IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012 2941

Nonsmooth Optimization for Efficient Beamformingin Cognitive Radio Multicast Transmission

Anh Huy Phan, Student Member, IEEE, Hoang Duong Tuan, Member, IEEE, Ha Hoang Kha, Member, IEEE, andDuy Trong Ngo, Student Member, IEEE

Abstract—It is known that the design of optimal transmit beam-forming vectors for cognitive radio multicast transmission can beformulated as indefinite quadratic optimization programs. Giventhe challenges of such nonconvex problems, the conventionalapproach in literature is to recast them as convex semidefiniteprograms (SDPs) together with rank-one constraints. Then, thesenonconvex and discontinuous constraints are dropped allowingfor the realization of a pool of relaxed candidate solutions, fromwhich various randomization techniques are utilized with thehope to recover the optimal solutions. However, it has been shownthat such approach fails to deliver satisfactory outcomes in manypractical settings, wherein the determined solutions are foundto be unacceptably far from the actual optimality. On the con-trary, we in this contribution tackle the aforementioned optimalbeamforming problems differently by representing them as SDPswith additional reverse convex (but continuous) constraints. Non-smooth optimization algorithms are then proposed to locate theoptimal solutions of such design problems in an efficient manner.Our thorough numerical examples verify that the proposed algo-rithms offer almost global optimality whilst requiring relativelylow computational load.

Index Terms—Beamforming, cognitive radio, multicast trans-mission, nonsmooth optimization.

I. INTRODUCTION

T HE deployment of numerous broadband wirelessapplications with different service requirements leads

to a huge demand on the expensive radio spectrum. As such,spectrum shortage becomes a significant challenge toward theimplementation of next-generation communication networks.On the other hand, it has been reported that much of the licensedradio spectrum lies idle at any given time and location, and thatthe spectrum shortage results from the spectrum managementpolicy rather than the physical scarcity of the usable frequencies[1]. Spectrum utilization can thus be substantially improvedby permitting secondary access to spectrum holes unoccupied

Manuscript received April 28, 2011; revised October 26, 2011 and February08, 2012; accepted February 16, 2012. Date of publication March 05, 2012; dateof current version May 11, 2012. The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Philippe Ciblat.A. H. Phan is with the School of Electrical Engineering and Telecommuni-

cations, University of New South Wales, UNSW Sydney, NSW 2052, Australia(e-mail: z3261071@student.unsw.edu.au).H. D. Tuan and H. H. Kha are with the Faculty of Engineering and Infor-

mation Technology, University of Technology Sydney, Broadway, NSW 2007,Australia (e-mail: Tuan.Hoang@uts.edu.au; khahoang.ha@uts.edu.au).D. T. Ngo is with the Department of Electrical and Computer Engineering,

McGill University, Montreal, QC HA32A7, Canada (e-mail: duy.ngo@mail.mcgill.ca).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2189857

by the primary (or licensed) users. In recent years, cognitiveradio technology [2]–[4] has been identified as promisingcandidate to effectively exploit the existence of the unoccupiedspectrum portions. Specifically, while the primary users havepriority access to the available radio frequency bands, thesecondary (or unlicensed or cognitive) users have restrictedaccess, subject to a constrained degradation on the primaryusers’ performance. In spectrum sharing environments, thekey design criteria include protecting the primary users fromexcessive interference introduced by the secondary users aswell as satisfying some quality-of-service (QoS) requirementsfor the latter [5]–[7].In multiple-antenna communication systems, transmit beam-

forming has been employed as an effective measure to controlthe level of interference by placing nulls at the direction of eachco-channel receiver. The study of [8] addresses the design ofsuboptimal beamformers in physical-layer multicasting that in-volves only one single multicast group of wireless users. Thework in [9] extends the results of [8] to the case of multiple mul-ticast groups. In the context of cognitive radio communicationswherein the interference introduced from secondary to primarynetworks is strictly regulated, transmit beamforming techniqueturns out to be particularly relevant. The issue of beamformingdesign for cognitive radio network coexisting with a primarysystem has been investigated in [10]. Here, the QoS require-ments of unlicensed network are guaranteed while the total in-terference induced to the primary system is kept under a prede-fined threshold.The optimal transmit beamforming design problems are

