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Optimal Dynamic Point Selection for PowerMinimization in Multiuser Downlink CoMP

Duy H. N. Nguyen,Member, IEEE, Long B. Le,Senior Member, IEEE, andTho Le-Ngoc,Fellow, IEEE

Abstract—This paper examines a CoMP system where multiplebase-stations (BS) employ coordinated beamforming to servemultiple mobile-stations (MS). Under the dynamic point selectionmode, each MS can be assigned to only one BS at any time. Thiswork then presents a solution framework to optimize the BSassociations and coordinated beamformers for all MSs. Withtarget signal-to-interference-plus-noise ratios at the MSs, thedesign objective is to minimize either the weighted sum transmitpower or the per-BS transmit power margin. Since the originaloptimization problems contain binary variables indicating theBS associations, finding their optimal solutions is a challengingtask. To circumvent this difficulty, we first relax the origin alproblems into new optimization problems by expanding theirconstraint sets. Based on the nonconvex quadratic constrainedquadratic programming framework, we show that these relaxedproblems can be solved optimally. Interestingly, with the firstdesign objective, the obtained solution from the relaxed problemis also optimal to the original problem. With the second designobjective, a suboptimal solution to the original problem isthenproposed, based on the obtained solution from the relaxed prob-lem. Simulation results show that the resulting jointly optimalBS association and beamforming design significantly outperformsfixed BS association schemes.

Index Terms—CoMP, multicell system, multiuser, coordinatedbeamforming, dynamic point selection, convex optimization,semidefinite programming.

I. I NTRODUCTION

To improve the spectral efficiency, current designs of wire-less networks adopt universal frequency reuse where all thecells can share the same radio spectrum resources. However,universal frequency reuse comes at the cost of severe in-tercell interference (ICI), especially at cell-edge mobile sta-tions (MS). In the 3GPP LTE-Advanced standard, coordinatedmulti-point transmission/reception (CoMP) is consideredasan enabling technique to actively deal with the ICI [2]. InCoMP, the coverage, throughput and efficiency of the multicellsystem can be significantly improved by fully coordinating andoptimizing the concurrent transmissions from multiple base-stations (BS) to the MSs [2], [3]. Depending on the level ofcoordination among the coordinated cells, a CoMP system canoperate under different modes, namely joint processing/joint

Duy H. N. Nguyen is with the Department of Electrical and ComputerEngineering, San Diego State University, San Diego, CA 92182, USA (e-mail: duy.nguyen@sdsu.edu).

Long B. Le is with INRS-EMT, Universite du Quebec, Montreal, QC,Canada, H5A 1K6 (email: long.le@emt.inrs.ca).

Tho Le-Ngoc is with the Department of Electrical and Computer Engineer-ing, McGill University, 3480 University Street, Montreal,QC, Canada, H3A0E9 (email: tho.le-ngoc@mcgill.ca).

transmission (JP/JT), dynamic point selection (DPS), and co-ordinated scheduling/coordinated beamforming (CS/CB) [2],[4], as illustrated in Fig. 1.

coordinated area

JP/JT

(a) Joint Processing/Joint Transmission (JP/JT): a MS is servedsimultaneously by two BSs.

coordinated area

CS/CB

(b) Coordinated Scheduling/Coordinated Beamforming (CS/CB):a MS’s signal (solid blue arrow) is transmitted by one BS andthe interference (red dashed arrow) coming from the other BSiscoordinated.

coordinated area

DPS

(c) Dynamic Point Selection (DPS): a MS is served by one singleBS at any time and MS-BS association can be changed accordinglyto the channel conditions.

Fig. 1: Example operations of different CoMP modes.

In the JP/JT mode (Fig. 1a), the antennas of a clusterof coordinated BSs form a large single antenna array [5],[6]. The signals intended for a particular MS are simultane-ously transmitted from multiple BSs across cell sites. Thus,JP/JT offers the benefit of large-scale BS cooperation [7],[8]. Asymptotic performance of JP/JT has been analyzed inrecent works through the large system analysis of coordinatedmulticell systems [7]–[9]. Although JP/JT can exploit the bestperformance from the CoMP system, it is the most complexmode in terms of signaling and synchronization among the

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BSs [2]. Per the 3GPP LTE-Advanced Release 11, the JP/JTmode is normally assumed to be “coherent”, meaning that co-phasing of the signs from different coordinated transmissionpoints is performed by means of precoding at the transmitter[2]. Thus, implementing JP/JT will need a high-resolutionadjustable analog delay to each coordinated BS to cope withthe delay variations. For this reason, it is difficult to fullyrealize the potential performance gains of coherent JP/JT,which may limit their applicability only to BSs connectedby a fast backhaul [10], [11]. In addition, coherent JP/JTrequires inter-point phase information as part of the channelstate information (CSI) feedback from multiple points [12].

The CS/CB mode accounts for the least complex CoMPmode. In CS/CB (Fig. 1b), the signal to a single MS is trans-mitted from the serving cell only [2]. However, the beamform-ing functionality is dynamically coordinated between the BSsto control/reduce the ICI [2], [4], [13]. Optimal beamformingdesign for CoMP system under CS/CB mode can be obtainedfrom joint optimization [7], [8], [13]–[17] or game theory [18].To effectively coordinate the inter-cell interference, CS/CBrequires CSI feedback from multiple points [12]. However,by exploiting channel reciprocality [12], optimal downlinkCS/CB can be implemented if a BS knows the CSI only to itsconnected MSs [13].

In DPS mode (Fig. 1c), the MS, at any one time, is beingassociated to a single BS. However, this single associatedBS can dynamically change from time-frame to time-framewithin a set of possible BSs inside the cluster [2], [4],[19]. CoMP DPS provides a good trade-off between thetransmission algorithm complexity, system performance andbackhaul overhead, in comparison to JP/JT and CS/CB [20].In fact, the synchronization issue and the requirement of fastbackhaul communications can be alleviated in the DPS mode,compared to the JP/JT mode. In DPS, each MS’s data hasto be available at all the possible BSs ready for selection.In addition, the beamforming functionality is still neededtocoordinate the transmission across the BSs for interferencecontrol [2]. To facilitate the interference control, DPS demandssimilar CSI feedback as CS/CB such that no inter-pointphase information is required [12]. In fact, when the user-BSassociation is determined, the DPS mode becomes the CS/CBmode. Compared to CS/CB, DPS offers the advantage of siteselection diversity, since DPS can provide a “soft-handoff”solution to among the coordinated BSs to quickly switch thebest BS for association for each MS. However, it is not clearhow a joint BS association strategy and beamforming designin the DPS mode can be optimally determined to maximizethe performance of the CoMP system. In this paper, we areinterested in jointly optimizing the BS association strategy andlinear beamforming design for a CoMP downlink system underthe DPS mode. With a set of target signal-to-interference-plus-noise ratios (SINR) at the MSs, our design objective is tominimize either: i.) the weighted sum transmit power acrossthe BSs or ii.) the per-BS transmit power margin.

A. Related Works

Designing multicell beamformers under the CS/CB modehas attracted a lot of research attention, such as [7], [8],

[13]–[18]. Uplink-downlink duality and iterative fixed-pointiteration have been successfully exploited in [13]–[17] toob-tain optimal beamformers to either minimize the sum transmitpower at the BSs or maximize the minimum SINR at the MSs.Different to these previous studies, part of this work examinesthe application of uplink-downlink duality to optimize themulticell beamformers under the DPS mode.

While the problem of joint BS association and beamformingdesign/power control in uplink transmission has been inten-sively studied [21]–[23], the counterpart problem in downlinktransmission is not well understood. There are few priorstudies in literature which deal with this downlink transmissionproblem. In [24], the problem for downlink transmission hasbeen investigated for the case of power control (not includingbeamforming design). It is stated in [24] that there is noPareto-optimal solution for the problem of joint BS associationand beamforming design in the downlink. The work in [25]tackled the problem of joint downlink beamforming, powercontrol, and access point allocation in a congested system.In[26], the joint optimization of BS association and beamformingdesign was examined and a relaxing-and-rounding techniquewas proposed as a suboptimal solution to the binary variablesindicating the BS association. Recent works in [27]–[29] pro-posed joint BS association and power allocation/beamformingdesign strategies to maximize the multicell system throughput.In another work [30], the problem of joint BS assignmentand power allocation for maximizing the minimum rate ina single-input single-output (SISO) interference channelwasinvestigated. A two-stage algorithm was proposed to iterativelyfind the BS assignment and power allocation for the users[30]. In contrast to these works, our formulation and solutionframework are to attain Pareto-optimal joint BS associationand beamforming design strategies with guaranteed SINRs atthe MSs.

In the context of finding the optimal beamforming designfor power minimization, the optimization can be formulatedas a nonconvex quadratic constrained quadratic programming(QCQP) problem [31]. The nonconvex QCQP is then solvedindirectly via convex semi-definite programming (SDP) re-laxation [31] or a transformation into a convex second-orderconic programming (SOCP) problem [13], [32], [33]. It willbe shown later in this paper that it is not possible to transformthe problems under consideration into a SOCP. Thus, we relyon recent developments in nonconvex QCQP [34] in joinlydevising the optimal BS association strategy and beamformingdesign.

B. Contributions of This Paper

In this work, we formulate the joint BS association andbeamforming design problems as mixed integer programs,which contain the binary variables indicating the BS associa-tions. To circumvent the difficulty in dealing with the binaryvariables and devise optimal joint BS association and beam-forming designs, our proposed solution approaches, which alsoaccount for the main contributions of this paper, are as follows:

• We propose a relaxation method to solve these originalmixed integer programs by relaxing all the binary vari-ables to1 and focusing on optimizing the beamformers.

