Conic Quadratic Formulations for Wireless Communications Design

Conic Quadratic Formulations for Wireless Communications Design

Quang-Doanh Vu, Markku Juntti, , Een-Kee Hong, , and Le-Nam Tran,  This work was supported in part by the Academy of Finland under project Message and CSI Sharing for Cellular Interference Management with Backhaul Constraints (MESIC) belonging to the WiFIUS program with NSF, and in part by a research grant from Science Foundation Ireland (SFI) and is co-funded under the European Regional Development Fund under Grant Number 13/RC/2077.Quang-Doanh Vu, and Markku Juntti are with Centre for Wireless Communications, University of Oulu, FI-90014, Finland. Email: {quang.vu, markku.juntti}@oulu.fi.Een-Kee Hong is with the School of Electronics and Information, Kyung Hee University, South Korea. Email: ekhong@khu.ac.kr.Le-Nam Tran is with the Department of Electronic Engineering, Maynooth University, Maynooth, Co. Kildare, Ireland. Email: lenam.tran@nuim.ie.
Abstract

As a wide class of resource management problems in wireless communications are nonconvex and even NP-hard in many cases, finding globally optimal solutions to these problems is of little practical interest. Towards more pragmatic approaches, there is a rich literature on iterative methods aiming at finding a solution satisfying necessary optimality conditions to these problems. These approaches have been derived under several similar mathematical frameworks such as inner approximation algorithm, concave-convex procedure, majorization-minimization algorithm, and successive convex approximation (SCA). However, a large portion of existing algorithms arrive at a relatively generic program at each iteration, which is less computationally efficient compared to a more standard convex formulation. This paper proposes numerically efficient transformations and approximations for SCA-based methods to deal with nonconvexity in wireless communications design. More specifically, the central goal is to show that various nonconvex problems in wireless communications can be iteratively solved by conic quadratic optimization. We revisit various examples to demonstrate the advantages of the proposed approximations. Theoretical complexity analysis and numerical results show the superior efficiency in terms of computational cost of our proposed solutions compared to the existing ones.

Resource management, sequential convex programming, second-order cone programming, multiuser multi-antenna communications, reduced complexity, large-scale systems.

I Introduction

The exponential increase in the number of portable devices which have powerful multimedia capabilities (e.g. smartphones) has given rise to tremendous demand on wireless data traffic. For example, Cisco’s projections of global mobile data traffic predicts smartphone will reach three-quarters of mobile data traffic by 2019 and global mobile data traffic will increase nearly tenfold between 2014 to 2019 [1]. In addition, more and more connected devices and the scarcity of bandwidth make the coordination of multiuser interference highly important while complicated. The evolution of wireless networks also posts other challenges including resource costs, environmental impact, security, fairness between subcribers, etc. [2, 3].

Over the years, advanced optimization techniques have been widely used and become vital tools for wireless communications design [4, 5, 6, 7]. Representative examples include semidefinite relaxation (SDR) technique [8], dual decomposition and alternating direction method of multipliers (ADMM) [9, 10], robust optimization[11], to name but a few. In general, the first choice of solving a resource management problem is to represent (or equivalently reformulate) it in a form of a convex program, if possible. A good example in this regard is the problem of minimizing the transmit power at the transmitter while the quality-of-service (QoS) of each individual receiver is guaranteed [12]. This power minimization problem can be solved optimally by transforming the original nonconvex problem into an equivalent second-order cone program (SOCP). Unfortunately, such efficient convex reformulation is impossible for many other design problems, e.g., weighted sum rate maximization [13, 14], energy efficiency maximization for multiuser systems [15], full-duplex communications [16], relay networks [17], etc. Generally, finding a globally optimal solution to these nonconvex problems is difficult and, more importantly, not practically appealing. Consequently, low-complexity suboptimal approaches are of particular interest.