formulated as indefinite quadratic optimization programs invarious settings [8]–[13]. Toward solving these difficult non-convex problems, the typical approach in literature involves theapplication of semidefinite program (SDP) relaxation techniquetogether with randomization search. Specifically, the quadraticoptimization problems are recast as SDPs with additionalconstraints which impose that the solution matrices must be ofrank one. Such nonconvex and discontinuous constraints arethen dropped resulting in SDP relaxations. From the set of allpossible solutions obtained by resolving these SDPs, differentrandomization techniques can be employed to generate feasiblesolutions to the original design problems. However, it is worthnoticing that in the scenario that involves multiple cochannelmulticast groups (see, e.g., [9]) or that requires nonzero interfer-ence on the primary system, the randomization procedure mustbe carried out in parallel with the resolution of a large number

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2942 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

of linear programs. This certainly implies a high computationalcomplexity that might not be afforded in many applications.On the other hand, a novel approach to solve rank-constrained

separable semidefinite programs has been recently proposed in[14]. Here, while the proposed relaxed SDP is able to offer op-timal rank-one solutions at three different categories, such op-timality is not always guaranteed. In our earlier work [15], asimple alternative approach for cognitive beamforming in thecase of single-group multicasting has been proposed. Essen-tially, such solution is a modification of the alternating projec-tion in [16] to directly tackle the original indefinite quadraticprogram. Not only have the numerical examples presented in[15] shown that the approach proposed there remarkably out-performs its conventional counterpart, they have also revealedthat the latter often yields solutions that are unacceptably farfrom the actual optimum.Motivated by the shortcomings of existing solutions, this

paper aims to develop an efficient approach which works effi-ciently and consistently in any scenario that involves multipleco-channel multicast groups of cognitive users. Specifically, thecritical indefinite quadratic constraints are expressed as reverseconvex ones, which effectively means that the original beam-forming problems are reformulated as SDPs with additionalreverse convex (but continuous) constraints. Next, the resultingproblems are converted into minimizing a nonsmooth (but con-tinuous) concave function over a set of linear matrix inequalityconstraints, an important class of nonconvex optimization (see,e.g., [17]). An iterative procedure is finally proposed to offeralmost optimal solutions. While the conventional method mightgive solutions that are very far from the lower bounds providedby relaxed SDPs, numerical results show that the our solutionsapproach global optimality in most cases. Moreover, this isachieved at an affordable computational complexity. It shouldalso be noted that indefinite quadratic program is among thehardest classes in algorithmic optimization. Given there is noeffective approach available to solve for the global optimum ofsuch problem [17]–[19], the results presented in this work arenovel even considered from an optimization perspective.The rest of this paper is organized as follows: Section II

presents the system model under investigation and also recallsvarious optimization formulations for the problem of transmitbeamforming design. Section III reviews the capacity of con-ventional randomization SDP approach available in literature.In Section IV, novel algorithms which aim at achieving globaloptimal solutions of the formulated problems are proposed.Next, Section V provides extensive numerical examplesto verify performance of the devised method. And finally,Section VI concludes the paper. Some preliminary results ofthe paper have been appeared in [20] and [21].Notations: Matrices and column vectors are denoted

by boldfaced uppercase and lowercase characters, respec-tively. For a Hermitian matrix , is its maximaleigenvalue, while is its spectral radius defined by

with , beingits eigenvalues. Furthermore, means is positive

Fig. 1. System model.

semi-definite.We denote for a square matrixand for matrices and of ap-

propriate dimension, where is the conjugate transpose of. Accordingly, for two complex vectors and of the samedimension, and accordingly,and . Also, denotes the expectationoperator in respect to random variable .

II. SYSTEM MODEL AND PROBLEM FORMULATIONS

Consider a communication scenario in which a primary basestation (BS) transmits to its primary users (PUs). To effi-ciently implement opportunistic spectrum access, an -antennasecondary BS is also deployed to send information-bearingsignals , each to individual multicast groups

of secondary users (SUs). Effectively, all SUswithin the same group will receive identical information fromthe secondary BS. Assume that each multicast group consistsof secondary receivers and that each SU belongs to only onegroup. Hence, the total number of SUs in the cognitive multi-cast network is indeed . For convenience, if thesecondary receiver belongs to the group , it is denoted as