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These relaxed optimization problems are shown to benonconvex QCQP. Our analysis based the QCQP solutionframework then shows that the relaxed problems can besolved optimally.

• Under the design objective of minimizing the weightedsum transmit power, the obtained solution from the re-laxed problem is also optimal to the original problem.Specifically, this solution indicates both the optimal BSassociation strategy and the optimal beamforming designfor all MSs. Our proposed framework also indicates thatany Pareto-optimal solution can be obtained by properlyadjusting the weight factors in the objective function ofsum transmit power.

• We propose two solution approaches based on the La-grangian duality and the dual uplink problem to find theoptimal solution. Via the dual uplink problem, we proposea distributed algorithm to obtain the optimal joint BSassociation and beamforming design. We show that theDPS can be optimally implemented when a BS knowsthe CSI only to users within its serving user set.

• Under the design objective of minimizing the per-BStransmit power margin, the optimality of the relaxedproblem’s solution to the original problem is not alwaysobserved. Nevertheless, based on the obtained solutionfrom the relaxed problem, a suboptimal solution to theoriginal problem is then proposed. We observe that theperformance gap between the suboptimal solution tothe optimal one is negligible in simulations. Simulationresults also show that the resulting optimal joint BSassociation and beamforming design can significantlyimprove the performance of the CoMP system.

Notations: Superscripts(·)T , (·)∗, (·)H stand for trans-pose, complex conjugate, and complex conjugate transposeoperations, respectively; upper-case bold face letters are usedto denote matrices whereas lower-case bold face letters areused to denote column vectors;diag(d1, d2, . . . , dM ) de-notes anM × M diagonal matrix with diagonal elementsd1, d2, . . . , dM ; [·]i,j denotes the(i, j) element of the matrixargument;x⋆ indicates the optimal value of the variablex;A � B (andA ≻ B) is to indicate the matrix inequality (andstrict matrix inequality) defined on the cone of nonnegativedefinite matrices;A ⊁ B is to denote thatA−B is a semi-definite and singular matrix;|x| denotes the absolute value ofthe scalar numberx whereas|X | denotes the cardinality of thesetX ; C andR denote the sets of complex and real numbers,respectively.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We consider the multiuser downlink transmission in amulticell network consisting ofQ BSs andK MSs operatingon a same frequency band. DenoteQ and K as the set ofBSs and MSs, respectively. It is assumed that each BS isequipped withM transmit antennas and each MS is equippedwith a single receive antenna. In each cell, the BS multiplexesand concurrently sends multiple data streams to multiple MSs.However, each MS can be only associated with one BS at any

time. In the downlink transmission to a particular MS, sayMS-i, its received signalyi can be modeled as

yi =

Q∑

q=1

hHiqxq + zi, (1)

wherexq ∈ CM×1 is the transmitted signal at BS-q, h∗iq ∈

CM×1 represents the channel from BS-q to MS-i, and zi isthe AWGN with a power spectral densityσ2.

Let Qi ⊂ Q be the cluster of coordinated BSs serving user-i, and letKq be the serving user set by BS-q. Let us definebinary variablesaiq, i ∈ K, q ∈ Qi to represent the associationbetween MS-i and BS-q. More specifically, the binary variableaiq = 1 if and only if BS-q ∈ Qi is assigned to serve MS-i. By means of linear beamforming, the transmitted signal atBS-q can be formulated as

xq =∑

i∈Kq

wiqui, (2)

where wiq ∈ CM×1 is the beamforming vector andui isa complex scalar representing the signal intended for MS-i.Without loss of generality, letE[|ui|] = 1. Clearly, if aiq = 0,wiq needs to be set at all-0 vector. If BS-q with q ∈ Qi isselected to serve MS-i, the SINR at MS-i is then given by

SINRiq =

∣

∣hHiqwiq

∣

∣

2

K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2+ σ2

. (3)

B. Problem Formulation

We first consider the joint BS association and beamformingdesign with the design objective of minimizing the weightedsum transmit power across the BSs with a set of target SINRsat the MSs. Letwq be the positive weight for the transmitpower at BS-q. The optimization problem is then stated as

P1 : minimize{aiq},{wiq}

Q∑

q=1

wq

∑

i∈Kq

‖wiq‖2 (4)

subject to∑

q∈Qi

aiqSINRiq ≥ γi, ∀i

aiq = {0, 1}, ∀q ∈ Qi, ∀i∑

q∈Qi

aiq = 1, ∀i,

where the last constraint is to ensure that only one BS inQi

will be associated with MS-i.Remark 1:With a predetermined BS association strategy

(known aiq ’s), problemP1 becomes the CS/CB design prob-lem, whose optimal solution is readily obtainable [13]. In thiscase, the SINR constraints can be cast as convex second-order conic (SOC) constraints, which effectively transformsthe optimization problem into a convex one. However, withthe dynamic BS association strategy, the presence of binaryvariables aiq, problem P1 is a nonconvex mixed integerprogram, which is NP-hard [35]. In fact, an exhaustive searchfor the optimal BS association has exponential complexity andis impractical for implementation.

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One common method to solve a mixed integer program isrelaxing the discrete variables into continuous ones [36].Inthis work, we take a completely different approach by settingthe binary variablesaiq ’s to 1s. More precisely, we considerthe following optimization problem:

P ′1 : minimize

{wiq}

Q∑

q=1

wq

∑

i∈Kq

‖wiq‖2 (5)

subject to∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2

K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2+ σ2

≥ γi, ∀i.

Theorem 1. The minimum weighted sum transmit powerobtained from solving problemP ′

1 is a lower-bound to thatobtained from solving problemP1.

Proof: Suppose that({a⋆iq}, {w⋆iq}) is the optimal solu-

tion to the original joint BS and beamforming problemP1.From the solution(a⋆iq,w

⋆iq), we denoteqi ∈ Qi as the BS

associated with MS-i. Since a MS can only be assigned toone BS, we havea⋆iqi = 1, a⋆iq = 0, ∀q 6= qi, andw

⋆iqi

6= 0

andw⋆iq = 0, ∀q ∈ Qi, q 6= qi. In addition,{w⋆

iqi} must be

an optimal solution to the following problem

P1 : minimize{wiqi

}

K∑

i=1

wqi ‖wiqi‖2 (6)

subject to

∣

∣hHiqi

wiqi

∣

∣

2

K∑

j 6=i

∣

∣hHiqj

wjqj

∣

∣

2+ σ2

≥ γi, ∀i,

where∣

∣hHiqj

wjqj

∣

∣

2is the interference induced by the BS

connected to MS-j, i.e., qj , to MS-i. If additional constraintswiq = 0, ∀q ∈ Qi, q 6= qi, ∀i are introduced to problemP1,we will have the following optimization problem

P1 : minimize{wiq}

Q∑

q=1

wq

∑

i∈Kq

‖wiq‖2 (7)

subject to

∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2

K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2+ σ2

≥ γi, ∀i

wiq = 0, ∀q ∈ Qi, q 6= qi, ∀i.Due to the additional constraintswiq = 0, ∀q ∈ Qi, q 6= qi, ∀i,the objective functions of problemsP1 and P1 are the same.In addition, the numerators in the SINR constraints in the twoproblems are the same, so are the denominators. Thus, problemP1 must yield the same solutionas problemP1, i.e., sameoptimal point. If the additional constraintswiq = 0, ∀q ∈Qi, q 6= qi, ∀i are now removed from problemP1, we willhave problemP ′

1. Since problemP ′1 has a larger feasibility

region than problemP1, the minimum point of problemP ′1

must not exceed that of problemP1. As a result, the optimalpoint of problemP ′

1 is a lower boundto the optimal point ofthe original problemP1.

Corollary 1. If problem P1 is feasible, problemP ′1 is also

feasible. Conversely, the infeasibility of problemP ′1 also

indicates the infeasibility of problemP1.

Corollary 2. Given that{w⋆iq} with i ∈ K, q ∈ Qi is the set

of optimal beamformers obtained from solving problemP ′1, if

there existsqi ∈ Qi such thatw⋆iqi

6= 0 andw⋆iq = 0, ∀q 6= qi,

then{w⋆iq} is also the optimal solution to problemP1.

Proof: This corollary comes directly from Theorem 1 andits proof. The optimal BS association for MS-i is then given bythe BS indexqi ∈ Qi corresponding tow⋆

iqi6= 0. Moreover,

w⋆iqi

is also the optimal beamforming vector for MS-i.In the following sections, we focus on solving problemP ′

1.It is noted the SINR constraint in problemP ′

1 can be restatedas

∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2 ≥ γi

K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2+ γiσ

2. (8)

If there is only one term on the left hand side of theabove inequality constraint, say

∣

∣hHiqi

wiqi

∣

∣

2, one can assume

hHiqi

wiqi to be real. The constraint then can be transformedinto a SOC form [32], which is convex. However, since wenow have the summation of multiple terms

∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2,

with |Qi| > 1, there is no known method to transform thenonconvex quadratic constraint (8) into a convex form,e.g.,SOC constraint. Thus, in order to devise an optimal solutionto problemP ′

1, we rely on the nonconvex QCQP frameworkpresented in Section III. Interestingly, it will be shown theoptimal solution to problemP ′

1 indeed meets the conditionsgiven in Corollary 2.