Among suboptimal solutions in the literature, SDR and successive convex approximation (SCA) techniques are the two that have been extensively used to tackle the nonconvexity in various wireless communication problems [8, 18, 19, 20]. Basically, instead of dealing with a design parameter, say , the SDR method defines a positive semidefinite (PSD) matrix and then lifts the design problem into the PSD domain. By omitting the rank-1 constraint on , we can arrive at a semidefinite program (SDP). For some special cases, a SDR-based solution can yield exact optimal solutions, see [8] and references therein. In other cases, randomization techniques are required to produce high-performance feasible solutions. The major disadvantage of the SDR is that the computational complexity of the resulting SDP increases quickly with the problem size. Moreover, randomization techniques might not always provide feasible points, and thus the SDR even fails to obtain a feasible solution. The two issues were investigated in [21, 19, 22].

This paper is centered on a class of suboptimal solutions which are based on SCA. In fact, SCA is a general term referred to similar algorithms such as inner approximation algorithm [23], concave-convex procedure [24], majorization-minimization algorithm [25], or difference of convex (DC) algorithm [26]. Essentially, the idea of a SCA-based approach is to safely and iteratively approximate the nonconvex feasible set (and/or nonconvex objective) of a nonconvex problem by a convex one [27, 23, 24, 28]. By a proper approximation, SCA is provably monotonically convergent to a stationary solution to the original nonconvex problem. Compared to SDR, SCA is more flexible and can be applied to a broader range of applications. However, since an SCA-based solution requires to solve a sequence of convex programs, the type of convex programs significantly affects the computation time of the SCA-based method. In other words, the overall complexity of SCA-based solutions strongly depends on that of the convex program arrived at each iteration.

In general, a convex problem can be efficiently solved, i.e., in polynomial time, by interior-point methods. However, the exponents of the polynomial vary significantly according to the structure of the convex program [29]. For example, solving a linear program is much more efficient than solving an SOCP in terms of both complexity and stability [29, Chap. 6]. In the same way, SOCP’s are much more computationally efficient than SDP’s. Thus one would consider an SOCP instead of an equivalent SDP, if possible, and many examples are given in [30, 31].

Motivated by the above discussions, we propose in this paper novel transformations and approximations particularly useful for wireless communications design where the problems of interest are nonconvex. Different from many seminal papers above in which general algorithmic frameworks for SCA are the main focus, we are interested in providing formulations that allow for solving a wide variety of problems by conic quadratic programming. The choice of conic quadratic programming (CQP) is affected by the fact that linear programming (LP) is nearly impossible, as far as beamforming techniques in multiple antennas systems are concerned. Our contributions include the following:

  • We first present a comprehensive review on the general framework of SCA. A possible method of finding a feasible point to start the SCA is also described. Then, we identify a class of common nonconvex constraints in wireless communications design and propose transformations and approximations to convert them to conic quadratic constraints.

  • In the second part of the paper, we demonstrate the efficiency and flexibility of the proposed formulations in dealing with various resource management problems in wireless communications. These include physical layer secure transmission, relay communications, cognitive radio, multicarrier management with different design criteria such as rate maximization, transmit power minimization, rate fairness, and energy efficiency fairness. Analytical and numerical results are provided to show the superior performance of the proposed solutions, compared to the existing ones.

The rest of the paper is organized as follows. Section II provides the preliminaries of the SCA. Section III presents the proposed low-complexity formulations. Section IV illustrates an example of using the proposed approach for the scenario of secrecy relay transmissions. The second application about cognitive radio is studied in Section V. Section VI shows how to particularize the proposed method to the case of MIMO relay communications. In Section VII, we consider the scenario of multiuser multicarrier system with two problems: weighted sum rate maximization and max-min energy efficiency fairness. Finally, Section VIII concludes the paper.

Notation: Standard notations are used in this paper. Bold lower and upper case letters represent vectors and matrices, respectively; represents the norm; represents the absolute value; represents the complex conjugate; and , represent the space of real and complex matrices of dimensions given in superscript, respectively; denotes the identity matrix; denotes a complex Gaussian random vector with zero mean and variance matrix ; and represents real and image parts of the argument; and are Hermitian and normal transpose of , respectively; is the vectorization operation that converts the matrix into a column vector; is the trace of ; denotes Kronecker product. For ease of description, we also use “MATLAB notation” throughout the paper. Specifically, when , …, are matrices with the same number of rows, denotes the matrix with the same number of rows obtained by staking horizontally , …, and . When , …, are matrices with the same number of columns, stands for the matrix with the same number of columns obtained by staking vertically , …, and . Other notations are defined at their first appearance.