. As well, let bethe channel vectors between the secondary BS and SU , and

be the channels betweenthe secondary BS and PU . The system model is depicted inFig. 1.The idea of transmit beamforming is for the secondary BS to

apply a beam weight to each infor-mation signal . Then, the resulting vectors are combinedto form the signal which shall be transmitted to allmulticast groups. Suppose that are indepen-dent, each of which has a flat power spectral density (PSD) with

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zero-mean and unit variance. The total transmit power at the sec-ondary BS is thus .The coexistence of PUs and multicast groups of SUs may

cause interference induced by the signals from primary BS,which are destined to its PUs, onto secondary receivers. Evenwithin the secondary network, the simultaneous transmissionsfrom secondary BS to multiple multicast groups also resultin intranetwork interference at each cognitive user. In fact, itcan be shown that the signal-to-interference-plus-noise ratio(SINR) at the secondary receiver is

where models the sum of total interference induced by pri-mary network plus additive noise at cognitive user .Similarly, the signals from secondary BS, which are intended

for its own serviced users, might interfere the reception at thePUs’ receivers. This amount of interference can be expressed as

A. Secondary BS Transmit Power Minimization

Here, the objective is to find optimal beamforming vectorsthat minimize the total radiated power at the secondary BS,constrained on meeting prescribed secondary SINR thresholdsand satisfying tolerable interference threshold at indi-

vidual PUs. Suppose that there is only secondary multi-cast group in the system. Then, by introducing a slack variable

and matrices and , the de-sign problem can be formulated as

(1a)

(1b)

In the case of several secondary multicast groups coexistingwith primary network, the optimization problem (1) can nowbe generalized to amounts to

(2a)

(2b)

B. Secondary User SINR Maximization

Also assuming , a related design is to maximize theminimum secondary receivers’ SINR subject to constraints ona fixed power budget as well as on the interference limitsinduced to the PUs. This involves solving the following opti-mization problem

which can be expressed by the following with additional slackvariables and :

(3a)

(3b)

Regarding the problem of maximizing the minimum secondarySINR which involves multiple multicast groups of SUs (i.e.,

), the denominator of the SINR expression makes theconstraints become highly nonlinear. Since such scenario is dif-ficult to deal with, it is not considered here in this paper.

III. CAPACITY OF CONVENTIONAL SDP RELAXATIONWITH RANDOMIZATION

Towards solving optimization problems in the forms of (1),(2), (3), the existing approach in literature (see, e.g., [8]–[11]) isto recast the formulated problems to relaxed semi-definite pro-grams and generate feasible solutions from the pool of possiblecandidates bymeans of randomization.While our earlier numer-ical results in [15] have revealed the numerical inconsistency ofsuch approach, we will discuss the capacity of this conventionalmethod in the following.Let us begin with (1). According to the conventional method,

the nonconvex but still continuous constraint (1a) is first substi-tuted by the equivalent discontinuous rank-one constraint. Thisresults in an equivalent formulation of (1) as

(4)

Next, the rank-one constraint in (4) is dropped, leading to thefollowing SDP relaxation

(5)

Obviously, (4) belongs to the class of rank constrained SDPs,which has been previously considered in robust control (see,e.g., [22]–[27]). SDP (5) is also the effective convex relaxation

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of the rank minimization problem (see, e.g., [22], [23], [26],and [27])

(6)

If the optimal solution of (5) is of rank one, i.e.,, then is the optimal solution of (1).

Otherwise, the optimal value of (5) simply offers lower-boundperformance. To circumvent this issue, the works in [8] and[10] make use of the following randomization technique inorder to generate feasible solutions to (1).Suppose that . Then, this matrix admits the

singular value decomposition (SVD)

(7)

where matrix is unitary constituting of theeigenvectors of and matrix is diagonal whose diagonalentries are arranged in decreasing order, so for. Then, the feasible solutions of (1) can be generated from

(8)

Here, it is assumed that either the elementsof are uniformly distributed indepen-dent random variables taken on the unit circle in the complexplan, or is a vector of zero-mean unit-variance complex circu-larly symmetric uncorrelated Gaussian random variables. Con-sequently, the randomization for searching suboptimal solutionsof (1) is indeed carried out only on the following -dimensionalsubspace formed by eigenvectors

(9)