III. N ONCONVEX QUADRATIC CONSTRAINED QUADRATIC

PROGRAMMING

This section presents a brief background on nonconvexQCQP and exposes relevant properties on strong duality ofnonconvex QCQP. We consider a generic nonconvex QCQPas follows:

QCQP : minimizex∈CN

f0(x) (9)

subject to fi(x) ≤ 0, i = 1, . . . , L,

wherefi(x), i = 0, . . . , L are quadratic, but not necessarilyconvex, functions onx ∈ CN . The Lagrangian of problem (9)is given as

L(x,λ) = f0(x) +L∑

i=1

λifi(x), (10)

where λ , [λ1, . . . , λL]T and λi ≥ 0 is the Lagrangian

multiplier associated with constraintfi(x) ≤ 0, i = 1, . . . , L.The dual function is then given by

g(λ) = infx

L(x,λ). (11)

By nature, the dual functiong(λ) is concave onλ ∈ RL+ [37].

Let p⋆ be the optimal value of problemQCQP and d⋆ bethe optimal value of the dual problemmaximizeλ≥0 g(λ). Bydefinition [37], one has

p⋆ = minx

supλ≥0

L(x,λ), (12)

and d⋆ = maxλ≥0

infx

L(x,λ). (13)

5

Weak duality dictates thatd⋆ ≤ p⋆ and the differencep⋆−d⋆

is called the duality gap (cf. Section 5 in [37]). If strongduality holds,i.e., zero duality gap withd⋆ = p⋆, the optimalsolution of the primal problem can be found through the dualproblem as in (13). While strong duality holds for any convexoptimization problem with Slater’s condition qualification,strong duality also obtains for nonconvex problems on rareoccasions [37]. In any case of having strong duality, a saddlepoint (x⋆,λ⋆) for functionL(x,λ), defined as

L(x⋆,λ)≤L(x⋆,λ⋆)≤L(x,λ⋆), ∀x ∈ CN , ∀λ ∈ RL+, (14)

must exist. The following property, presented in Section 5.4of [37], underlines the connection between the existence ofasaddle point forL(x,λ) and strong duality.

Property 1. If the functionL(x,λ) possesses a saddle point(x⋆,λ⋆) on CN × RL

+, then strong dualityd⋆ = p⋆ holds.Conversely, ifd⋆ is finite withλ⋆ = argmaxλ≥0 g(λ), and theoriginal problem has an optimal solution atx⋆, then(x⋆,λ⋆)is a saddle point ofL(x,λ).

The following property concerning the conditions on theexistence of a saddle point has been presented in [34] and itsproof was partially sketched in Page 1063 of the work.

Property 2. The existence of a saddle point ofL(x,λ) onCN ×RL

+ is equivalent to the following condition: there existsλ⋆ such that the functionL(x,λ⋆) is convex onCN and has aminimizerx⋆ onCN satisfyingλ⋆

i fi(x⋆) = 0, fi(x

⋆) ≤ 0, i =1, . . . , L.

Thus, in order to prove strong duality in a nonconvex QCQPproblem and obtain its optimal solution via its Lagrangiandual problem, it suffices to show that the condition givenin Property 2 is fulfilled [34]. Strong duality in nonconvexQCQP is also guaranteed under the following property, whichis presented as Theorem 6 in [34].

Property 3. Assume that the concave dual functiong(λ) =infx L(x,λ) attains its maximum at a pointλ⋆ ∈ RL

+. IfL(x,λ⋆) is strictly convex onCN , then strong duality holds.

IV. QCQP SOLUTION APPROACH TOPROBLEM P ′1

This section presents an analytical approach to obtain anoptimal solution to problemP ′

1. It is noted that problemP ′1

is a nonconvex QCQP, which is NP-hard in general [34]. Ourapproach is to prove strong duality of this particular problem

P ′1. First, the Lagrangian of problemP ′

1 can be stated as

L1({wiq},λ)

=

Q∑

q=1

wq

∑

i∈Kq

‖wiq‖2

−K∑

i=1

λi

(

1

γi

∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2 −K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2 − σ2

)

=

K∑

i=1

λiσ2

+

K∑

i=1

∑

q∈Qi

wHiq

(

wqI−λi

γihiqh

Hiq+

K∑

j 6=i

λjhjqhHjq

)

wiq. (15)

The dual function is then given byg1(λ) =min{wiq} L1({wiq},λ). Clearly, if any matrixwqI − λi

γihiqh

Hiq +

∑K

j 6=i λjhjqhHjq is not positive semi-

definite, it is possible to findwiq to makeg1(λ) unboundedbelow. Thus, the dual problem is given by

maximizeλ≥0

K∑

i=1

λiσ2 (16)

subject to wqI+K∑

j 6=i

λjhjqhHjq � λi

γihiqh

Hiq , ∀q ∈ Qi, ∀i.

Remark 2: The dual problem is an SDP and a convexproblem by nature. Its optimal solution can be easily obtainedby the interior point method or standard SDP solvers, suchas cvx [38]. However, a closer look on the dual problem(16) can analytically establish an optimal solution to problemP ′1 as well as its feasibility. Note that the dual problem

(16) is always feasible (for instance,λi = 0, ∀i satisfies allthe constraints). However, its feasibility does not necessarilyindicate the feasibility of the primal problemP ′

1. It mayhappen thatλi → ∞ at optimality and all constraints in (16)are still satisfied,i.e., the dual problem is unbounded above.In this case, the primal problemP ′

1 is infeasible thanks to theweak duality properties [37].

We now focus on the case where the optimal value of thedual problem (16) is finite,i.e., the primal problemP ′

1 isfeasible.

Theorem 2. If the nonconvex QCQPP ′1 is feasible, then

strong duality holds.

Proof: Denoteλ⋆ as the optimal solution of the dualproblem. At λ⋆, the functionL1({wiq},λ⋆) is convex in{wiq}. Thus, in order to satisfy the conditions in Property2 as presented in Section III, it is left to find{w⋆

iq} ∈argmin{wiq} L({wiq},λ⋆) such that{w⋆

iq} is feasible toproblemP ′

1 and moreover

λ⋆i

1

γi

∑

q∈Qi

∣

∣hHiqw

⋆iq

∣

∣

2 −K∑

j 6=i

∑

r∈Qj

∣

∣hHirw

⋆jr

∣

∣

2−σ2

= 0, ∀i.

(17)

6

Consider the set of constraints related to MS-i with optimalλ⋆ in the dual problem (16). Suppose that

wqI+K∑

j 6=i

λ⋆jhjqh

Hjq ≻ λ⋆

i

γihiqh

Hiq , ∀q ∈ Qi. (18)

We can increaseλ⋆i to some valueλi such that

wqI+

K∑

j 6=i

λ⋆jhjqh

Hjq � λi

γihiqh

Hiq , ∀q ∈ Qi. (19)

By setting λj = λ⋆j , ∀j 6= i, one yields a feasible solution

λ = [λ1, . . . , λK ]T that improves the objective function ofproblem (16),i.e.,

∑K

i=1 λiσ2 >

∑K

i=1 λ⋆i σ

2. Thus,λ⋆ cannotbe an optimal solution of problem (16) by contradiction. Asa result, there must exist a non-empty subsetQi ⊂ Qi, suchthat

wqI+

K∑

j 6=i

λ⋆jhjqh

Hjq ⊁

λ⋆i

γihiqh

Hiq , ∀q ∈ Qi. (20)

Otherwise,λ⋆i can be further increased. Due to the above set of

inequalities (with⊁), λ⋆i , ∀i must be positive. Forq ∈ Qi\Qi,

strict inequality applies,i.e.,

wqI+K∑

j 6=i

λ⋆jhjqh

Hjq ≻ λ⋆

i

γihiqh

Hiq , ∀q ∈ Qi\Qi. (21)

Since strict inequality (21) is enforced∀q ∈ Qi\Qi,the corresponding beamforming vectorw⋆

iq must be set toall-0 vector in order to have the Lagrangian function (15)minimized. On the other hand, since inequality (20) happensfor q ∈ Qi, there exists an eigenvectorwiq ∈ CM , ‖wiq‖ = 1corresponding to the0 eigenvalue, such that

wHiq

(

wqI+

K∑

j 6=i

λ⋆jhjqh

Hjq−

λ⋆i

γihiqh

Hiq

)

wiq= 0, ∀i, ∀q ∈ Qi.

(22)To minimize the LagrangianL1({wiq},λ⋆), w

⋆iq can be

chosen as a scaled version ofwiq. For each MS, say MS-i, a BS indexed asqi ∈ Qi is randomly chosen and thecorresponding beamforming vectorw⋆

iqiis set as

√δiwiqi ,

where the scaling coefficientδi > 0 will be determinedshortly. For all BSs,q ∈ Qi, q 6= qi, w

⋆iq is purposely

set at 0. Thus, we obtain a set of beamforming vector{w⋆

iq}∀q∈Qi∈ argmin{wiq} L({wiq},λ⋆). The next step is

to determineδi’s such thatw⋆iqi

’s satisfies condition (17).Note thatλ⋆

i > 0, ∀i andw⋆iq = 0, ∀q 6= qi. By substituting

w⋆iqi

=√δiwiqi into (17), we obtain a set of equations

δiγi

∣

∣hHiqi

wiqi

∣

∣

2 − δj

K∑

j 6=i

∣

∣hHiqj

wjqj

∣

∣

2= σ2, i = 1, . . . ,K, (23)

Equivalently,Gδ = 1σ2, whereδ = [δ1, . . . , δK ]T andG ∈RK×K is defined as[G]i,i = (1/γi)

∣

∣hHiqi

wiqi

∣

∣

2and [G]i,j =

−∣

∣hHiqj

wjqj

∣

∣

2.

