex combinatorially difficult problem by using a trace norm as a convex surrogate for the rank-one functions, yielding the following SDP problem, minimize Qm

subject to

M X m=1 M X

tr (Qm ) tr (Clm Qm ) ≤ Pl , ∀l ∈ A,

(23)

m=1

tr Gm hk hH ≥ γk σk2 , k Qm 0, ∀k ∈ Gm , m ∈ M. We will apply Gaussian randomization method [10] to obtain an approximate solution if the rank-one constraints fail to satisfy.

tr (Clm Qm ) ≤ Pl , ∀l ∈ A? ,

(24)

m=1

tr Gm hk hH ≥ γk σk2 , k Qm 0, ∀k ∈ Gm , m ∈ M.

(20)

m=1

subject to

subject to

l∈A M X

Since we drop the rank-one constraints, it is necessary to check whether the optimal Q?m satisfy the rank-one constraints. The Gaussian randomization approach [10] will be adopted to find an approximated rank-one solution if it fails. Thus, the proposed semidefinite programming based multicast group sparse beamforming algorithm for network power minimization in Cloud-RAN is presented in Algorithm 1. Algorithm 1: Multicast Group Sparse Beamforming Algorithm Step 0: Solve the group sparse inducing optimization problem in Eq. (18) via alternating algorithm introduced in Section III-A. 1) If it is infeasible, go to End. 2) If it is feasible, calculate the ordering criterion in Eq. (21) and sort θl in the ascending order, go to Step 1. Step 1: Initialize Jlow = 0, Jup = L, J = 0. Step 2: J +J 1) Set J ← b low 2 up c. 2) Solve problem in Eq. (23) with the given active RRH set A: set Jlow = J if it is feasible; set Jup = J otherwise. 3) If Jup − Jlow 6= 1, repeat Step 2; else, go to Step 3. Step 3: Set J0 = Jlow and obtain the optimal active RRH set A? with A? ∪ Z = L and Z = {π1 , . . . , πJ0 }. Step 4: Solve problem in Eq. (24) with the obtained A? . End IV. PARALLEL S EMIDEFINITE P ROGRAMMING S OLVING VIA ADMM A LGORITHM To solve the SDP problems generated in the multicast group sparse beamforming procedure, one may utilize the interiorpoint method, which is implemented in most advanced off-theshelf SDP solvers like SeDuMi and SDPT3. This commonly used method, however, maintains a worst-case complexity of O(N 4 log( 1 )) to achieve a -accuracy solution with > 0 [10]. In addition, such second-order method cannot handle problems with a high dimension and is not effective to enable parallel computation so as to utilize the shared computation resources at the BBU pool. Instead, the first-order method,

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ΠC (Q) =

n X (λi )+ ui uTi ,

(25)

i=1

wherePC = Sn+ is a semidefinite cone, (x)+ = max(x, 0), n T and i=1 λi ui ui is the eigenvalue decomposition of Q. Nevertheless, the computational cost of the semidefinite cone projection is much higher than the one performing secondorder cone projection, e.g., Eq. (32) in [13]. To further enhance the scalability of the first-order method for SDP problems, we need to investigate how to speedup the semidefinite cone projection. In this paper, we mainly emphasize the parallel computation capability of the ADMM algorithm to take advantage of the powerful and shared computational platform at the BBU pool, thereby enhancing scalability. Furthermore, the main motivation for the standard cone programming form transformation is to provide the certificate of infeasibility via solving the corresponding homogeneous self-dual embedding, while most existing works applied the ADMM algorithm, e.g. [16], either assume that the underlying problem is always feasible or only provide a heuristic way to detect infeasibility. In this paper, we use the numeric-based modeling framework CVX [14] to transform the original problem into a standard form, which will be solved by the ADMM algorithm based optimization solver SCS [9]. V. S IMULATION R ESULTS In this section, we simulate the proposed multicast group sparse beamforming algorithm. Denote the channel propagation from the l-th RRH to the k-th MU as hlk = Dlk glk with glk ∼ CN (0, INl ) as the small-scale fading and Dlk as the large-scale fading coefficient.

1 SDPT3 SCS

Empirical Probability of Feasbility

e.g., the ADMM algorithm [8], which has recently been implemented in the solver SCS [9], can provide a flexible framework for efficient computation via performing parallel cone projection and subspace projection, i.e., Eq. (30) and Eq. (29) in [13]. To use the general solver, the original problem needs to be transformed to the standard form that supported by the solver. Specifically, one may either use the numeric-based modeling framework like CVX [14] or symbolic-based framework based on the matrix stuffing technique [13] to transform the original problem in Eq. (24), to the standard conic programming form as Eq. (1) in [9]. To enable infeasibility detection and parallel computation, in the second stage, the ADMM algorithm can be adopted to solve the homogeneous self-dual embedding of the primal-dual pair of the standard cone program. Compared with the second-order cone programming problems addressed in [13], the main difference for the underlying SDP problems in this paper is the cone projection, i.e., Eq. (30) in [13]. Specifically, we need to perform the following semidefinite cone projection, i.e., [Section 6.3.3] in [15],

9 RRHs, 6 MUs

0.8

15 RRHs, 10 MUs 0.6

0.4

0.2

0

0

2

4

6

8

Target SINR [dB]

Fig. 1. The empirical probability of feasibility versus target SINR with different network size using SDPT3 and SCS.