It is true that one uses (5) with the hope of arriving at an op-timal solution of as small as possible rank . Clearly, sucha small number actually restricts the space within which therandomization procedure is carried out. We will see that as faras the rank of is not less than 2, i.e., it has at least two dif-ferent strictly positive eigenvalues that make any its rank-oneapproximation poor, the randomization turns out to be both in-consistent and inefficient. In fact, our numerical results, whichwill be presented in later section, reveal that the best (lowest)beamforming power obtained by randomization technique inthe case of nonzero interference on the primary system is evenhigher than that in the complete absence of interference.It is important to note that the generated according to (8) is,

by no means, guaranteed to satisfy constraints (1b). Therefore,it must be first rescaled to meet (1b) as

(10)

After this step, if (1b) cannot be satisfied, it is claimed that therandomization procedure in (8) fails to generate a feasible solu-tion to (1b). Our simulation results confirm that the failure rateof this approach in certain scenarios is quite high.

The foregoing approach can be easily extended to the generalcase of (2) as follows. As before, the equivalent problem of (2)can be expressed as

(11)

where the rank-one constraint can be relaxed to obtain the fol-lowing SDP:

(12)

Denote the optimal solution of (12) and suppose that. It is now possible to perform the SVDs

(13)

where are unitary and are diag-onal with diagonal elements arranged in decreasing order, i.e.,

. Similarly, a feasible solution of (2) canbe obtained according to

(14)

The solutions defined by (14) are contained in-dimensional subspaces formed by

for . In this case, the task of recovering feasiblesolutions from such suboptimal solutions is far more unreliableand computationally costly, compared to that for (1). Indeed,it involves finding the optimal solution of thefollowing linear program:

(15a)

(15b)

and computing the feasible solutions as

(16)

It is imperative to point out that the linear program (15) is notguaranteed to be feasible either; hence, this approach may failto deliver a feasible solution for (2).In brief, the conventional approach is an approximation

method that tries to locate an suboptimal solution throughrandom search on low-dimensional subspace defined (9). Iteven becomes less effective when the number of SUs tendsto be large, as in such cases the rank of matrix oris easily different to 1 but is still low. Apparently, there is no

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reason for the optimal solution to belong to the low dimensionalspace defined by (9) and thus even the search over the wholespace does not bring a good suboptimal solution. Our latersimulation is able also to show that if the tolerable interferencethresholds at PUs are more flexible, the solutions obtained byrandomization procedure are unacceptably far from optimality.

IV. NONSMOOTH OPTIMIZATION APPROACH TO COGNITIVERADIO MULTICAST TRANSMIT BEAMFORMING DESIGN

As can already be seen, the main drawback of the rank-oneconstraints in (4) and (11) are their discontinuous nature thatprevents efficient relaxation. While it is true that the optimalsolutions of (5) and (12) would be rank-dropped, rank-one isactually the lowest for nonzero matrices. Therefore, obtaining arank-one solution immediately after resolving (5) and (12) is notquite expected. On the other hand, the study of [16] has provenin theory that the optimal solutions of (5) and (12) are of ranktwo in most cases. Therefore, it is possible that the random-ization approach may fail to provide desirable results as theirrandom search becomes too narrow.

A. Single-Group Cognitive Radio Multicasting

The previous works [24], [27]–[31] have addressed therank- constrained SDP by smooth optimization,which nevertheless do not seem to work appropriately forrank-one constraint. Motivated by the shortcomings of existingsolutions, in this contribution we develop a novel nonsmoothoptimization approach to resolve the desired problems at hand.Let us start with (1). First, constraint (1a) can be expressed as

(17)

This follows from the fact that

(18)

holds true for any , so (17) implies

which means that there is only one nonzero eigenvalue of .Thus, it is possible to require

where is the unit-norm eigenvector (i.e., ) ofcorresponding the maximal eigenvalue .Based on (17), (1) can now be equivalently expressed as

(19)

Note that function is convex on the set of Hermitianmatrices [32, p. 147]. Then, it is obvious that isa concave function in , meaning that (17) is a reverse convex

constraint [17]. Consequently, (19) is a convex program withadditional reverse convex constraint, an important class of non-convex global optimization [17].It is also essential to point out that for small

enough

(20)

thus allowing to satisfy (1b). Therefore, our aimhere is to make as small as possible. For thispurpose, we incorporate this objective into the cost function,resulting in the following alternative formulation to (19):

(21)

where is a large enough weight to achieve small value of. Clearly, the objective of (21) is to minimize

both and so that the optimum of (1) canbe achieved. As such, this can be regarded as a penalty functionapproach [33], [34]. However, the following result shows that(19) and (21) are equivalent and thus the later provides the exactpenalty optimization for the former.Theorem 1: Suppose that the optimal value of (19) is finite

while the convex feasibility set of (21) is bounded. There issuch that whenever problems (19) and

(21) are equivalent in the sense that they share the same optimalsolution as well as the same optimal value.