It is noted that the set of equations in (22) can be cast as

GTλ⋆ = [wq1 , . . . , wqK ]T > 0. (24)

SinceGT is a Z-matrix and there existsλ⋆ > 0 such that

GTλ⋆ > 0, G

T is an M-matrix by its characterization(Condition I28, Theorem 6.2.3 in [39]).1 Thus,G, also anM-matrix, is invertible and its inverse is a positive matrix [39].As a result,δ > 0 can be determined by

δ = G−1

1σ2. (25)

Since{w⋆iqi

=√δiwiqi}∀i now satisfies the set of equations

(22), we yield a feasible solution to the problemP ′1 where

each constraint is met with equality. The qualification ofcondition (17) by{w⋆

iqi} then guarantees the satisfaction of

all conditions in Property 2. Strong duality for problemP ′1

then follows. Furthermore,{w⋆iqi

} must be a globally optimalsolution of the nonconvex problemP ′

1.We now relate the optimal solution{w⋆

iqi} of problemP ′

1

to the original mixed-integer problemP1 as follows.

Proposition 1. The obtained optimal solution{w⋆iqi

} of prob-lemP ′

1 is also optimal to the original mixed-integer problemP1. Furthermore,qi indicates an optimal BS association forMS-i.

Proof: In solving problemP ′1, we derived an optimal

solution wherew⋆iqi

6= 0 and w⋆iq = 0, ∀q 6= qi. Thus,

Corollary 2 is applicable. The optimality of{w⋆iqi

} to problemP1 and the association of MS-i to BS-qi follow.

Through numerous numerical simulations, we observe thatinequality (20) is met at only one BS in the setQi, i.e., |Qi| =1, except for the extremely rare cases where the channels fromtwo BSs are exactly symmetric or two BSs are co-located. Weaddress the cases when|Qi| > 1 in the following proposition.

Proposition 2. If |Qi| > 1, MS-i can be associated to eitherone of the BSs in the setQi without affecting minimumweighted sum transmit power across the BSs.

Proof: If |Qi| > 1, we can first select any BS, sayqi ∈ Qi, such that the corresponding beamformerw

⋆qi

isset to be non-zero andw⋆

q = 0, ∀q 6= qi. The derivationsteps (22)–(25) can be sequentially applied to determine thescaling factor and the beamformers{w⋆

iq} for all the users.Interestingly, different association schemes (withqi ∈ Qi

and their corresponding beamforming designs) might yielddifferent globally optimal solutions to problemP ′

1 with thesame optimal value. The reason for this result is because theobtained solutions(w⋆

iq ,λ⋆) will satisfy the set of equations

(17) and other conditions in Property 2 to be globally optimal.In addition, different BS assignments for MS-i in Qi will alsoyield the same minimum weighted sum power across the BSs,which must equal to the optimal value of the dual problem(16),

∑K

i=1 λ⋆i σ

2. In spite of that, individual transmit powers atthe BSs might not be the same with different BS assignmentsfor MS-i.

Since theP1 can be optimally solved, any Pareto-optimalsolution of the problem can be obtained by properly adjustingthe weight factorwq ’s in the objective function.

1 A square matrixX is a Z-matrix if all its off-diagonal elements arenonpositive. A square matrixX is a P-matrix if all its principle minors arepositive. A square matrix that is both aZ-matrix and aP-matrix is called aM-matrix [39], [40].

7

V. I NTERPRETATION VIA UPLINK-DOWNLINK DUALITY

In the previous section, we have presented an analyticalapproach to solve problemP ′

1 via its Lagrangian dual problem.In this section, we provide an alternative approach for solvingproblemP ′

1 via the well-known uplink-downlink duality. Itwill be shown shortly that the Lagrangian dual problem (16) isindeed the power minimization problem with SINR constraintsin the uplink. We note that uplink-downlink duality is apowerful tool which has been studied in different contextsof multicell beamforming designs [13]–[17]. Fixed-point it-erative algorithms were proposed to find the correspondingoptimal beamforming solutions [13]–[16]. Herein, we showthat uplink-downlink duality is also applicable to the joint BSassociation and beamforming design problem under consider-ation. We then propose an iterative fixed-point algorithm toeffectively solve the problem.

A. Dual Uplink System Model

We consider the dual uplink system with the same settingas in Section II. Specifically, the dual uplink system is derivedfrom the downlink system by transposing the channel matricesand by interchanging the input and the output vectors. Inaddition, the noise at each BS, say BS-q, is assumed to bezero mean AWGN with the covariance matrixσ2wqI. Herein,the single-antenna MS-q is transmitting at powerpi andhiq isthe uplink channel from MS-i to BS-q. If MS-i is associatedwith BS-q where q ∈ Qi, the BS then applies the receivebeamforming vectorwiq to decode MS-i’s signal. In theconsidered uplink system, the BS association is performedby selecting a BS inQi such that MS-i needs to transmitat minimum power to obtain the SINR targetγi at the veryBS. Thus, the design objective now is to jointly optimizethe power allocationpi’s, the receive beamforming vectorwiq, q ∈ Qi, ∀i, and the BS association to satisfy the set ofSINR constraintsγi’s. The joint uplink optimization can bestated as

minimizep1,...,pK ,{wiq}

K∑

i=1

pi (26)

subject to maxq∈Qi

pi∣

∣wHiqhiq

∣

∣

2

K∑

j 6=i

pj∣

∣wHiqhjq

∣

∣

2+ σ2wqw

Hiq wiq

≥ γi, ∀i.

We now underline the connection between the downlinkproblemP ′

1 and the dual uplink problem (26).

Proposition 3. The optimal downlink beamforming problemP ′1 can be solved via a dual uplink problem in which the SINR

constraints remain the same. Specifically, the Lagrangian dualproblem(16) of problemP ′

1 is the following problem

minimizeλ1,...,λK ,{wiq}

K∑

i=1

λiσ2 (27)

subject to maxq∈Qi

λiσ2∣

∣wHiqhiq

∣

∣

2

K∑

j 6=i

λjσ2∣

∣wHiqhjq

∣

∣

2+ σ2wqw

Hiq wiq

≥γi, ∀i,

where pi = λiσ2 is dual uplink power of MS-i. If the dual

uplink problem(27) is feasible, its optimal solution is alsooptimal to the Lagrangian dual problem(16). Otherwise, theLagrangian dual problem(16) is unbounded above.

Proof: For given uplink power allocationpi = λiσ2, the

optimal receive beamforming vector at BS-q, q ∈ Qi is theminimum mean-squared error (MMSE) receiver

wiq =

(

K∑

j=1

λjhjqhHjq + wqI

)−1

hiq. (28)

By substituting the above MMSE receiverwiq, the SINRconstraint for MS-i in (27) becomes

λi

(

1 +1

γi

)

·maxq∈Qi

hHiq

(

K∑

j=1

λjhjqhHjq + wqI

)−1

hiq ≥ 1. (29)

Note that the above set of constraints fori = 1, . . . ,K mayconstitute an empty set, which then renders the dual uplinkproblem infeasible. However, if the dual uplink problem (27)is feasible, at optimality the set of inequality constraints (29)must meet at equality,i.e.,

λi

(

1+1

γi

)

·maxq∈Qi

hHiq

(

K∑

j=1

λjhjqhHjq + wqI

)−1

hiq = 1, ∀i.

(30)Thanks to Lemma 1 in [33] as provided following this proof,

the constraint in the Lagrangian dual problem (16) can berecast as

λi

(

1+1

γi

)

hHiq

(

K∑

j=1

λjhjqhHjq + wqI

)−1

hiq ≤ 1, ∀q ∈ Qi,

or equivalently,

λi

(

1 +1

γi

)

·maxq∈Qi

hHiq

(

K∑

j=1

λjhjqhHjq+wqI

)−1

hiq ≤ 1. (31)

Unlike the dual uplink problem (27), the Lagrangian dualproblem (16) is always feasible, thanks to its nonempty con-straint set. In case of having a finite optimal value, it is clearthat at optimality the set of inequality constraints (31) must bemet at equality, as given in (30). Thus, the power minimizationproblem (27) of

∑K

i=1 λiσ2 with minimum SINR constraints

(29) and the power maximization problem (16) of∑K

i=1 λiσ2

with maximum SINR constraints in (31) are equivalent sinceλi’s in both problems are the fixed point of the equations (30).It will be shown shortly that this fixed point is unique if itexists. In that case, the fixed point is the optimal solution forboth problems. If a fixed point does not exist, the dual uplinkproblem (27) is not feasible and equivalently the Lagrangiandual problem (16) is unbounded above.

For completeness, Lemma 1 in [33] is presented as follows:“Let A be ann × n symmetric positive semidefinite matrixand b be ann × 1 vector. Then,A � bb

H if and only ifbHA

−1b ≤ 1.”

8

B. An Iterative Algorithm for Solving ProblemP ′1

Having established the equivalence between the Lagrangiandual problem (16) and the dual uplink problem (27), thissection focuses on obtaining the solution to both problemsby finding the fixed point to the set of equations (30). Byrearranging (30) into a fixed point iteration, one has

λ(n+1)i = min

q∈Qi

fiq(

λ(n))

, (32)

wherefiq(λ) is defined as

fiq(λ) =γi

1 + γi· 1

hHiqΣ

−1q hiq

. (33)

andΣq =∑K

j=1 λjhjqhHjq + wqI.

Proposition 4. If a fixed point of(30) exists, it is unique andthe iterative function evaluation(32) converges geometricallyfast to the fixed point.

Proof: It is proven in [32], [33] thatfiq(λ) satisfies thethree properties (positivity, monotonicity, and scalability) tobe a standard function. Moreover, the point-wise minimumof a set of standard function,i.e., minq∈Qi

fiq(λ), is alsoa standard function [41]. Thus, the iteration (32) convergesgeometrically fast to the fixed point, if it exists.