Ii Preliminaries

Ii-a Successive Convex Approximation

We first briefly review some preliminaries of SCA which are central to the discussions presented in the rest of the work. The interested reader is referred to [27, 23, 28] for further details. Note that the SCA presented in this paper can include concave-convex procedure [24], majorization-minimization (MM) algorithm [25, 32], or DC algorithm [26] as special cases.111More precisely, when the feasible set of the problem being considered is convex, MM algorithms are different from SCA in the sense that the surrogate function in MM framework can be nonconvex. The interested reader is referred to [32] for a sharp comparison between SCA and MM algorithms. Let us consider a general nonconvex optimization problem of the following form

(1a)
(1b)
(1c)

where , and , are assumed to be convex and noncovex functions, respectively. We also assume that all the functions in (1) are continuously differentiable. Note that (1) is in the complex domain as it appears naturally in wireless communications design (the case of the real domain or mixed real-complex domain will be elaborated later on). In (1), represents the cost function to be optimized, and , are the constraints related to, e.g., quality of service (minimum rate requirement), radio resources (transmit power, system bandwidth, backhaul), etc. At first, the assumption of convexity on seems to be strong, but as we will see below, the cost functions in many design problems are originally convex. Moreover, by proper transformations, e.g., using the epigraph form, we can bring the nonconvexity of the objective into the feasible set.222It is not difficult to see that the epigraph form also has the same KKT points as (1). We refer the interested reader to [28, 20] for a direct way to deal with problems with a nonconvex objective. Herein the assumption on the convexity of is mainly to simplify the general description of SCA.

The difficulty of solving (1) is obviously due to the nonconvex constraints in (1c). An SCA-based approach is an iterative procedure which tries to seek a stationary solution (i.e. a Karush-Kuhn-Tucker (KKT) point) of (1) by sequentially approximating the nonconvex feasible set by inner convex ones. To do so, the nonconvex functions , are replaced by their convex upper bounds. We denote by the solution obtained in the th iteration of the iterative process. At the iteration , let denote a convex upper estimate of which is continuous on , where is a fixed feasible point depending on the solution of the problem in the th iteration. That is , where is a continuous function [27]. Then the problem considered in the th iteration is given by

(2a)
(2b)
(2c)

Note that problem (2) produces an upper bound of (1) due to the replacement of by . To guarantee that the sequence of objective value is nonincreasing and a limit point of the iterates , if converge, is a stationary solution of (1), the mapping function must satisfy the following properties

(3a)
(3b)
(3c)

for all , where denotes the gradient of with respect to the complex conjugate of (we for brevity refer to this gradient as the term ‘conjugate gradient’ in the rest of the paper). We note that the gradient in (3c) can be replaced by the directional derivative in general. In this regard, the definition of stationarity can be made more specific (cf. [33, 34] for more details). The main steps of the SCA procedure are outlined in Algorithm 1.

0:  Set and choose a feasible point .
1:  repeat
2:     Solve (2) to obtain optimal solution .
3:     Update: and .
4:  until Convergence.
4:  .
Algorithm 1 Generic framework of an SCA-based approach

The real and mixed real-complex domain

Before going forward we remark that for the case when is a real valued vector, the conjugate gradient of in (3c) is to be replaced by the normal gradient. If where and , , then is defined as . Furthermore, we define the inner product of two vectors and , denoted by , for several cases. If and are real valued, then . On the other hand, when and are complex valued, then .