Proof: Obviously, each feasible solution of (19) is alsofeasible to (21) for which so the optimal value of(19) is not less than the optimal value of (21) for all . It alsomeans the optimal value of (21) is bounded by the optimal valueof (19) for all . Thus, it suffices to show the existence of

such that for , all optimal solution of (21)must satisfy so they are feasible to(19), implying that the optimal value of (21) is not less than theoptimal value of (19).Note that the feasibility set of (21) is compact. Assume to the

contrary that there is no such . By taking a subsequence ifnecessary, it follows that as with

, i.e., asfor some . This means

, a contradiction with their boundedness.Since its cost function is concave, problem (21) in-

volves minimization of a concave function over a convex set;thus, it belongs to the class of concave programming [17], [25].Moreover, as the function is not smooth (i.e., not dif-ferentiable), is not smooth either. On the other hand, asub-gradient of is because [35], [36]

(22)

Therefore, based on an iteratively feasible of (21) withmaximum eigenvalue and its corresponding unit-

2946 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 6, JUNE 2012

norm eigenvector , the following SDP gives an improvedsolution of (21):

(23)

which actually is the SDP

To see this, suppose that is the optimal solution of (23).As is feasible to (23), it is obvious that

Therefore, by using (22)

(24)

Ultimately, after initializing with a fixed and with any fea-sible to the linear matrix (convex) inequality constraints(1b), we are able to iterate a convergent sequence of im-proved solutions of (21) through solving (23). As for a decentstep procedure, it can be easily shown that its cluster satisfies thefirst order optimality condition of (21). Being a penalty functionalgorithm per se, the efficiency of the proposed iterative proce-dure depends very much upon the proper selection of . Also,as with any local optimization algorithm, the choice of initialfeasible point is of equal importance. It is therefore desir-able to have that is feasible to (1b) and (19). Obviously,if (19) is infeasible then there is no reason to move further withoptimization procedure.In this paper, we propose a two-stage penalty function

method, which shall be referred to as PenFun algorithm, toprovide efficient solution to the design problem (1). Specif-ically, in the Initialization stage a value of weight and afeasible solution are determined. Then, from theOptimization stage searches for improved solutions in an itera-tive manner. The key steps of our newly devised approach areoutlined in Algorithm 1. Notice that due to the initial condition

, it is very likely that the Optimizationstage is terminated at some where .As will be seen later on, our extensive numerical results con-firm this important fact which is consistent with our previousexperience in [25].

Algorithm 1: PenFun Algorithm to Compute Optimal Solutionof (1)

% Initialization stage:

% Initial step: Initialize proper and to satisfy (1b). Set.

% Step : Solve (23) to obtain its optimal solution .

if (i.e., rank-one solution found)then

Reset . Terminate, and output , and .else if (i.e., no improved solution found, norank-one result) then

Reset and return to the Initial step.

else

Reset and for the next iteration.

end if

% Optimization stage:

Set . Solve (23) to obtain its optimal solution .

if (i.e., convergence) then

Terminate, and output .

else

Reset and . Continue to thenext iteration.

end if

Output the final solution .

B. Extensions to Other Cases

We will now show that the foregoing derived approach canbe readily adapted to resolve other formulated problems in thispaper, namely (2) and (3). For (2), we first express its nonconvexconstraints (2a) by the following reverse convex constraint:

(25)

As such, (2) is actually a convex program with an additionalreverse convex constraint

(26)

For iterative purpose, (26) can be converted to the followingconcave program [cf. (21)]:

(27)

Again, after initializing from a feasible point[which satisfies (2b)] whose maximum eigenvalue is

PHAN et al.: NONSMOOTH OPTIMIZATION FOR EFFICIENT BEAMFORMING 2947

and with the corresponding normalized eigenvector

, the following SDP program [cf. (23)] provides an optimalsolution that is better than of (27):

(28)

By incorporating the above modifications to Algorithm 1, wepropose in Algorithm 2 a nonsmooth approach to resolve thetransmit beamforming problem in the presence of multiple cog-nitive radio multicast groups.