The iteration (32) accounts for the first step of the iterativealgorithm to solve problemP ′

1. The second step is to findthe optimal receive beamformerwiqi , where qi is the BSassociation with MS-i. Then, the final step is to obtain theoptimal transmit beamformerwiqi . We summarize these threesteps in the following Algorithm 1.

Algorithm 1: Iterative Algorithm for Minimizing TheWeighted Sum Transmit Power

1 Initialize λi > 0, ∀i;2 repeat3 Updateλi as given in equation (32),4 until convergence toλ⋆;5 Setqi = argminq∈Qi

fiq(λ⋆),∀i;

6 Find the receive beamformerwqi as given in equation (28);7 Find the transmit beamformerw⋆

iqi=

√δiwqi , whereδi is

given by (25);

C. Distributed Implementation

An interesting development from the above uplink-downlinkduality interpretation is that all the three steps in the iterativealgorithm proposed in the previous section can be imple-mented distributively. Herein, it is assumed that the systemis operating in the time division duplex (TDD) mode wherethe uplink and downlink channels are reciprocal. It is alsoassumed that the weightwq is known at BS-q.

In the first step, the iteration (32) on the uplink powerλi involves only its channel vectorshiq ’s and the matricesΣq ’s obtained from the BSs inQi. With known wq, BS-qcan compensate the background noise towqσ

2. Then Σq,as the covariance matrix of the total received signal at BS-q in the uplink, can be estimated locally by the BS. Thus,the transmit powersλi’s can be updated as in (32) on aper-user basis without inter-BS or inter-user coordination.

Should the acquisition of the channelhiq ’s or the matricesΣq ’s be challenging at the MSs, BS-q can simply calculatefiq = γi

1+γi· 1hH

iqΣ−1

q hiq

for i ∈ Kq as the required transmit

power at MS-i to obtain its target SINRγi at the very BS.Subsequently,fiq ’s are passed to MS-i, who will choose thelowest uplink powerλi = minq∈Qi

fiq. The BS that canachieve the SINRγi with the uplink powerλi is then the oneassociated with MS-i. Thanks to Proposition 4, these iterativesteps always converge to a fixed point if it exists.

While the second step to determine the MMSE receivers(28) is straightforward at the BSs, the final step to calculatethe scaling factorsδi’s is more involved. In particular, althoughδi’s are found as in (25), this matrix inversion process requirescentralized implementation. On the other hand, findingδi’sis equivalent to the downlink power control problem forachieving a set target SINRsγi’s. One solution approach isthe Foschini-Miljanic’s algorithm where the optimal downlinkpowers can be found iteratively in a fully distributed mannerusing per-user power updates [42].

Remark 3:In downlink CS/CB, it is shown in [13] that theoptimal downlink beamforming can be obtained even if eachBS only knows the CSI to its connected MSs by exploitingchannel reciprocality. Via the distributed implementation pre-sented in this section, we show that the optimal BS associationand beamforming design in the DPS mode can be devisedif each BS only knows the CSI to the MSs in its servinguser set,i.e., BS-q needs the CSIhiq to the MSs inKq. Byexploiting channel reciprocality, BS-q can listen to the trainingsignal from MS-i in the uplink transmission for estimatinghiq. The only signaling or feedback involved is passing of therequired transmit powerfiq for MS-i to connect to BS-q in theuplink. MS-i is then required to make the recommendation ofits selected BSqi, i.e., qi = argminq∈Qi

fiq. This selectionrecommendation by the MSs is consistent with the CoMPimplementation in the LTE Release 11 [12].

VI. SEMIDEFINITE PROGRAMMING RELAXATION

In this section, we present the SDP relaxation approachto find an optimal solution to problemP ′

1. It is well knownthat SDP relaxation can be successfully exploited to find theoptimal multiuser beamforming design for single-cell systems[31], [32], [43]. To apply the SDP relaxation to the multicellsystem model under consideration, we first replacewiqw

Hiq by

Xiq andhiqhHiq by Hiq and recast problemP ′

1 into an SDP

minimize{Xiq}

Q∑

q=1

wq

∑

i∈Kq

Tr{Xiq} (34)

subject to1

γi

∑

q∈Qi

Tr{HiqXiq}−K∑

j 6=i

∑

r∈Qj

Tr{HirXjr}≥σ2, ∀i

Xiq � 0, rank{Xiq} = 1.

Since the rank constraint is nonconvex, we remove it andrelax problem (34) into a convex SDP. Once we have a convexSDP, the interior-point method can be applied to find itsoptimal solution. Through numerous numerical simulations,we found a similar result reported [31], [43] that a rank-1 solution of the SDP relaxation problem can always be

9

found. Thus, it is possible to retrievewiq from the obtainedsolution inXiq. In addition, it is even more interesting thatamong the optimal solution set related to a particular MS,e.g., {Xiq}, q ∈ Qi, there is onlyone non-zero (and rank-1) matrix. As a result, solving the SDP relaxation version ofproblem (34) provides the optimal solution not only to thebeamforming problemP ′

1, but also to the original joint BSassociation and beamforming design problemP1.

It is noted that the obtained optimal result from the SDPrelaxation approach can be proved analytically. First, it can beshown that the Lagrangian dual of the SDP relaxation versionof problem (34) is the same as problem (16). Second, problem(16) is the dual problem of the QCQPP ′

1 and strong dualityholds. Thus, our proposed framework via nonconvex QCQPin Section IV provides a rigorous analytical confirmation tothe numerical results obtained here by the SDP relaxationapproach. Nonetheless, the drawback of the SDP relaxationapproach is the complexity in solving the relaxed version ofproblem (34) due to the expanded set of variables. Certainly,solving the convex SDP in the Lagrangian dual problem (16)is much simpler.

VII. M INIMIZATION OF THE PER-BASE-STATION

TRANSMIT POWER MARGIN

In the optimization problemP1, the adjustment of theweight factorswq ’s provides a trade-off among the powerconsumptions at different BSs. In this section, we considera practical scenario of minimizing the transmit power marginacross the BSs, in which the weights are implicitly determined.To jointly optimize the BS association and the beamformingdesign, the optimization problem can be formulated as follows:

P2 : minimize{aiq},{wiq},α

α

Q∑

q=1

Pq (35)

subject to∑

q∈Qi

aiqSINRiq ≥ γi, ∀i

aiq = {0, 1}, ∀q ∈ Qi, ∀i∑

q∈Qi

aiq = 1, ∀i,∑

i∈Kq

‖wiq‖2 ≤ αPq .

Herein,α represents the margin between the transmit powerof a BS, say BS-q, and its maximum power valuePq. Byminimizing α, the multicell system tries to balance the powerconsumptions across the BSs and does not overuse any ofthem. This formulation is especially beneficial to heteroge-neous multicell systems wherePq ’s can be different by oneor two orders of magnitude. The resulting optimalα⋆ fromproblem P2 is also important to verify the compliance ofindividual power constraints at the BSs. Specifically, ifα⋆ ≤ 1,then it is feasible to find an optimal BS association andbeamforming design to meet all the SINR constraints and theper-BS power constraint

∑

i∈Kq‖wiq‖2 ≤ Pq, ∀q.

Similar to problemP1, problemP2 is a difficult nonconvexmixed integer program. Thus, we take a similar approach in

solving problemP1 by relaxing problemP2 into the followingoptimization problem:

P ′2 : minimize

{wiq},αα

Q∑

q=1

Pq (36)

subject to∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2

K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2+ σ2

≥ γi, ∀i

∑

i∈Kq

‖wiq‖2 ≤ αPq.

In other words, we let all the binary variablesaiq to be1. Letus denoteα⋆ andα− as the optimal solutions in problemsP2

andP ′2, respectively.

Remark 4:Theorem 1 is also applicable to the relaxation ofproblemP2 into problemP ′

2. Corollary 2 is also applicableto the optimal solution of problemP ′

2. Unfortunately, solvingproblemP ′

2 does not always provide us a solution that meetsthe conditions in Corollary 2. To illustrate this observation, letus consider a simple system setting withK = 1 andQ = 2.In solving problemP ′

1, the MS will be assigned to the BSthat requires the lowest transmit power. However, under theproblem formulationP ′

2, the obtained solution will result innon-zerotransmit powers at both BSs to haveα minimized,i.e., the transmit powers are split and balanced at the both BSs.Nevertheless, in solving problemP ′

2, one can obtain the lowerbound on the optimal value of problemP2.

Remark 5:Suppose that one has obtained{w−iq} as the

optimal solution to problemP ′2. Let q+i = argmaxq∈Qi

SINRiq = argmaxq∈Qi

∣

∣hHiqw

−iq

∣

∣

2be the BS association with

MS-i. Then, for a known BS association profile{q+i }, i =1, . . . ,K, an optimal beamforming design for minimizing thetransmit power margin across the BSs can be easily found [44]by solving the following optimization

minimize{wiqi

},αα

Q∑

q=1

Pq (37)

subject to

∣

∣hHiqi

wiqi

∣

∣

2

∑K

j 6=i

∣

∣hHiqj

wjqj

∣

∣

2+ σ2

≥ γi, ∀i∑

i∈Kq,qi=q

‖wiqi‖2 ≤ αPq.

We denote the obtained per-BS transmit power margin asα+.Certainly,α− andα+ serve as a lower bound and an upperbound onα⋆, i.e.,

α− ≤ α⋆ ≤ α+. (38)

We observe through numerous simulations that the gap be-tweenα− andα+ is nonexistent for most of the simulations(with K > 1). For these cases, solving problemP ′

2 doesprovide the optimal solution of problemP2 too. For othercases, the BS association profile{qi}, i = 1, . . . ,K and itscorresponding beamforming design can be employed as asuboptimal solution to problemP2.

