Nonsmooth Optimization for Efficient Beamforming in Cognitive Radio ...

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

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Nonsmooth Optimization for Efficient Beamforming in Cognitive Radio Multicast Transmission Anh Huy Phan, Student Member, IEEE, Hoang Duong Tuan, Member, IEEE, Ha Hoang Kha, Member, IEEE, and Duy Trong Ngo, Student Member, IEEE

Abstract—It is known that the design of optimal transmit beamforming vectors for cognitive radio multicast transmission can be formulated as indefinite quadratic optimization programs. Given the challenges of such nonconvex problems, the conventional approach in literature is to recast them as convex semidefinite programs (SDPs) together with rank-one constraints. Then, these nonconvex and discontinuous constraints are dropped allowing for the realization of a pool of relaxed candidate solutions, from which various randomization techniques are utilized with the hope to recover the optimal solutions. However, it has been shown that such approach fails to deliver satisfactory outcomes in many practical settings, wherein the determined solutions are found to be unacceptably far from the actual optimality. On the contrary, we in this contribution tackle the aforementioned optimal beamforming problems differently by representing them as SDPs with additional reverse convex (but continuous) constraints. Nonsmooth optimization algorithms are then proposed to locate the optimal solutions of such design problems in an efficient manner. Our thorough numerical examples verify that the proposed algorithms offer almost global optimality whilst requiring relatively low computational load. Index Terms—Beamforming, cognitive radio, multicast transmission, nonsmooth optimization.

I. INTRODUCTION

T

HE deployment of numerous broadband wireless applications with different service requirements leads to a huge demand on the expensive radio spectrum. As such, spectrum shortage becomes a significant challenge toward the implementation of next-generation communication networks. On the other hand, it has been reported that much of the licensed radio spectrum lies idle at any given time and location, and that the spectrum shortage results from the spectrum management policy rather than the physical scarcity of the usable frequencies [1]. Spectrum utilization can thus be substantially improved by permitting secondary access to spectrum holes unoccupied

Manuscript received April 28, 2011; revised October 26, 2011 and February 08, 2012; accepted February 16, 2012. Date of publication March 05, 2012; date of current version May 11, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Ciblat. A. H. Phan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, UNSW Sydney, NSW 2052, Australia (e-mail: [email protected]). H. D. Tuan and H. H. Kha are with the Faculty of Engineering and Information Technology, University of Technology Sydney, Broadway, NSW 2007, Australia (e-mail: [email protected]; [email protected]). 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, cognitive radio technology [2]–[4] has been identified as promising candidate to effectively exploit the existence of the unoccupied spectrum portions. Specifically, while the primary users have priority access to the available radio frequency bands, the secondary (or unlicensed or cognitive) users have restricted access, subject to a constrained degradation on the primary users’ performance. In spectrum sharing environments, the key design criteria include protecting the primary users from excessive interference introduced by the secondary users as well as satisfying some quality-of-service (QoS) requirements for the latter [5]–[7]. In multiple-antenna communication systems, transmit beamforming has been employed as an effective measure to control the level of interference by placing nulls at the direction of each co-channel receiver. The study of [8] addresses the design of suboptimal beamformers in physical-layer multicasting that involves only one single multicast group of wireless users. The work in [9] extends the results of [8] to the case of multiple multicast groups. In the context of cognitive radio communications wherein the interference introduced from secondary to primary networks is strictly regulated, transmit beamforming technique turns out to be particularly relevant. The issue of beamforming design for cognitive radio network coexisting with a primary system has been investigated in [10]. Here, the QoS requirements of unlicensed network are guaranteed while the total interference induced to the primary system is kept under a predefined threshold. The optimal transmit beamforming design problems are formulated as indefinite quadratic optimization programs in various settings [8]–[13]. Toward solving these difficult nonconvex problems, the typical approach in literature involves the application of semidefinite program (SDP) relaxation technique together with randomization search. Specifically, the quadratic optimization problems are recast as SDPs with additional constraints which impose that the solution matrices must be of rank one. Such nonconvex and discontinuous constraints are then dropped resulting in SDP relaxations. From the set of all possible solutions obtained by resolving these SDPs, different randomization techniques can be employed to generate feasible solutions to the original design problems. However, it is worth noticing that in the scenario that involves multiple cochannel multicast groups (see, e.g., [9]) or that requires nonzero interference on the primary system, the randomization procedure must be carried out in parallel with the resolution of a large number