Algorithm 2: PenFun Algorithm to Compute Optimal Solutionof (2)

% Initialization stage:

% Initial Step: Initialize proper and a solutionto satisfy (2b). Set .

% Step : Solve (28) to obtain its optimal solutions.

if (i.e., rank-onesolution found) then

Reset . Terminate, and outputand .

else if (i.e., no improvedsolution found, no rank-one result) then

Reset and return to the Initial step.

else

Reset and for the next iteration.

end if

% Optimization stage:

Set . Solve (28) to obtain its optimal solution.

if (i.e., convergence) then

Terminate, and output

else

Reset and .Continue to the next iteration

end if

Output the final solution .

It is noteworthy that in principle we are not required to findthe global optimum of the SDP routines (23) and (28). By uti-lizing any interior-point SDP solver (e.g., SeDuMi [37]), theseroutines can be terminated as soon as a better feasible solution

or is found. To determine localoptimum of (23) and (28), it has been shown that nonsmooth

Fig. 2. Single group of SUs—Normalmulticast: Transmit power minimization.

local optimization algorithms (see, e.g., [38]) are much fasterthan interior-point SDP solvers. We shall address the applica-tion of such methods in our future developments.Finally, the above proposed algorithms can be adapted in an

obvious way for the computation of optimal solution of (3). In-stead of SDP (23), the following SDP can be used in Algorithm1 to generate iterative solution for the final resolutionof (3)

(29)

V. NUMERICAL RESULTS

This section presents numerical results to verify the per-formance of our proposed nonsmooth optimization approach.In each example, the frequency-flat channels are generatedaccording to Rayleigh distribution with normalized channelgains, while , , are set in all optimiza-tion formulations (1), (2), and (3). The final results are thenobtained by averaging over 1000 Monte Carlo simulation runs.For comparison purposes, a total of 5000 randomization roundsare performed in the conventional method to extract the bestfeasible beamforming vector. Also, the lower bounds obtainedby SDP relaxation are provided to serve as the performancebaselines.

A. Single Group of SUs

Let us assume that a secondary BS equipped withantenna elements transmits the same information to onlygroup of SUs. The achievement of our proposed

solutions in the design problem (1) is verified for two differentsettings, namely with and without the presence of PUs.1) Multicast With no PU: This scenario is referred to as

“normal multicast,” as there is PU present. A compar-ison of both conventional and proposed solutions for [see (1)]is illustrated in Fig. 2, whereas that for the minimum SNR max-imization problem (3) is plotted in Fig. 3. As can be clearly

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TABLE INUMBER OF FAILURES IN 1,000 SIMULATION TRIALS

Fig. 3. Single group of SUs—Normal multicast: Minimum SNR maximiza-tion.

seen from these figures, PenFun algorithm performs substan-tially better than the conventional method. Further, as our re-sultant optimal values are very close to the lower bounds pro-vided by SDP, it implies that the newly developed algorithm isable to locate the global optimums with tolerance accuracy inmost instances. In particular, Fig. 2 shows that performance ofthe conventional method gets less effective at regions of highSU’s prescribed SNRs wherein the rate of attaining rank-onesolutions of (5) is considerably low. It has been noticed thatPenFun method requires merely tens of iterations to converge,a certainly attractive feature in terms of computational com-plexity. However, each its iteration requires a SDP solver, whichis much more computationally demanded than randomizationrounds. Regarding the convergence performance of Algorithm1, the average iterations for its initialization stage is 6.3 whilethe average iterations for its optimization stage is 14.2) Multicast in the Presence of PUs: This scenario is referred

to as “cognitive multicast,” as it is assumed that there arePUs coexisting with the secondary network. Fig. 4 compares thetotal beamforming power for different QoS requirements (referto (1)), in which two cases of interference thresholds inducedto PUs are considered, namely, (zero interference) and

. Apparently, it is more advantageous to adopt the pro-posed PenFun algorithm, where the performance gain over itsconventional counterpart is more pronounced at higher valuesof minimum SNRs required by SUs. Additionally, the respec-tive lower-bound and PenFun curves show that when there isno interference allowed on PUs (i.e., ), the beamformingpower is higher than that in the nonzero interference case (i.e.,