Ii-B Initial Feasible Points for Algorithm 1: A Relaxed Algorithm

We can see that Algorithm 1 requires an initial feasible point to start the iterative procedure of the SCA. In some problems such as those where all and are homogeneous, generating a feasible point can be done easily via scaling/rescaling operation. More specifically, we can randomly generate a vector and then multiply the vector by a proper scalar such that all the constraints are satisfied. An example will be shown in Section IV. However, such simple manipulations cannot apply to more sophisticated cases. To overcome this issue, we present a relaxed version of Algorithm 1 which was used in [35, 19, 36, 37, 38]. Consider a relaxed problem of (2) given by

(4a)
(4b)
(4c)

where is the newly introduced slack variables and is a positive parameter. The purpose of introducing is to make (4) feasible for any . Indeed, given some , we can always find with sufficiently large elements satisfying (4b) and (4c). In addition, let be a feasible point of (4). Then is also feasible for (1) if . That is to say, successively solving (4) may produce a feasible solution to (1) because is encouraged to be zero due to the minimization of the objective in (4). Moreover, if is strongly convex and the Mangasaran-Fromovitz constraint qualification is satisfied at any such that , then it guarantees that after a finite number of iterations if is larger than a finite lower bound which can be determined based on the Lagrangian multipliers of (2) [38]. In general this lower bound is difficult to find analytically and the choice of can be heuristic in practice. For example, can be set to be sufficiently large [35], or it can be increased after each iteration [35, 37]. Note that these results still hold if we replace the variable by in (4b) for , by in (4c) for , and the term by in (4a) [38]. We summarize this relaxed procedure in Algorithm 2.

0:  Randomly generate .
1:  repeat
2:     Solve (4) to obtain optimal solution .
3:     Update and .
4:  until 
4:  .
Algorithm 2 A general initialization for Algorithm 1

Ii-C Convergence Results

There are several convergence results of the SCA [23, 24, 25, 27, 35, 39, 26, 28, 40], among them [23, 24, 27, 35, 26, 28, 40] consider nonconvex constraints. We briefly summarize these results herein for the sake of completeness. First, due to the use of upper bounds in (2c), is feasible to the convexified subproblem (2) at iteration . Thus and the sequence is nonincreasing, possibly to negative infinity. If is coercive on the feasible set or the feasible set is bounded, then converges to a finite value. However, this does not necessarily imply the convergence of the iterates to a stationary point or a local minimum in general.333If converges, the limit point is a stationary solution of (1) [27, 40, Th. 1].

To establish the convergence of the iterates, the key point is to ensure that a strict descent can be obtained after each iteration, i.e., for all , unless . If is strongly convex and the feasible set is convex as assumed in [27], the strict descent property and convergence of the iterates to a stationary point are guaranteed. When strict convexity does not hold for , the iterative process might not make the objective sequence strictly decreasing. To overcome this issue, we can add a proximal term as done in [35]. In particular, instead of (2), we consider a regularized problem of (2) given by

(5a)
(5b)

where is a regularization parameter. By adding proximal term , is strictly decreasing and the sequence converges to a stationary point due to the following result [35]. In fact, the proximal term makes each subproblem of SCA-based methods strongly convex, which is the main idea behind the work of [28]. Note that parameter should not be large, otherwise, the algorithm converges slowly. In practice, we should only consider (5) if a strict descent is not achieved at the current iteration [35]. Stronger convergence results for problems where the objective and constraints are nonconvex were reported in [28]. Particularly, if the approximation function of the nonconvex cost function is strongly convex (but not necessarily a tight global upper bound) and some other mild conditions are satisfied, the SCA procedure is guaranteed to converge to a stationary point with appropriate rules of updating operation points. Moreover, possibilities for distributed solutions under the framework of SCA were also discussed in [28, Section IV].

Ii-D Desired Properties of Approximation Functions

The main point of an SCA-based method is to find a convex upper approximation for a nonconvex function that satisfies the three conditions in (3). There are in fact several ways to do this. If is concave, constraint is called a reverse one and a convex upper bound of can be easily found from the first order Taylor series. In many cases, is a DC function, i.e., where both and are convex. In such a case, a convex upper bound can be given by , which is usually done in the context of the convex-concave procedure [24, 37]. Note that there are infinitely many DC decompositions for , e.g, by adding a quadratic term to both and . In particular, if the gradient of is -Lipschitz continuous, a surrogate function is given by which is a convex quadratic function. In some problems, the range of may be useful to find a convex upper bound of .