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of linear programs. This certainly implies a high computational complexity 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 optimal rank-one solutions at three different categories, such optimality is not always guaranteed. In our earlier work [15], a simple alternative approach for cognitive beamforming in the case of single-group multicasting has been proposed. Essentially, such solution is a modification of the alternating projection in [16] to directly tackle the original indefinite quadratic program. Not only have the numerical examples presented in [15] shown that the approach proposed there remarkably outperforms its conventional counterpart, they have also revealed that the latter often yields solutions that are unacceptably far from the actual optimum. Motivated by the shortcomings of existing solutions, this paper aims to develop an efficient approach which works efficiently and consistently in any scenario that involves multiple co-channel multicast groups of cognitive users. Specifically, the critical indefinite quadratic constraints are expressed as reverse convex ones, which effectively means that the original beamforming problems are reformulated as SDPs with additional reverse convex (but continuous) constraints. Next, the resulting problems are converted into minimizing a nonsmooth (but continuous) concave function over a set of linear matrix inequality constraints, an important class of nonconvex optimization (see, e.g., [17]). An iterative procedure is finally proposed to offer almost optimal solutions. While the conventional method might give solutions that are very far from the lower bounds provided by relaxed SDPs, numerical results show that the our solutions approach global optimality in most cases. Moreover, this is achieved at an affordable computational complexity. It should also be noted that indefinite quadratic program is among the hardest classes in algorithmic optimization. Given there is no effective approach available to solve for the global optimum of such problem [17]–[19], the results presented in this work are novel 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 recalls various optimization formulations for the problem of transmit beamforming design. Section III reviews the capacity of conventional randomization SDP approach available in literature. In Section IV, novel algorithms which aim at achieving global optimal solutions of the formulated problems are proposed. Next, Section V provides extensive numerical examples to verify performance of the devised method. And finally, Section VI concludes the paper. Some preliminary results of the paper have been appeared in [20] and [21]. Notations: Matrices and column vectors are denoted by boldfaced uppercase and lowercase characters, respectively. For a Hermitian matrix , is its maximal is its spectral radius defined by eigenvalue, while with , being its eigenvalues. Furthermore, means is positive


Fig. 1. System model.

semi-definite.We denote for a square matrix and for matrices and of apis the conjugate transpose of propriate dimension, where . Accordingly, for two complex vectors and of the same and accordingly, dimension, and . Also, denotes the expectation operator in respect to random variable . II. SYSTEM MODEL AND PROBLEM FORMULATIONS Consider a communication scenario in which a primary base primary users (PUs). To effistation (BS) transmits to its ciently implement opportunistic spectrum access, an -antenna secondary BS is also deployed to send information-bearing , each to individual multicast groups signals of secondary users (SUs). Effectively, all SUs within the same group will receive identical information from the secondary BS. Assume that each multicast group consists of secondary receivers and that each SU belongs to only one group. Hence, the total number of SUs in the cognitive multicast network is indeed . For convenience, if the secondary receiver belongs to the group , it is denoted as . As well, let be the channel vectors between the secondary BS and SU , and be the channels between the secondary BS and PU . The system model is depicted in Fig. 1. The idea of transmit beamforming is for the secondary BS to apply a beam weight to each information signal . Then, the resulting vectors are combined which shall be transmitted to all to form the signal multicast groups. Suppose that are independent, each of which has a flat power spectral density (PSD) with

PHAN et al.: NONSMOOTH OPTIMIZATION FOR EFFICIENT BEAMFORMING

zero-mean and unit variance. The total transmit power at the secondary 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. Even within the secondary network, the simultaneous transmissions from secondary BS to multiple multicast groups also result in intranetwork interference at each cognitive user. In fact, it can 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 primary 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 the PUs’ receivers. This amount of interference can be expressed as

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(2b)

B. Secondary User SINR Maximization , a related design is to maximize the Also assuming minimum secondary receivers’ SINR subject to constraints on a fixed power budget as well as on the interference limits induced to the PUs. This involves solving the following optimization problem

which can be expressed by the following with additional slack and : variables (3a)

(3b) Regarding the problem of maximizing the minimum secondary SINR which involves multiple multicast groups of SUs (i.e., ), the denominator of the SINR expression makes the constraints become highly nonlinear. Since such scenario is difficult to deal with, it is not considered here in this paper. III. CAPACITY