10

A. QCQP Solution Approach to ProblemP ′2

In this section, we apply the QCQP solution approachpresented in Section IV to devise the globally optimal solutionto problemP ′

2. The Lagrangian of problemP ′2 can be stated

as

L2({wiq}, α,λ,µ)

= α

Q∑

q=1

Pq +

Q∑

q=1

µq

(

∑

i∈Kq

‖wiq‖2 − αPq

)

−K∑

i=1

λi

(

1

γi

∑

q∈Qi

∣

∣hHiqwiq

∣

∣

2 −K∑

j 6=i

∑

r∈Qj

∣

∣hHirwjr

∣

∣

2 − σ2

)

=

K∑

i=1

λiσ2 + α

( Q∑

q=1

Pq −Q∑

q=1

µqPq

)

+K∑

i=1

∑

q∈Qi

wHiq

(

µqI−λi

γihiqh

Hiq+

K∑

j 6=i

λjhjqhHjq

)

wiq, (39)

where λi’s and µq ’s are the Lagrangian multipliers associ-ated with the SINR and the power constraints andλ ,

[λ1, . . . , λK ]T andµ , [µ1, . . . , µQ]T . The dual function is

then given byg2(λ,µ) = min{wiq},α L2({wiq}, α,λ,µ). Ifany matrixµqI − λi

γihiqh

Hiq +

∑K

j 6=iλjhjqhHjq is not positive

semi-definite or∑Q

q=1 Pq <∑Q

q= µqPq, it is possible to findwiq or α > 0 to makeg2(λ,µ) unbounded below. Thus, thedual problem is defined as

maximizeλ≥0,µ≥0

K∑

i=1

λiσ2 (40)

subject to µqI+

K∑

j 6=i

λjhjqhHjq � λi

γihiqh

Hiq , ∀q ∈ Qi, ∀i

Q∑

q=1

µqPq ≤Q∑

q=1

Pq.

Compared to problem (16) with pre-determined weightfactorswq ’s, the variableµq, functioning as the weight for thetransmit power at BS-q, have to be optimized in problem (40).Since the dual problem (40) is convex, its optimal solutioncan be efficiently obtained by standard convex optimizationtechniques.

Let λ⋆ and µ⋆ be the optimal solution of problem (40).Except an extremely rare case where the channels from theBSs to the MSs are exactly symmetric, it is not possible tohaveµ⋆

qI +∑K

j 6=iλ⋆jhjqh

Hjq ⊁

λ⋆i

γihiqh

Hiq , ∀q ∈ Qi, ∀i. Thus,

L2({wiq}, α,λ⋆,µ⋆) is a strictly convex function. Accordingto Property 3 in Section III, strong duality holds,i.e., theoptimal solution of problemP ′

2, can be found through thedual problem (40). It is noted thatw−

iq must be set to0 to have the Lagrangian function (39) minimized shouldµ⋆qI +

∑K

j 6=iλ⋆jhjqh

Hjq ≻ λ⋆

i

γihiqh

Hiq . By applying the same

argument as in the proof of Theorem 2, the optimal solution{w−

iq} to problemP ′2 typically has a sparse structure.

In one special case, among the set of constraints related to

a MS, say MS-i, if there existsonly oneqi ∈ Qi such that

µ⋆qiI+

K∑

j 6=i

λ⋆jhjqih

Hjqi

⊁λ⋆i

γihiqih

Hiqi

, (41)

and

µ⋆qI+

K∑

j 6=

λ⋆jhjqh

Hjq ≻ λ⋆

i

γihiqh

Hiq , ∀q ∈ Qi, q 6= qi, (42)

then one hasw−iqi

6= 0 and w−iq = 0, ∀q 6= qi, ∀i as the

optimal solution to problemP ′2. Since the requirements in

Corollary 2 are now satisfied, the optimal solution of problemP2 is also found. In other cases, the steps given in Remark 5can be applied to generate a suboptimal BS association andbeamforming design solution of problemP2. We summarizethe steps to obtain lower and upper bounds on the optimalper-BS transmit power marginα⋆ and a suboptimal solutionof problemP2 in Algorithm 2.

Algorithm 2: Iterative Algorithm for Minimizing The Per-BS Transmit Power Margin

1 Solve the dual problem (40) to obtain the optimal dualsolutionsλ⋆ andµ⋆;

2 Obtain the optimal primal solution{w−

iqi}, α− of problemP ′

2;3 Setα− as the lower bound onα⋆;4 Verify if conditions (41)–(42) are satisfied;5 if yes;6 then {w−

iqi} is optimal to problemP2;

7 otherwise setqi = argmaxq∈Qi

∣

∣hHiqw

−

iqi

∣

∣

2;

8 Set{qi} as a suboptimal BS association strategy;9 Solve problem (37) to obtain the upper boundα+ on α⋆ and

the corresponding beamforming design;

B. A Comparison to the Relaxation-and-Rounding Techniquesin [26]

In a prior work [26], we proposed two relaxation-and-rounding techniques to solve the joint BS association andbeamforming design problemP2. Of the two techniques, thebetter performing ‘relaxation-based-2’ approach first relaxesall BS association variables{aiq} to 1 and finds the beam-forming design through the optimization

P2 : minimize{wiq},α

α

Q∑

q=1

Pq (43)

subject to

∣

∣

∣

∣

∑

q∈Qi

hHiqwiq

∣

∣

∣

∣

2

K∑

j 6=i

∣

∣

∣

∣

∑

r∈Qj

hHirwjr

∣

∣

∣

∣

2

+ σ2

≥ γi, ∀i

∑

i∈Kq

‖wiq‖2 ≤ αPq.

The obtained solution, denoted as{wiq}, is then utilized forgenerating a BS association profile in a similar fashion as givenin Remark 5. Subsequently, the beamforming design can befound accordingly to the generated BS association profile [26].

11

It is noted that there is a subtle difference between problemsP ′2 and P2 in expressing the SINR constraints. In fact, with

the SINR constraint expression in (43), problemP2 mimicsthe optimization of the beamformers in a single-cell systemwith power constraints per groups of antennas. It is noted thatproblem P2 can be recast into a convex second-order conicprogram (SOCP) [26]. A quick verification on the dual prob-lem of the optimizationP2 would indicate that the obtainedoptimal solution{wiq} does not have a sparse structure,i.e.,wiq 6= 0, ∀i, ∀q. Hence,{wiq} cannot be an optimal solutionof the original problemP2, unlike the solution{w−

iq} obtainedfrom solving problemP ′

2. In addition, the obtained optimalvalue from solving problem (43), denoted asα, is typicallymuch smaller than the one from solving problemP ′

2, i.e., α−.Thus, the relaxation-and-rounding approach in [26] usuallygenerates a large gap between the lower bound and upperbound onα⋆, unlike the proposed QCQP solution approachproposed in this work. A numerical comparison between thetwo approaches will be presented in the simulation to verifythis observation.

C. SDP Relaxation

In order to obtain a globally optimal solution to the opti-mization problemP ′

2, we can also apply the SDP approach.Let Xiq = wiqw

Hiq andHiq = hiqh

Hiq , the QCQPP ′

2 can berecast as an SDP

minimize{Xiq},α

α

Q∑

q=1

Pq (44)

subject to1

γi

∑

q∈Qi

Tr{HiqXiq}−K∑

j 6=i

∑

r∈Qj

Tr{HirXjr}≥σ2, ∀i

∑

i∈Kq

Tr{Xiq} ≤ αPq

Xiq � 0.

Herein, the rank constraintrank{Xiq} = 1 is dropped torender problem (44) as a convex SDP. This convex relaxationSDP then can be optimally solved by the interior-point methodand a convex SDP solver likecvx [38].

Since strong duality holds for the QCQPP ′2 and the dual

problem of the SDP (44) is also (40), a rank-1 solutionof {Xiq}, ∀q, ∀i can always be found. Should the obtainedsolution of problem (44) meet all the requirements in Corollary2, it is also the optimal solution to the joint optimization of BSassociation and beamforming design problemP2. Otherwise,the approximation steps in Remark 5 can be applied togenerate a suboptimal solution of problemP2.

VIII. N UMERICAL RESULTS

This section presents the numerical evaluations on thepower consumption of a multiuser multicell system employingdynamic BS association. In all simulations, we assume thatthe locations of the BSs are fixed and the distance betweenany two nearby BSs is normalized to1, whereas the MSs arerandomly located. Each BS is equipped withM = 4 transmitantennas. The channel from a BS to a MS is generated from

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Transmit Power at BS−1

Tra

nsm

it P

ower

at B

S−

2

All MSs assigned to BS−2

Pareto optimal tradeoff curve

Minimum sum power

All MSs assigned to BS−1

Fig. 2: The Pareto-optimal tradeoff curve in power consumption between thetwo BSs with optimal joint BS association and beamforming design.

0 40 80 120 160 20010

−5

10−4

10−3

10−2

10−1

100

101

Number of iterations

Nor

m r

esid

ue

SINR Target at 10 dB

SINR Target at 12 dB

SINR Target at 14 dB

SINR Target at 16 dB

Fig. 3: Convergence of the proposed iterative algorithm to solve ProblemP ′1

with different SINR targets. The speed of convergence is slower with higherSINR targets.

i.i.d. Gaussian random variables (Rayleigh fading) using thepath loss model with the path loss exponent of4 and thereference distance of1. The transmit power at each BS islimited at1 W (0 dB). The AWGN power spectral densityσ2

is assumed to be0.01 W while the target SINRs at the MSsare set the sameγ.