). On the contrary, because of a high failure rate ofconventional method, the total transmit power required by such

Fig. 4. Single group of SUs—Cognitive multicast: Power minimization withand interference threshold at PUs.

approximation solution for zero PU interference is indeed lowerthan that for the nonzero PU interference. In fact, it is the in-equality and equality constraints in the optimization problem(1) that limit the success of scaling step, making the processof recovering feasible solutions in the conventional method im-possible in many instances. Moreover, as the search space forrandomization vectors gets narrow, the radiated power actuallygrows. To further illustrate this point, assumingTable I shows the failure rate of conventional approach whichranges from about 12% to more than 50%. In contrast, our de-rived method succeeds in all simulation trials. The advantagesof the newly proposed approach are further confirmed in Fig. 5for the minimum SNR maximization criterion (3).3) Further Comparison: As raised by one of the reviewers,

we compare the PenFun method with a novel sequential second-order cone programing (SOCP) [39], [40], which also performsbetter than conventional SDP plus randomization. The perfor-mance is summarized in Fig. 6. In both scenarios ( and

with ), PenFun method always overperformsSOCP in terms of total beamforming power minimization at anySINR thresholds. In terms of computational time, an SOCP con-verges faster than a PenFun procedure in average.

B. Multiple Groups of SUs

Let us now assume that there is groups, each of whichconsists of an equal number of SUs. Consider a secondary BSequipped with antenna elements transmits indepen-dent information bursts to each group; however, all SUs in oneparticular group receive the same information. Similarly, the su-perior performance of our proposed solution for the transmitpower minimization problem (2) is confirmed in two settings,namely with and without the presence of PUs. Recall that the

PHAN et al.: NONSMOOTH OPTIMIZATION FOR EFFICIENT BEAMFORMING 2949

TABLE IIMULTIPLE GROUPS OF SUS—NORMAL MULTICAST: PERFORMANCE OF PENFUN ALGORITHM 2 WITH

TABLE IIIMULTIPLE GROUPS OF SUS—NORMAL MULTICAST: PERFORMANCE OF PENFUN ALGORITHM 2 WITH

Fig. 5. Single group of SUs—Cognitive multicast: Minimum SNR maximiza-tion with interference threshold at PUs.

conventional method is computationally expensive since it in-volves the resolution of linear program (15) for each randomlychosen . As such, one finds it impossible to run enough sim-ulation trials for reliable statistics to verify its performance (see[9]). In our examples, a pre-step is thus carried out for each trialwhich essentially check the feasibility of SDP relaxation (12).If this problem (12) is infeasible then so is (2), in which case thetrial shall be terminated.1) Multicast With no PU: Table II summarizes the numerical

results for this case of “normal multicast.” In particular, row 4shows the feasibility rate of the relaxed SDP (12) (of course,without constraint (2c) as here) whereas that of thePenFun Algorithm 2 is indicated in row 5. As can be observed,these feasibility rates are almost the same, whereas the ratiobetween the feasibility rates of the conventional method and therelaxed SDP (12) is as low as 30% [9, Table II]. Additionally,

Fig. 6. Comparison of PenFun and SOCP: Transmit power minimization.

row 6 displays the ratio between the optimal values found bythe PenFun Algorithm 2 and the lower bounds given by SDPrelaxation. The fact that most of the ratios in row 6 tend to 1.0implies that the derived approach is very likely to offer globallyoptimal solutions. The advantages of the novel nonsmooth op-timization solution are further confirmed in Fig. 7, where totalbeamforming powers for different SU’s SINR requirements areplotted for two cases of , , and , .As before, the difference between relaxed SDP and PenFuncurves are relatively small, especially in the former case.2) Multicast in the Presence of PUs: Assuming there are

PUs, the tolerable interference threshold at PUs is nowset to be for , , and for

, and for . Numerical results of thisexample are presented in Table III. Notice that since there are in-terference constraints to be met in this case, the infeasibility rate

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Fig. 7. Multiple group of SUs—Normal multicast: Transmit power minimiza-tion.