A. Secondary BS Transmit Power Minimization Here, the objective is to find optimal beamforming vectors that minimize the total radiated power at the secondary BS, constrained on meeting prescribed secondary SINR thresholds and satisfying tolerable interference threshold at indisecondary multividual PUs. Suppose that there is only cast group in the system. Then, by introducing a slack variable and matrices and , the design problem can be formulated as (1a)

OF CONVENTIONAL SDP RELAXATION WITH RANDOMIZATION

Towards solving optimization problems in the forms of (1), (2), (3), the existing approach in literature (see, e.g., [8]–[11]) is to recast the formulated problems to relaxed semi-definite programs and generate feasible solutions from the pool of possible candidates by means of randomization. While our earlier numerical results in [15] have revealed the numerical inconsistency of such approach, we will discuss the capacity of this conventional method in the following. Let us begin with (1). According to the conventional method, the nonconvex but still continuous constraint (1a) is first substituted by the equivalent discontinuous rank-one constraint. This results in an equivalent formulation of (1) as

(1b)

(4)

In the case of several secondary multicast groups coexisting with primary network, the optimization problem (1) can now be generalized to amounts to

Next, the rank-one constraint in (4) is dropped, leading to the following SDP relaxation (5)

(2a)

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])

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

(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-bound performance. To circumvent this issue, the works in [8] and [10] make use of the following randomization technique in order to generate feasible solutions to (1). Suppose that . Then, this matrix admits the singular value decomposition (SVD) (7) where matrix is unitary constituting of the eigenvectors of and matrix is diagonal whose diagonal entries 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 elements of are uniformly distributed independent random variables taken on the unit circle in the complex plan, or is a vector of zero-mean unit-variance complex circularly symmetric uncorrelated Gaussian random variables. Consequently, the randomization for searching suboptimal solutions of (1) is indeed carried out only on the following -dimensional subspace formed by eigenvectors (9) It is true that one uses (5) with the hope of arriving at an optimal solution of as small as possible rank . Clearly, such a small number actually restricts the space within which the randomization procedure is carried out. We will see that as far as the rank of is not less than 2, i.e., it has at least two different strictly positive eigenvalues that make any its rank-one approximation poor, the randomization turns out to be both inconsistent and inefficient. In fact, our numerical results, which will be presented in later section, reveal that the best (lowest) beamforming power obtained by randomization technique in the case of nonzero interference on the primary system is even higher 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 the randomization procedure in (8) fails to generate a feasible solution to (1b). Our simulation results confirm that the failure rate of this approach in certain scenarios is quite high.

(11) where the rank-one constraint can be relaxed to obtain the following SDP: (12) Denote

the optimal solution of (12) and suppose that . It is now possible to perform the SVDs (13)

are unitary and are diagwhere onal with diagonal elements arranged in decreasing order, i.e., . Similarly, a feasible solution of (2) can be obtained according to

(14) defined by (14) are contained in The solutions -dimensional subspaces formed by for . In this case, the task of recovering feasible solutions from such suboptimal solutions is far more unreliable and computationally costly, compared to that for (1). Indeed, it involves finding the optimal solution of the following linear program:

(15a) (15b)

and computing the feasible solutions as (16) It is imperative to point out that the linear program (15) is not guaranteed to be feasible either; hence, this approach may fail to deliver a feasible solution for (2). In brief, the conventional approach is an approximation method that tries to locate an suboptimal solution through random search on low-dimensional subspace defined (9). It even becomes less effective when the number of SUs tends to be large, as in such cases the rank of matrix or is 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 dimensional space defined by (9) and thus even the search over the whole space does not bring a good suboptimal solution. Our later simulation is able also to show that if the tolerable interference thresholds at PUs are more flexible, the solutions obtained by randomization procedure are unacceptably far from optimality.

(20)

IV. NONSMOOTH OPTIMIZATION APPROACH TO COGNITIVE RADIO MULTICAST TRANSMIT BEAMFORMING DESIGN As can already be seen, the main drawback of the rank-one constraints in (4) and (11) are their discontinuous nature that prevents efficient relaxation. While it is true that the optimal solutions of (5) and (12) would be rank-dropped, rank-one is actually the lowest for nonzero matrices. Therefore, obtaining a rank-one solution immediately after resolving (5) and (12) is not quite expected. On the other hand, the study of [16] has proven in theory that the optimal solutions of (5) and (12) are of rank two in most cases. Therefore, it is possible that the randomization approach may fail to provide desirable results as their random search becomes too narrow. A. Single-Group Cognitive Radio Multicasting The previous works [24], [27]–[31] have addressed the rankconstrained SDP by smooth optimization, which nevertheless do not seem to work appropriately for rank-one constraint. Motivated by the shortcomings of existing solutions, in this contribution we develop a novel nonsmooth optimization 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

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

.