From the above discussion, one would be interested in finding the best approximation function for a given nonconvex function. However, the solution to this problem is not unique as it is problem specific. In general, a good approximation function will provide at least two features: tightness and numerical tractability. The tightness property is obvious as we want the approximate convex set and the original nonconvex one are as close as possible. This has a huge impact on the convergence rate of the iterative procedure. The numerical tractability properties means that the chosen approximation should yield a convex program that can be solved very efficiently, e.g., by analytical solution. In case an analytical solution is impossible to find, we may prefer to seek an approximation such that the convex subproblem in (2) belongs to a class of convex programs for which numerical methods are known to be more efficient. For example, an SOCP is much easier to solve than a generic convex problem consisting of a mix of SOC and exponential cone constraints. To see our arguments, let us consider an exemplary problem where and , , , are the problem data. An easy and straightforward approximation of would be by linearizing the term , which was actually considered in [41, 42]. However, this results in a generic nonlinear program (NLP). By exploiting the problem structure, we may find a positive such that is convex, and thus is in a DC form. As a result, a convex quadratic program (QP) is obtained if we linearize the term . Obviously, a QP is easier to solve than a generic NLP in terms of solution efficiency. An interesting example for this is provided in Section VII.

Iii Proposed Conic Quadratic Approximate Formulations

As discussed above, there are infinitely many approximations for a given nonconvex function or a nonconvex constraint, which have crucial impact on the convergence speed, numerical efficiency, etc. of SCA-based algorithms. From the standpoint of numerical optimization methods, it is probably best to arrive at a linear program for each subproblem in a SCA-based method. Unfortunately, this is hard to achieve in many wireless communications related problems, especially in view of beamforming techniques for multiantenna systems. As a result, conic quadratic optimization is a good choice due to its broad modeling capabilities and computational stability and efficiency. In this section, we will present some nonconvex constraints widely seen in wireless communications design problems and introduce novel convex approximations to deal with their nonconvexity.

The nonconvex constraints considered in this paper are given in a general form as

(6)

where and are affine or convex quadratic functions. We assume that , , and , which hold in numerous practical problems in wireless communications. The upper and lower limits and may be constants or optimization variables, depending on the specific problem. The cases where and/or are optimization variables mostly result from considering the epigraph form of the original design problem.

Iii-a Case 1: and Are Affine

We note that, when and are affine, if or is a constant, the associated constraint becomes a linear one, thus approximation is not needed. Here we are interested in the case where both and are optimization variables. In this case, is a real-valued vector. This class of constraints usually occurs in power control problems [43]. To handle such nonconvex constraints, [43] applied SCA so that the nonconvex problem is approximated as geometric programming (GP). We now show that this constraint can be approximated as a conic quadratic formulation.

Let us consider the constraint first, which is equivalent to with . A convex upper bound can be found

(App1)

and the SCA parameter update in Step 3 of Algorithm 1 is . Note that is a convex quadratic function for a given and a generalization of a result in [27]. The gradient of the upper bound function at some is given as which reduces to when substituting by . That is, the condition (3c) is satisfied by the bound in (App1).

In another way, we can rewrite in a DC form, i.e. , and the convex upper estimate can be found as

(App2)

In this case the SCA parameter update is simply

The constraint , , can be handled similarly. Specifically, in light of (App1), a convex approximation of can be found as

(App3)

where the SCA parameter update is . Following the same steps as for (App1), it is straightforward to check that the bound in (App3) satisfies the condition (3c). Note that is not a quadratic function but the constraint can be easily expressed by the following two SOC ones

(7)
(8)

Note that (8) is a rotated SOC constraint, since is affine. Alternatively, we can simply use the approximations similar to (App2).