Fig. 8. Multiple group of SUs—Cognitive multicast: Transmit power mini-mization.

of relaxed SDP (12) is more diverse as evidenced by the num-bers in row 5. Nonetheless, row 6 shows that as long as (12) isfeasible, the PenFun Algorithm 2 is very likely to successfullydeliver a solution. Again, the mean ratios in row 7 indicate thatthe total radiated power by the devised method is comparableto its respective lower bound by SDP relaxation. Finally, it isapparent from Fig. 8 that the performance of our proposed so-lution approaches the corresponding lower bound.

VI. CONCLUSION

This paper has revisited the problems of designing optimaltransmit beamformers for cognitive radio multicast networks.In particular, it has been shown that the conventional SDP re-laxation algorithms cannot in general provide optimal solutionswhich typically are matrices of rank one. Even after employingvarious randomization techniques, such approach is still unableto offer satisfactory outcomes in many applications. Alterna-tively, we have equivalently expressed the rank-one constraintsas reverse convex constraints. Efficient iterative algorithms are

then proposed which yield significantly better and more reli-able solutions that those obtained by the conventional method.Numerical results have confirmed the superiority of our pro-posed algorithms. In future work, we shall investigate the ap-plication of the approach devised here to the resolution of otherrank-constrained problems in multiinput-multioutput (MIMO)communications.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their helpful comments and suggestions, which greatly im-proved the presentation of this paper.

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Anh Huy Phan (S’10) was born in Nghe An, Vietnam, in 1981. He receivedthe Bachelor’s degree in physics from Hanoi University of Science, Hanoi,Vietnam, in 2003, and the M.Eng. degree in telecommunications from the Uni-versity of Melbourne, Melbourne, Australia, in 2007.He is currently working toward the Ph.D. degree with the School of Electrical

Engineering and Telecommunications, University of New SouthWales, Sydney,Australia. His research interests are in signal processing for communications,currently on optimization problems in cognitive radio, wireless relay networks,and MIMO detection.

Hoang Duong Tuan (M’94) was born in Hanoi, Vietnam, in 1964. He receivedthe diploma and the Ph.D. degree in applied mathematics from Odessa StateUniversity, Ukraine, in 1987 and 1991, respectively.From 1991 to 1994, he was a Researcher with the Optimization and Systems

Division, Vietnam National Center for Science and Technologies. He was anAssistant Professor in the Department of Electronic-Mechanical Engineering,Nagoya University, Japan, from 1994 to 1999 and an Associate Professor inthe Department of Electrical and Computer Engineering, Toyota Technolog-ical Institute, Nagoya, from 1999 to 2003. He was a Professor in the School ofElectrical Engineering and Telecommunications, the University of New SouthWales, Sydney, Australia, from 2003 to 2011. Currently, he is a Professor andthe core member of the Centre for Health Technologies, Faculty of Engineeringand Information Technology, University of Technology Sydney. His research in-terests include theoretical developments and applications of optimization basedmethods in many areas of control, signal processing, communication, and bioin-formatics.

Ha Hoang Kha (S’05–M’09) was born in Dong Thap, Vietnam, in 1977. Hereceived the B.Eng. and M.Eng. degrees from HoChiMinh City University ofTechnology, in 2000 and 2003, respectively, and the Ph.D. degree from the Uni-versity of New South Wales, Sydney, Australia, in 2009, all in electrical engi-neering and telecommunications.From 2000 to 2004, he was a Research and Teaching Assistant with the

Department of Electrical and Electronics Engineering, HoChiMinh CityUniversity of Technology. He was a Visiting Research Fellow at the School ofElectrical Engineering and Telecommunications, the University of New SouthWales, from April 2009 to March 2011. He is currently a Research PostdoctoralFellow at the Faculty of Engineering and Information Technology, Universityof Technology Sydney. His research interests are in digital signal processingand wireless communications, with a recent emphasis on convex optimizationtechniques in signal processing for wireless communications.

Duy Trong (Danny) Ngo (S’08) received the B.Eng. (with first-class honorsand the University Medal) degree in telecommunication engineering from theUniversity of New South Wales, Sydney, Australia, in 2007, and the M.Sc. de-gree in electrical engineering (communication) from the University of Alberta,Edmonton, Canada, in 2009.He is currently working toward the Ph.D. degree in electrical engineering

with the Department of Electrical and Computer Engineering, McGill Univer-sity, Montréal, Canada. His research interest is in the area of resource allocationfor wireless communications systems with special emphasis on heterogeneousnetworks.