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

thus allowing to satisfy (1b). Therefore, our aim here is to make as small as possible. For this purpose, 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) can be achieved. As such, this can be regarded as a penalty function approach [33], [34]. However, the following result shows that (19) and (21) are equivalent and thus the later provides the exact penalty 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 is such that whenever problems (19) and (21) are equivalent in the sense that they share the same optimal solution as well as the same optimal value. Proof: Obviously, each feasible solution of (19) is also feasible to (21) for which so the optimal value of (19) is not less than the optimal value of (21) for all . It also means the optimal value of (21) is bounded by the optimal value of (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 the optimal 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 if necessary, it follows that as with , i.e., as for some . This means , a contradiction with their boundedness. Since its cost function is concave, problem (21) involves 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 differentiable), is not smooth either. On the other hand, a sub-gradient of is because [35], [36]

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

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

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norm eigenvector solution of (21):


, the following SDP gives an improved

Algorithm 1: PenFun Algorithm to Compute Optimal Solution of (1) % Initialization stage:

(23)

% Initial step: Initialize proper .

and

to satisfy (1b). Set

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

which actually is the SDP

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

if then

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

.

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

and return to the Initial step.

else and

Reset

for the next iteration.

end if % Optimization stage: Therefore, by using (22)

Set

. Solve (23) to obtain its optimal solution

.

(i.e., convergence) then

if

.

Terminate, and output else

(24)

Reset next iteration.

and

. Continue to the

end if Ultimately, after initializing with a fixed and with any feasible to the linear matrix (convex) inequality constraints (1b), we are able to iterate a convergent sequence of improved solutions of (21) through solving (23). As for a decent step procedure, it can be easily shown that its cluster satisfies the first order optimality condition of (21). Being a penalty function algorithm per se, the efficiency of the proposed iterative procedure depends very much upon the proper selection of . Also, as with any local optimization algorithm, the choice of initial feasible point is of equal importance. It is therefore desirable to have that is feasible to (1b) and (19). Obviously, if (19) is infeasible then there is no reason to move further with optimization procedure. In this paper, we propose a two-stage penalty function method, which shall be referred to as PenFun algorithm, to provide efficient solution to the design problem (1). Specifically, in the Initialization stage a value of weight and a feasible solution are determined. Then, from the Optimization stage searches for improved solutions in an iterative manner. The key steps of our newly devised approach are outlined in Algorithm 1. Notice that due to the initial condition , it is very likely that the Optimization stage is terminated at some where . As will be seen later on, our extensive numerical results confirm this important fact which is consistent with our previous experience in [25].

Output the final solution

.

B. Extensions to Other Cases We will now show that the foregoing derived approach can be readily adapted to resolve other formulated problems in this paper, namely (2) and (3). For (2), we first express its nonconvex constraints (2a) by the following reverse convex constraint: (25) As such, (2) is actually a convex program with an additional reverse convex constraint (26) For iterative purpose, (26) can be converted to the following concave program [cf. (21)]:

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


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and with the corresponding normalized eigenvector , the following SDP program [cf. (23)] provides an optimal solution that is better than of (27):

(28) By incorporating the above modifications to Algorithm 1, we propose in Algorithm 2 a nonsmooth approach to resolve the transmit beamforming problem in the presence of multiple cognitive radio multicast groups. Algorithm 2: PenFun Algorithm to Compute Optimal Solution of (2)

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

% Initialization stage: % Initial Step: Initialize proper and a solution to satisfy (2b). Set . % Step : Solve (28) to obtain its optimal solutions . if solution found) then Reset and

(i.e., rank-one . Terminate, and output

local optimization algorithms (see, e.g., [38]) are much faster than interior-point SDP solvers. We shall address the application 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). Instead of SDP (23), the following SDP can be used in Algorithm 1 to generate iterative solution for the final resolution of (3)

.

else if (i.e., no improved solution found, no rank-one result) then Reset

and return to the Initial step. V. NUMERICAL RESULTS

else Reset

and

for the next iteration.

end if % Optimization stage: Set

(29)