In the first simulation setting, we consider a two-cell systemwith 4 randomly located MSs between the two BSs. The targetSINR γ is set at16 dB. For a randomly generated channelrealization, we plot in Fig. 2 the Pareto-optimal tradeoff curvein the transmit powers at the two BSs employing dynamic BSassociation. To obtain each tradeoff point, we varyw1 in theinterval [0, 1] and setw2 = 1−w1. Depending on the weights,our proposed framework can obtain the corresponding Pareto-optimal joint BS association and beamforming design. In fact,it is impossible to find a joint BS association and beamformingdesign that results in a power allocation profile below theplotted Pareto-optimal tradeoff curve. Note that at the extremepoints of the tradeoff curves, the MSs are all assigned to eitherone of the two BSs.

12

BS� BS�

BS�BS�

BS�

BS� BS�

Fig. 4: A seven-cell network grouped into three clusters with ten randomlylocated MSs. A MS can only be associated to one of the BSs in itsassignedcluster.

Fig. 3 illustrates the convergence of the iterative algorithmin Section V-B, which allows us to obtain the optimal solutionto problem P ′

1. In the figure, we plot the norm residue‖λ(n) − λ⋆‖ (whereλ⋆ is the optimal uplink power vector)versus the number of iterations with different SINR targets. Itis observed that the fixed-point iteration (32) converges veryfast. Interestingly, the speed of convergence becomes slowerwith increasing SINR targets.

In the second simulation setting, we compare the resultsobtained from the optimal BS association (with differentclustering sizes) to that obtained from fixed BS associationschemes. Examples of fixed BS association schemes for a MSare the channel-based scheme (assigned to the BS with thestrongest downlink channel) and the location-based scheme(assigned to the closest BS). With fixed BS association, thebeamforming vectors for the MSs and the transmit powerat the BSs are optimally obtained by means of coordinatedbeamforming [13]. We consider a multicell system with7 BSs(each equipped with four antennas) and10 MSs, as illustratedin Fig. 4. Of the7 cells, we consider two clustering scenarios:i.) universal clustering with all7 cells and ii.)3-cell clusteringwith cluster #1 (cell #1, #2, and #3), cluster #2 (cell #1, #4,and#5), and cluster #3 (cell #1, #6, and #7). In the3-cell clusteringscenario, a MS, say MS-i, is first assigned to a cluster basedon its relative distance to the center of the cluster. MS-i thencan only be associated to one of the3 BSs within its assignedcluster.

Fig. 5 displays the percentage of finding a feasible beam-forming strategy to meet the target SINR at the MSs withdifferent BS association schemes. As the target SINR varies,10, 000 channel realizations at each SINR value are used toobtain the ratios in Fig. 6. Unlike the first simulation settingwith M = K = 4, it is not always possible to find a feasiblebeamforming strategy in the second simulation setting whereM = 4 and K = 10. It is observed from the figure thatthe chance of finding a feasible beamformer design can bedoubled by the proposed DPS strategy, thanks to the optimaland dynamic association of the MSs to the BSs. In contrast, bypre-determining the associations, an optimal CS/CB strategyusing [13] may not be found at high probability. Interestingly,

10 11 12 13 14 15 160

20

40

60

80

100

Target SINR γ in dB

Per

cent

age

of F

indi

ng a

Fea

sibl

e S

olut

ion

7−cell cluster − Optimal DPS

3−cell cluster − Optimal DPS

Channel−based with CS/CB

Location−based with CS/CB

Fig. 5: Percentage of finding a beamforming design to meet thetarget SINRwith different BS association schemes. The optimal DPS can double thechance of finding a feasible solution, compared to the fixed BSassociationschemes with optimal CS/CB [13].

10 11 12 13 14 15 1612

15

18

21

24

27

30

Target SINR γ in dB

Sum

Tra

nsm

it P

ower

in d

Bm

7−cell cluster − Optimal DPS

3−cell cluster − Optimal DPS

Channel−based with CS/CB

Location−based with CS/CB

Fig. 6: Average sum transmit powerversustarget SINR with different BSassociation schemes. The optimal DPS can save more than5 dB in the sumtransmit power, compared to the fixed BS association schemeswith optimalCS/CB [13].

by grouping the cells into clusters of3 cells, one can obtainnearly the same optimal performance achieved by the largercluster of7 cells.

Fig. 6 illustrates the average sum transmit power across the7 BSs (with equal weights)versusthe target SINR at the MSs(each MS is set at the same SINR target). As observed fromthe figure, more transmit power is required to meet the highertarget SINR. Out of the considered BS association schemes,it is clearly shown that the optimal joint BS association andbeamforming design significantly outperforms the fixed BSassociation schemes (location-based and channel-based).Inparticular, the optimal joint schemes can save the transmitpower at each BS up to5 dB over the fixed BS associationschemes with optimal CS/CB [13]. It is also observed that theoptimal joint scheme with 3-cell clustering only imposes apenalty of0.5 dB in power usage, compared to the full 7-cellclustering. Clearly, a small cluster size is much more beneficial

13

10 11 12 13 14 15 16−25

−20

−15

−10

−5

0

5

Target SINR γ in dB

Per

−B

S T

rans

mit

Pow

er M

argi

n α in

dB

7−cell cluster − Upper−bound DPS

7−cell cluster − Lower−bound DPS

3−cell cluster − Upper−bound DPS

3−cell cluster − Lower−bound DPS

Channel−based with CS/CB

Location−based with CS/CB

Fig. 7: Per-BS transmit power marginversustarget SINR with different BSassociation schemes. The optimal DPS can provide a much better balance intransmit powers across the BSs and reduce the peak transmit power as muchas7 dB, compared to the fixed BS association schemes with optimalCS/CB[13].

10 11 12 13 14 15 16

−25

−20

−15

−10

−5

Target SINR γ in dB

Per

−B

S T

rans

mit

Pow

er M

argi

n α in

dB

7−cell cluster − Algorithm 2

3−cell cluster − Algorithm 2

7−cell cluster − [23]

3−cell cluster − [23]

Fig. 8: Comparing the per-BS transmit power margins obtained from Algo-rithm 2 and the approach in [26]:solid lines are for the upper bounds anddashed-dottedlines are for the lower bounds. The proposed DPS scheme inAlgorithm 2 provides a much smaller gap between its upper bound and lowerbound, compared to the approach in [26].

for practical implementation.Fig. 7 shows the performance of the joint BS association and

beamforming design for minimizing per-BS transmit powermargin. As observed from the figure, the per-BS transmitpower margin is reduced by at least5 dB to 10 dB by thedynamic BS association schemes proposed in Algorithm 2,compared to the fixed BS association schemes with optimalCS/CB in [13]. Herein, the lower bound was generated bysolving problemP ′

2, whereas the upper bound was generatedby the BS association profile{q+i } accordingly to the solutionof problemP ′

2. It is also observed from the figure that thegap between the two bounds on the transmit power marginas given in (38) is very tight for both3-cell and 7-cellclustering schemes. Hence, the proposed joint BS associationand beamforming design in Algorithm 2 can generate anexceptionally well-performed and near-optimal solution to the

10 11 12 13 14 15 16−30

−25

−20

−15

−10

−5

0

Target SINR γ in dB

Per

−B

S T

rans

mit

Pow

er M

argi

n α in

dB

Minimization of Total Transmit Power

Minimization of Per−BS Transmit Power Margin − Upper−bound

Minimization of Per−BS Transmit Power Margin − Lower−bound

Fig. 9: Comparing the per-BS transmit power margin with3-cell clusteringfor the sum power minimization and the per-BS transmit powermarginminimization. The latter design criterion can reduce the peak transmit powerby 1-2 dB, compared to the former one.

original problemP2.In Fig. 8, we compare the performance between the pro-

posed Algorithm 2 in this work and the prior work in [26]. Asobserved from the figure, by relying on the solution of problemP2, the approach in [26] generates a very large gap betweenthe lower bound and upper bound on the optimal value ofproblemP2. In contrast, the tight gap generated by Algorithm2 allows us to determine the minimum per-BS transmit powermargin more properly. In addition, coupled with a closer upperbound, Algorithm 2 also generates a better suboptimal BSassociation and beamforming design than the approach in [26].

Finally, Fig. 9 compares the per-BS transmit power marginswith 3-cell clustering obtained from the two design objectives:sum power minimization and per-BS power margin minimiza-tion. It is observed from the figure that the per-BS powermargin can be reduced around1-2 dB by the latter designcriterion.

IX. CONCLUSION

This paper has presented a solution framework to obtainan optimal joint BS association and beamforming design fordownlink transmission. The design objective was to minimizeeither the weighted transmit power across the BSs or the per-BS transmit power margin with a set of target SINRs at theMSs. By properly relaxing the nonconvex joint BS associationand beamforming design problems, we have shown that theiroptimal solutions can be obtained via the relaxed problems.Under the first design objective, such optimality is alwaysguaranteed. Two solution approaches based on the Lagrangianduality and the dual uplink problem have been then proposedto find an optimal solution. Under the second design objective,based on the obtained solution from the relaxed problem, anear-optimal solution to the original problem is then proposed.Simulation results have shown the superior performance ofthe optimal joint BS association and beamforming designover fixed BS association schemes. In addition, simulationshows that3-cell clustering is sufficient to obtain a very closeperformance to the universal clustering.

14

REFERENCES

[1] D. H. N. Nguyen, L. B. Le, and T. Le-Ngoc, “Optimal joint basestation association and beamforming design for downlink transmission,”in Proc. IEEE Int. Conf. Commun., London, UK, Jun. 2015, pp. 4966–4971.

[2] 4G Americas,4G Mobile Broadband Evolution: 3GPP Release 11 &Release 12 and Beyond, Feb. 2014.