1) Infeasibility Detection: Consider two scenarios with different network sizes, one with L = 15 single-antenna RRHs and K = 10 single-antenna MUs, and the other with L = 9 single-antenna RRHs and K = 6 single antenna MUs. We assume that the MUs are divided into multicast groups with 2 MUs in each group in both scenario. All the channels are spatially uncorrelated. For each MU k, we set Dlk = α1 = 1, ∀l ∈ Ω1 with |Ω1 | = 5; Dlk = α2 = 0.6, ∀l ∈ Ω2 with |Ω2 | = 5; and Dlk = α3 = 0.2, ∀l ∈ Ω3 with |Ω3 | = 5. In particular, Ωi is uniformly drawn from the RRH set index L = {1, . . . , L} with Ω1 ∪ Ω2 ∪ Ω3 = L. Moreover, we set σk2 = 1, ∀k ∈ K and Pl = 1, ∀l ∈ L. The empirical probability of feasibility with different SINRs of the multicast beamforming optimization problem in Eq. (24) is demonstrated in Fig. 1, where ηl = 1, Plc = 0, ∀l ∈ L and A = L. Each point is averaged over 200 randomly and independently generated channel realizations. It can be seen that SCS shows the same capability of infeasibility detection as the interior-point method based solver SDPT3. This feature is essential to implement the proposed multicast group sparse beamforming algorithm, as it requires to solve a sequence of feasibility problems to determine the active RRH set. 2) Transmit Power Minimization: We shall demonstrate the quality of solutions of SCS by solving the multicast beamforming problem in Eq. (24). Consider the same network setting and constraints parameters as scenario one in last subsection, but we set α1 = 1, α2 = 0.8, and α3 = 0.5. We randomly and independently generate 100 channel realizations, and solve the corresponding transmit power minimization problem. The comparison of the average optimal values is illustrated in Table I. From Table I, we can see that the optimal values obtained by the SCS are within 0.011% of that attained by SDPT3. Form Fig. 1 and Table I, we conclude that SCS achieves high-accuracy solutions in terms of providing infeasibility certificates and quality of solutions. In the following subsections, we will use SCS to implement the proposed algorithm. TABLE I C OMPARISON OF Q UALITY OF S OLUTIONS FOR D IFFERENT S OLVERS

A. Comparison of SCS and SDPT3 In this subsection, we shall demonstrate the infeasibility detection capability and the quality of solutions of SCS, compared with SDPT3.

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SINR[dB] SDPT3 SCS

-2 1.0620 1.0622

0 1.8224 1.8224

2 3.1580 3.1579

4 5.4783 5.4777

6 9.5080 9.5091


Target SINR=5 Target SINR=4

Objective Value

1,000

search. Moreover, compared with the `1 /`∞ -norm based algorithm [6], which replaces the objective function of the problem (18) with maxm maxnl1 maxnl2 |Qm (nl1 , nl2 )|, the proposed algorithm with the variational quadratic weighted `1 /`2 -norm provides better performance in minimizing the network power consumption.

900

800

700

600 0

5

10

15

20

Iteration

Average Network Power Consumption [W]

Fig. 2. Convergence of the alternating optimization algorithm with different target SINR. Coordinated Beamforming [3] `1 /`∞ -Norm Algorithm [6] Proposed Algorithm Exhaustive Search

50

40

30

R EFERENCES 20 0

1

2

3

4

5

6

Target SINR [dB]

Fig. 3.

VI. C ONCLUSION In this paper, we proposed a systematic framework based on semidefinite programming for network power minimization via group sparse beamforming in multicast Cloud-RAN. A novel quadratic variational weighted `1 /`2 -norm has been proposed to induce group-sparsity for the beamforming vector. The active RRH set is selected based on the induced groupsparsity structure and the key system parameters. Moreover, the ADMM algorithm aided solver SCS was applied with regard of solving the SDP problems in parallel. Simulation results indicated the infeasibility detection ability and quality of solutions of the parallel semidefinite programming algorithm, as well as the superiority of the proposed multicast group sparse beamforming algorithm.

Average network power consumption versus target SINR.

B. Convergence of the Alternating Optimization Algorithm Consider a network consisting of L = 6 RRHs with 2 antennas and M = 2 multicast groups with 2 single-antenna MUs in each group. All the channels are spatially uncorrelated. For each MU k, we set Dlk = α1 = 1, ∀l ∈ Ω1 with |Ω1 | = 3; and Dlk = α2 = 0.8, ∀l ∈ Ω2 with |Ω2 | = 3. In particular, Ωi is uniformly drawn from the RRH set L with Ω1 ∪ Ω2 . We set Pl = 1, Plc = 5.6 + l − 1, ηl = 25%, ∀l ∈ L, σk2 = 1, ∀k ∈ K and ε = 0.01 N . The convergence of the alternating optimization algorithm with target SINR = 4 and target SINR = 5 are shown in Fig. 2, where we only simulate a typical channel realization for illustrative purpose. We see that the alternating algorithm converges very fast (in less than 20 iterations for this simulation). C. Network Power Minimization Consider a network with L = 6 RRHs equipped with 2 antennas and M = 3 multicast groups with 2 single-antenna MUs in each group. For each MU k, we set Dlk = α1 = 1, ∀l ∈ Ω1 with |Ω1 | = 2; Dlk = α2 = 0.7, ∀l ∈ Ω2 with |Ω2 | = 2; and Dlk = α3 = 0.5, ∀l ∈ Ω3 with |Ω3 | = 2. In particular, Ωi is uniformly drawn from the RRH set L with Ω1 ∪ Ω2 ∪ Ω3 = L. Let Pl = 1, Plc = 3 + l − 1, ηl = 25%, ∀l ∈ L, σk2 = 1, ∀k ∈ K and ε = 0.01 N . The average network power consumption under different target SINRs is demonstrated in Fig. 3, where each point is averaged over 200 randomly and independently generated channel realizations. It is observed that the proposed multicast group sparse beamforming algorithm approaches near-optimal values attained from exhaustive

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