Iii-B Case 2: and Are Convex Quadratic Functions

We now turn our attention to the case where and are convex quadratic functions. In wireless communications this form of constraint occurs in the problems related to precoder designs and is a vector of complex variables. Let us consider the constraint first, which is equivalent to , . Note that the term is convex with respect to , and thus a convex approximation of is given by

(App4)

where the SCA parameter update is taken as Another approximation can be found by introducing a slack variable, i.e. we have

(9)

The first constraint in the equivalent formulation can be rewritten as where , and is a quadratic function with respect to . Then the approximation can be given by

(App5)

The SCA parameter is updated as .

For the constraint we can equivalently write it as , and then a convex upper estimator is simply given by

(App6)

with the SCA parameter update is This constraints can also be approximated using the same approach as that in (App5).

To conclude this section we now show the conjugate gradient appearing in (App4), (App5) and (App6). Let us consider the conjugate gradient of . As mentioned earlier, we can write , where and are the elements of corresponding to and respectively. That means, it requires the conjugate gradient of a quadratic function. Suppose where is a PSD matrix, , and . Then . The conjugate gradients in (App5) and (App6) follow immediately.

Regarding the use of the above proposed algorithms we have the following remarks

  • When applying to a specific problem, an approximation function may be better than another. Thus, we can consider all applicable approximations to choose the best one for on-line design.

  • Many problems in wireless communications may not naturally express the design constraints in the forms written in this paper, for which cases equivalent transformations are required. In doing so, the number of newly introduced variables should be kept minimal. This issue will be further elaborated by an example in Section VII-B1.

In the following sections, we apply the above approximations to address four specific problems, which are chosen to cover a wide range of scenarios in wireless communications. However, we note that the proposed approximations also find applications in other contexts not considered herein. For the numerical experiments to follow, we use the modeling language YALMIP [44] with MOSEK [45] being the inner solver for SOCP, SDP, and GP. The proposed iterative method stops when the increase (or decrease) of the last 5 consecutive iterations is less than . The average run time reported in all figures takes into account the total number of iterations for the iterative algorithm to converge.

Iv Application I: Secure Beamforming Designs for Amplify-and-Forward Relay Networks

In the first application, we revisit the problem of secure beamforming for amplify-and-forward (AF) relay networks which was studied in[46].

Iv-a System Model and Problem Formulation

Figure 1: A diagram of secure AF relay networks with one source, one destination, relays, and eavesdroppers.

In the considered scenario, a source sends data to a destination through the assistance of relays that operate in the AF mode. In addition, there are eavesdroppers who want to intercept the information intended for the destination. It is assumed that there is no direct link between the source and the destination. The system model is illustrated in Fig. 1. We adopt the notations used in [46] for ease of discussion. Specifically, the channel between the source and relay , and that between relay and the destination are denoted by and , respectively. The channel between relay and eavesdropper is denoted by . Let be the complex weight used at relay . For notational convenience, the following vectors are defined: , , , and . Let be the noise vector at the relays. With the above notations, the signals received at the destination and eavesdropper are

(10)

and

(11)

respectively, where is the transmit power at the source, denotes the diagonal matrix with elements , and are the noise at the destination and eavesdropper , respectively. Then, the SINRs at the destination and eavesdropper are given by

(12)

respectively, where , , , and . Now the problem of maximizing the secrecy rate reads

(13a)
(13b)

where , , is the maximum total transmit power for all the relays, and is the maximum transmit power for relay .

To solve (13), [46] introduced the PSD matrix and arrived at a relaxation of this problem where the rank-1 constraint on was dropped for tractability. Then the relaxed program was solved by a method involving two-stage optimization; a one-dimensional search was performed at the outer-stage and SDPs were solved at the inner-stage.

Iv-B Proposed SOCP-based Solution

We now solve (13) based on the proposed approximations presented in the previous section. To do so, we first transform (13) into an equivalent formulation as

(14a)
(14b)
(14c)
(14d)

In fact, (14) is an epigraph form of (13) assuming the optimal value of the latter is strictly positive. Regarding (14), the objective function can be approximated using the upper bound in (App3), while the constraints (14b) and (14c) can be approximated by (App4) and (App6), respectively. The resulting convexified subproblem is an SOCP.