. 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 find the global optimum of the SDP routines (23) and (28). By utilizing any interior-point SDP solver (e.g., SeDuMi [37]), these routines can be terminated as soon as a better feasible solution or is found. To determine local optimum of (23) and (28), it has been shown that nonsmooth

This section presents numerical results to verify the performance of our proposed nonsmooth optimization approach. In each example, the frequency-flat channels are generated according to Rayleigh distribution with normalized channel gains, while , , are set in all optimization formulations (1), (2), and (3). The final results are then obtained by averaging over 1000 Monte Carlo simulation runs. For comparison purposes, a total of 5000 randomization rounds are performed in the conventional method to extract the best feasible beamforming vector. Also, the lower bounds obtained by SDP relaxation are provided to serve as the performance baselines. A. Single Group of SUs Let us assume that a secondary BS equipped with antenna elements transmits the same information to only group of SUs. The achievement of our proposed solutions in the design problem (1) is verified for two different settings, 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 comparison of both conventional and proposed solutions for [see (1)] is illustrated in Fig. 2, whereas that for the minimum SNR maximization problem (3) is plotted in Fig. 3. As can be clearly

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TABLE I NUMBER OF FAILURES IN 1,000 SIMULATION TRIALS

Fig. 3. Single group of SUs—Normal multicast: Minimum SNR maximization.

seen from these figures, PenFun algorithm performs substantially better than the conventional method. Further, as our resultant optimal values are very close to the lower bounds provided by SDP, it implies that the newly developed algorithm is able to locate the global optimums with tolerance accuracy in most instances. In particular, Fig. 2 shows that performance of the conventional method gets less effective at regions of high SU’s prescribed SNRs wherein the rate of attaining rank-one solutions of (5) is considerably low. It has been noticed that PenFun method requires merely tens of iterations to converge, a certainly attractive feature in terms of computational complexity. However, each its iteration requires a SDP solver, which is much more computationally demanded than randomization rounds. Regarding the convergence performance of Algorithm 1, the average iterations for its initialization stage is 6.3 while the 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 are PUs coexisting with the secondary network. Fig. 4 compares the total beamforming power for different QoS requirements (refer to (1)), in which two cases of interference thresholds induced to PUs are considered, namely, (zero interference) and . Apparently, it is more advantageous to adopt the proposed PenFun algorithm, where the performance gain over its conventional counterpart is more pronounced at higher values of minimum SNRs required by SUs. Additionally, the respective lower-bound and PenFun curves show that when there is no interference allowed on PUs (i.e., ), the beamforming power is higher than that in the nonzero interference case (i.e., ). On the contrary, because of a high failure rate of conventional method, the total transmit power required by such

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

approximation solution for zero PU interference is indeed lower than that for the nonzero PU interference. In fact, it is the inequality and equality constraints in the optimization problem (1) that limit the success of scaling step, making the process of recovering feasible solutions in the conventional method impossible in many instances. Moreover, as the search space for randomization vectors gets narrow, the radiated power actually grows. To further illustrate this point, assuming Table I shows the failure rate of conventional approach which ranges from about 12% to more than 50%. In contrast, our derived method succeeds in all simulation trials. The advantages of the newly proposed approach are further confirmed in Fig. 5 for 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 secondorder cone programing (SOCP) [39], [40], which also performs better than conventional SDP plus randomization. The performance is summarized in Fig. 6. In both scenarios ( and with ), PenFun method always overperforms SOCP in terms of total beamforming power minimization at any SINR thresholds. In terms of computational time, an SOCP converges faster than a PenFun procedure in average. B. Multiple Groups of SUs Let us now assume that there is groups, each of which consists of an equal number of SUs. Consider a secondary BS equipped with antenna elements transmits independent information bursts to each group; however, all SUs in one particular group receive the same information. Similarly, the superior performance of our proposed solution for the transmit power minimization problem (2) is confirmed in two settings, namely with and without the presence of PUs. Recall that the