[3] D. H. N. Nguyen and T. Le-Ngoc,Wireless Coordinated MulticellSystems: Architectures and Precoding Designs, ser. SpringerBriefs inComputer Science. Springer, 2014.

[4] H.-L. Maattanen, K. Hamalainen, J. Venalainen, K. Schober, M. Enescu,and M. Valkama, “System-level performance of LTE-advancedwith jointtransmission and dynamic point selection schemes,”EURASIP J. Adv.Signal Process., vol. 2012, no. 1, pp. 1–18, 2012.

[5] M. Karakayali, G. Foschini, and R. Valenzuela, “Networkcoordinationfor spectrally efficient communications in cellular systems,” IEEE Wire-less Commun., vol. 13, no. 4, pp. 56–61, Aug. 2006.

[6] D. Gesbert, S. Hanly, H. Huang, S. S. Shitz, O. Simeone, and W. Yu,“Multi-cell MIMO cooperative networks: a new look at interference,”IEEE J. Select. Areas in Commun., vol. 28, no. 9, pp. 1380–1408, Dec.2010.

[7] R. Zakhour and S. V. Hanly, “Base station cooperation on the downlink:Large system analysis,”IEEE Trans. Inform. Theory, vol. 58, no. 4, pp.2079–2106, Apr. 2012.

[8] L. Sanguinetti, R. Couillet, and M. Debbah, “Large system analysis ofbase station cooperation for power minimization,”IEEE Trans. WirelessCommun., vol. 15, no. 8, pp. 5480–5496, Aug. 2016.

[9] J. Zhang, C. K. Wen, S. Jin, X. Gao, and K. K. Wong, “Large systemanalysis of cooperative multi-cell downlink transmissionvia regularizedchannel inversion with imperfect CSIT,”IEEE Trans. Wireless Commun.,vol. 12, no. 10, pp. 4801–4813, Oct. 2013.

[10] D. Lee, H. Seo, B. Clerckx, E. Hardouin, D. Mazzarese, S.Nagata,and K. Sayana, “Coordinated multipoint transmission and reception inLTE-advanced: deployment scenarios and operational challenges,”IEEECommun. Mag., vol. 50, no. 2, pp. 148–155, Feb. 2012.

[11] J. Lee, Y. Kim, H. Lee, B. L. Ng, D. Mazzarese, J. Liu, W. Xiao, andY. Zhou, “Coordinated multipoint transmission and reception in LTE-Advanced systems,”IEEE Commun. Mag., vol. 50, no. 11, pp. 44–50,Nov. 2012.

[12] 3GPP Technical Report TR36.819,Coordinated multi-point operationfor LTE physical layer aspects. 3rd Generation Partnership Project,2012.

[13] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicellmulti-antenna wireless system,”IEEE Trans. Wireless Commun., vol. 9,no. 5, pp. 1748–1759, May 2010.

[14] Y. Huang, G. Zheng, M. Bengtsson, K. K. Wong, L. Yang, andB. Ottersten, “Distributed multicell beamforming with limited intercellcoordination,”IEEE Trans. Signal Process., vol. 59, no. 2, pp. 728–738,Feb. 2011.

[15] S. He, Y. Huang, L. Yang, A. Nallanathan, and P. Liu, “A multi-cellbeamforming design by uplink-downlink max-min SINR duality,” IEEETrans. Wireless Commun., vol. 11, no. 8, pp. 2858–2867, 2012.

[16] Y. Huang, G. Zheng, M. Bengtsson, K. K. Wong, L. Yang, andB. Ot-tersten, “Distributed multicell beamforming design approaching paretoboundary with max-min fairness,”IEEE Trans. Wireless Commun.,vol. 11, no. 8, pp. 2921–2933, Aug. 2012.

[17] Y. Huang, C. W. Tan, and B. D. Rao, “Joint beamforming andpowercontrol in coordinated multicell: Max-min duality, effective network andlarge system transition,”IEEE Trans. Wireless Commun., vol. 12, no. 6,pp. 2730–2742, Jun. 2013.

[18] D. H. N. Nguyen and T. Le-Ngoc, “Multiuser downlink beamforming inmulticell wireless systems: A game theoretical approach,”IEEE Trans.Signal Process., vol. 59, no. 7, pp. 3326–3338, Jul. 2011.

[19] R. Agrawal, A. Bedekar, R. Gupta, S. Kalyanasundaram, H. Kroener, andB. Natarajan, “Dynamic point selection for lte-advanced: Algorithms andperformance,” inProc. IEEE Wireless Commun. and Networking Conf.,Apr. 2014, pp. 1392–1397.

[20] M. Feng, X. She, L. Chen, and Y. Kishiyama, “Enhanced dynamic cellselection with muting scheme for dl comp in lte-a,” inProc. IEEE Veh.Technol. Conf., May 2010, pp. 1–5.

[21] S. Hanly, “An algorithm for combined cell-site selection and powercontrol to maximize cellular spread spectrum capacity,”IEEE J. Select.Areas in Commun., vol. 13, no. 7, pp. 1332–1340, Sep. 1995.

[22] F. Rashid-Farrokhi, L. Tassiulas, and K. Liu, “Joint optimal powercontrol and beamforming in wireless networks using antennaarrays,”IEEE Trans. Commun., vol. 46, no. 10, pp. 1313–1324, Oct. 1998.

[23] R. Yates and C.-Y. Huang, “Integrated power control andbase stationassignment,”IEEE Trans. Veh. Technol., vol. 44, no. 3, pp. 638–644,Aug. 1995.

[24] F. Rashid-Farrokhi, K. Liu, and L. Tassiulas, “Downlink power controland base station assignment,”IEEE Commun. Letters, vol. 1, no. 4, pp.102–104, Jul. 1997.

[25] R. Stridh, M. Bengtsson, and B. Ottersten, “System evaluation ofoptimal downlink beamforming with congestion control in wirelesscommunication,”IEEE Trans. Wireless Commun., vol. 5, no. 4, pp. 743–751, Apr. 2006.

[26] D. H. N. Nguyen and T. Le-Ngoc, “Joint beamforming design and base-station assignment in a coordinated multicell system,”IET Communica-tions, vol. 7, no. 10, pp. 942–949, Jul. 2013.

[27] M. Fallgren, H. Oddsdottir, and G. Fodor, “An optimization approachto joint cell and power allocation in multicell networks,” in Proc. IEEEInt. Conf. Commun., Kyoto, Japan, Jun. 2011, pp. 1–6.

[28] M. Hong, R. Sun, H. Baligh, and Z.-Q. Luo, “Joint base stationclustering and beamformer design for partial coordinated transmissionin heterogeneous networks,”IEEE J. Select. Areas in Commun., vol. 31,no. 2, pp. 226–240, Feb. 2013.

[29] M. Sanjabi, M. Razaviyayn, and Z.-Q. Luo, “Optimal joint base stationassignment and beamforming for heterogeneous networks,”IEEE Trans.Signal Process., vol. 62, no. 8, pp. 1950–1961, Apr. 2014.

[30] R. Sun, M. Hong, and Z.-Q. Luo, “Optimal joint base station assignmentand power allocation in a cellular network,” inProc. IEEE Int. Work. onSignal Process. Advances for Wireless Commun., Jun. 2012, pp. 234–238.

[31] M. Bengtsson and B. Ottersten,Optimal and suboptimal transmitbeamforming. L. C. Godara, ed., CRC Press, 2001.

[32] A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precodingvia conicoptimization for fixed MIMO receivers,”IEEE Trans. Signal Process.,vol. 54, no. 1, pp. 161–176, Jan. 2006.

[33] W. Yu and T. Lan, “Transmitter optimization for the multi-antennadownlink with per-antenna power constraints,”IEEE Trans. SignalProcess., vol. 55, no. 6, pp. 2646–2660, Jun. 2007.

[34] H. Tuy and H. D. Tuan, “Generalized S-lemma and strongduality in nonconvex quadratic programming,”Journal of GlobalOptimization, vol. 56, no. 3, pp. 1045–1072, 2013. [Online]. Available:http://dx.doi.org/10.1007/s10898-012-9917-0

[35] K. Aardal, R. Weismantel, and L. A. Wolsey, “Non-standard approachesto integer programming,” inDiscrete Applied Mathematics, 2002, pp. 5–74.

[36] S. Joshi and S. Boyd, “Sensor selection via convex optimization,” IEEETrans. Signal Process., vol. 57, no. 2, pp. 451–462, Feb. 2009.

[37] S. Boyd and L. Vandenberghe,Convex Optimization. United Kingdom:Cambridge University Press, 2004.

[38] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convexprogramming (web page and software). http://stanford.edu/˜boyd/cvx,”Jun. 2009.

[39] A. Berman and R. J. Plemmons,Nonnegative matrices in the mathemat-ical sciences. New York: Academic, 1979.

[40] R. W. Cottle, J.-S. Pang, and R. E. Stone,The linear complementarityproblem. Cambridge, UK: Academic, 1992.

[41] R. D. Yates, “A framework for uplink power control in cellular radiosystems,”IEEE J. Select. Areas in Commun., vol. 13, no. 7, pp. 1341–1347, Sep. 1995.

[42] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous powercontrol algorithm and its convergence,”IEEE Trans. Veh. Technol.,vol. 42, no. 4, pp. 641–646, Nov. 1993.

[43] A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz), “Zero-forcing precodingand generalized inverses,”IEEE Trans. Signal Process., vol. 56, no. 9,pp. 4409–4418, Sep. 2008.

[44] D. H. N. Nguyen and T. Le-Ngoc, “Efficient coordinated multicellbeamforming with per-base-station power constraints,” inProc. IEEEGlobal Commun. Conf., Houston, TX, USA, Dec. 2011, pp. 1–5.