To complete the first application, we now provide the worst-case computational complexity of the proposed SOCP-based method and the SDP-based solution in [46], using the results in [31]. For the former, the worst-case arithmetic cost per iteration is which reduces to when . For the latter, the worst-case per-iteration computational cost is reducing to when .444We omit the constant related to the desired solution accuracy for the complexity analysis. The analysis implies that the per-iteration complexity of the proposed solution is much less sensitive to than that in [46]. We recall that the optimization method in [46] is also an iterative procedure. In addition, the complexity of each subproblem in the proposed method is much less than that in [46] (in orders of magnitude). Thus, we can reasonably expect that the proposed solution is superior to the SDP-based method in [46] in terms of numerical efficiency, which will be elaborated by numerical experiments in the following.

Iv-C Numerical Results

(a) Average secrecy rate (in bps).
(b) Average run time (in seconds).
Figure 2: Average secrecy rate and average run time of the proposed solution and the solution in [46] versus different numbers of relays, . We take the number of eavesdroppers, , as , , and .

We now numerically evaluate the performances of the proposed solution in terms of achieved secrecy rate and computational complexity (i.e. run time). The considered simulation model follows the one in [46]. Specifically, the channels are independent Rayleigh fading with zero means and unit variances. The noise variance is set to . The transmit power at the source is dB, the maximum total transmit power at the relays is dB, and the power budget at antenna is determined as if is odd and otherwise. We note that a feasible can be easily generated as follows. First, a random (but small enough) is generated satisfying (14d). Then, the left sides of the constraints (14b) and (14c) are computed accordingly, from which feasible and can be found easily.

In Fig. 2(a), we plot the achieved secrecy rate (in bits per second) of the considered schemes with different numbers of relays and eavesdroppers. We can observe that, in all cases, the proposed SOCP-based solution achieves nearly the same performance as the SDP-based solution in [46], but with much lower computation time as shown in Fig. 2(b). As expected, the run time of the proposed solution increases slowly with and compared to that of the existing method. In particular, the computation time improvement achieved by the proposed solution is huge for large numbers of relays and eavesdroppers (approximately 10 times faster in favor of the proposed method at ). This observation is consistent with the theoretical bounds of the arithmetical cost reported in the previous subsection.

V Application II: Beamforming Designs for Cognitive Radio Multicasting

We now turn our attention to the cognitive radio which has been considered as one of the most promising techniques to improve the spectrum utilization. In particular, we revisit the problem of transmit power minimization for secondary multicasting investigated in [21].

V-a System Model and Problem Formulation

The considered system model consists of a secondary multi-antenna base station transmitting data to groups of secondary single-antenna users where users in the same group receive the same information content. Let , , denote the set of users in group . The total number of secondary users is , and each user belongs to only one group. In addition, there exist primary single-antenna users who are interfered by the secondary transmission. A diagram of the considered system is shown in Fig. 3. Let be the number of antennas equipped at the secondary BS, be the channel (row) vector between the secondary BS and secondary user (SU) , be the channel vector between the secondary BS and primary user (PU) , and be the multicast transmit beamforming vector at the secondary BS for group . The problem of minimizing the transmit power at the secondary BS is stated as [21]

(15a)
(15b)
(15c)

where is the predefined interference threshold at PU , and are the QoS level and noise variance corresponding to SU , respectively. The constraints in (15b) guarantee the QoSs of the SUs while those in (15c) ensure that the interference generated by the secondary transmission at the PUs are smaller than predefined thresholds.

Figure 3: An example of cognitive multicast transmissions with PUs and multiple multicast groups including total SUs.