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

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

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

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

conventional method is computationally expensive since it involves the resolution of linear program (15) for each randomly chosen . As such, one finds it impossible to run enough simulation trials for reliable statistics to verify its performance (see [9]). In our examples, a pre-step is thus carried out for each trial which essentially check the feasibility of SDP relaxation (12). If this problem (12) is infeasible then so is (2), in which case the trial shall be terminated. 1) Multicast With no PU: Table II summarizes the numerical results for this case of “normal multicast.” In particular, row 4 shows the feasibility rate of the relaxed SDP (12) (of course, without constraint (2c) as here) whereas that of the PenFun Algorithm 2 is indicated in row 5. As can be observed, these feasibility rates are almost the same, whereas the ratio between the feasibility rates of the conventional method and the relaxed SDP (12) is as low as 30% [9, Table II]. Additionally,

row 6 displays the ratio between the optimal values found by the PenFun Algorithm 2 and the lower bounds given by SDP relaxation. The fact that most of the ratios in row 6 tend to 1.0 implies that the derived approach is very likely to offer globally optimal solutions. The advantages of the novel nonsmooth optimization solution are further confirmed in Fig. 7, where total beamforming powers for different SU’s SINR requirements are plotted for two cases of , , and , . As before, the difference between relaxed SDP and PenFun curves 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 now set to be for , , and for , and for . Numerical results of this example are presented in Table III. Notice that since there are interference constraints to be met in this case, the infeasibility rate

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then proposed which yield significantly better and more reliable solutions that those obtained by the conventional method. Numerical results have confirmed the superiority of our proposed algorithms. In future work, we shall investigate the application of the approach devised here to the resolution of other rank-constrained problems in multiinput-multioutput (MIMO) communications. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments and suggestions, which greatly improved the presentation of this paper. REFERENCES

Fig. 7. Multiple group of SUs—Normal multicast: Transmit power minimization.

Fig. 8. Multiple group of SUs—Cognitive multicast: Transmit power minimization.

of relaxed SDP (12) is more diverse as evidenced by the numbers in row 5. Nonetheless, row 6 shows that as long as (12) is feasible, the PenFun Algorithm 2 is very likely to successfully deliver a solution. Again, the mean ratios in row 7 indicate that the total radiated power by the devised method is comparable to its respective lower bound by SDP relaxation. Finally, it is apparent from Fig. 8 that the performance of our proposed solution approaches the corresponding lower bound. VI. CONCLUSION This paper has revisited the problems of designing optimal transmit beamformers for cognitive radio multicast networks. In particular, it has been shown that the conventional SDP relaxation algorithms cannot in general provide optimal solutions which typically are matrices of rank one. Even after employing various randomization techniques, such approach is still unable to offer satisfactory outcomes in many applications. Alternatively, we have equivalently expressed the rank-one constraints as reverse convex constraints. Efficient iterative algorithms are

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Anh Huy Phan (S’10) was born in Nghe An, Vietnam, in 1981. He received the Bachelor’s degree in physics from Hanoi University of Science, Hanoi, Vietnam, in 2003, and the M.Eng. degree in telecommunications from the University 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 South Wales, 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 received the diploma and the Ph.D. degree in applied mathematics from Odessa State University, 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 an Assistant Professor in the Department of Electronic-Mechanical Engineering, Nagoya University, Japan, from 1994 to 1999 and an Associate Professor in the Department of Electrical and Computer Engineering, Toyota Technological Institute, Nagoya, from 1999 to 2003. He was a Professor in the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, Australia, from 2003 to 2011. Currently, he is a Professor and the core member of the Centre for Health Technologies, Faculty of Engineering and Information Technology, University of Technology Sydney. His research interests include theoretical developments and applications of optimization based methods in many areas of control, signal processing, communication, and bioinformatics.

Ha Hoang Kha (S’05–M’09) was born in Dong Thap, Vietnam, in 1977. He received the B.Eng. and M.Eng. degrees from HoChiMinh City University of Technology, in 2000 and 2003, respectively, and the Ph.D. degree from the University of New South Wales, Sydney, Australia, in 2009, all in electrical engineering and telecommunications. From 2000 to 2004, he was a Research and Teaching Assistant with the Department of Electrical and Electronics Engineering, HoChiMinh City University of Technology. He was a Visiting Research Fellow at the School of Electrical Engineering and Telecommunications, the University of New South Wales, from April 2009 to March 2011. He is currently a Research Postdoctoral Fellow at the Faculty of Engineering and Information Technology, University of Technology Sydney. His research interests are in digital signal processing and wireless communications, with a recent emphasis on convex optimization techniques in signal processing for wireless communications.

Duy Trong (Danny) Ngo (S’08) received the B.Eng. (with first-class honors and the University Medal) degree in telecommunication engineering from the University of New South Wales, Sydney, Australia, in 2007, and the M.Sc. degree 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 University, Montréal, Canada. His research interest is in the area of resource allocation for wireless communications systems with special emphasis on heterogeneous networks.