Problem (15) is a nonconvex program, and the prevailing approach is to lift the problem into a SDP [47, 48], i.e. PSD matrices , , are introduced and the rank-1 constraints on are ignored. However, this approach cannot guarantee even a feasible solution, since the relaxed problem generally does not yield rank-1 solutions and randomization procedures are inefficient. To overcome this shortcoming, [21] proposed an iterative approach where the PSD matrices were still introduced. However, the noncovex rank-1 constraints (which were expressed a reverse convex constraint, i.e. where is the maximal eigenvalue of ) were sequentially approximated in a manner similar to the SCA framework presented in this paper.

V-B Proposed SOCP-based Solution

We observe that the objective function in (15a) and constraint in (15c) are convex. Thus the difficulty of solving the problem comes from the nonconvex constraints in (15b). To handle these constraints, let us introduce and equivalently rewrite (15) as

(16a)
(16b)
(16c)

where , , (for ), and . Now, we can easily use (App4) to deal with (16b), and the convex approximated problem is actually an SOCP. We remark that, for (15), it is nontrivial to find a feasible to start the SCA procedure. Thus the relaxed version, i.e. Algorithm 2, is invoked for some first iterations, until a feasible point is found.

We now compare the complexity of the proposed solution and the method in [21]. In particular, the worst-case complexity for solving the SOCP in (16) is , while that for the SDP in [21] is . When the number of transmit antennas at the secondary BS is large, the two bounds reduce to and , respectively.

V-C Numerical Results

(a) Average transmit power at the secondary BS (in dB).
(b) Average run time (in seconds).
Figure 4: Average transmit power at the secondary BS and average run time of the proposed solution and the solution in [21] versus the number of equipped antennas, . Two sets of are taken those are and . Each multicast group includes 4 secondary users.

We follow the simulation model considered in [21] for performance comparisons. Particularly, the channels are generated as , . The QoS levels at the SUs and the interference thresholds at the PUs are set to , and . The stopping criterion of the method in [21] is when the decrease in the last 5 iterations is smaller than .

In Fig. 4(a), we plot the average required transmit power at the secondary BS for the proposed method and the one in [21] as functions of the number of transmit antennas . As can be seen, the transmit powers required by the two schemes are almost the same for all considered scenarios. In other words, the proposed solution is similar to the one in [21] in terms of power efficiency. However, the proposed method is much more computationally efficient as demonstrated in Fig. 4(b), in which we plot the run time of the two methods. We can clearly see that the average run time of both schemes increases with , but slowly for the proposed solution and very rapidly for the method in [21]. As a result, the computation time of the method in [21] is much higher than that of the proposed solution, especially for large . This numerical observation is consistent with the complexity analysis provided in the preceding subsection. In summary, the numerical results in Fig. 4 demonstrate that the proposed SOCP-based solution is superior to the existing one presented in [21].

Vi Application III: Precoding Design for MIMO Relaying

Relay-assisted wireless communications is expected to play a key role in improving coverage and spectral efficiency for the current and future generations of cellular networks. In this section, we apply the proposed approximations to the scenario of multiuser MIMO relaying which was investigated in [49].

Figure 5: An example of relaying communications with source-destination pairs and MIMO relays. The transmissions follow the two-phase precode-and-forward principle.

Vi-a System Model and Problem Formulation

We consider the wireless communication scenario in which the transmission of source-destination pairs are simultaneously assisted by a set of relays. Each of the sources and destinations is equipped with single antenna, whereas each relay is equipped with antennas (the total number of antennas at the relays is ). Suppose there are no direct links between the sources and destinations, and the relays operate according to the AF protocol. That is, the sources transmit their information to the relays in the first phase, and then the relays process the received signals and retransmit them to the destinations in the second phase. The considered system model is illustrated in Fig. 5. We reuse the notations introduced in [49]. Specifically, let , where , be the vector of messages sent by the sources, and be the vectors of channels from source to the relays and from the relays to destination , respectively. Let be the processing matrix at relay , . For ease of description, we also define , , and as in [49]. Then the signal vectors received at the relays and destination are given by

(17)

and

(18)

respectively, where and are the noise vectors at the relays and destination . Accordingly, the SINR at the th destination is written as

(19)

The power consumption at antenna , , is given by

(20)

where , and