Optimization Algorithms Combining (Meta)heuristics and Mathematical Programming and Its Application in Engineering
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Generalized Beamforming Design for Cooperative MIMO Multirelay Networks with Infinite Constraints and Imperfect CSI
Abstract
A generalized base stationrelayuser equipment (BSRelayUE) beamforming design is investigated for a cooperative multipleinput multipleoutput (MIMO) multirelay networks with imperfect channel state information (CSI). In order to minimize the worstcase mean square error (MSE) which is subject to a semiinfinite (SI) relay power constraints, a generalized optimal beamforming structure for the relay amplifying matrix is effectively proposed, and then the SI relay power constraints are converted into linear matrix inequalities (LMIs) version. In such conversion, the objective problem recasts as a decoupled biconvex semidefinite programming (SDP) one which can be efficiently solved by the proposed alternating algorithm. The system performance has been verified in terms of worstcase MSE using a set of qualitative analyses. The results show us that the proposed beamforming method outperforms the conventional schemes and can also effectively reduce the computational complexity when it is compared to the cuttingset schemes and also to the nonrobust ones.
1. Introduction
Recently, cooperative multiinput multioutput (MIMO) relay network approaches are popularized to increase the system capacity and to improve the transmission reliability by leveraging spatial diversity. Since the channel estimation is important to the wireless communications, the authors in [1] investigated the main challenges faced by highdimensional channel state information (CSI) acquisition in massive MIMO systems which was first shown that the narrow angular spread or the spatial sparsity is crucial for all lowrank approaches. Cooperative relay network technology has attracted significant interests due to the superior spectral efficiency. Various cooperative relaying schemes have been proposed, such as denoiseandforward (DNF) [2–4], decodeandforward (DF) [5, 6], and amplifyandforward (AF) [7–12]. Specifically, the spatially correlated fading channels are considered in [13], which is more practical and challenging. Training designs for estimation of spatially correlated multipleinput multipleoutput (MIMO) AF twoway multirelay channels are studied in [13], where an optimal training structure is initially derived to minimize total mean square error (MSE) of the channel estimation.
Considering inaccurate channel estimation and feedback delay, the perfect channel state information (CSI), which are proposed in the above works, is usually hard to obtain in practice. In order to circumvent these problems, by taking into account the channel uncertainties, the imperfect CSI scenario has been studied in [7, 14–20]. The authors in [14] considered robust transceiver design in a multiuser MIMO multirelay cognitive radio network with interference power constraints and individual transmission power constraints. In [15], the authors have investigated the global optimal transceiver design in the MIMO link under a deterministic CSI uncertainty model. Specifically, a robust minimum meansquareerror (MMSE) based beamforming scheme for an amplifyandforwarding multiantenna relay network is considered in [16]. Here, the authors divided the original problem into subproblems which are formulated into a convex optimization framework. By taking the imperfect channel state information into consideration, in [17], the robust AFMIMO transceiver optimization is developed in order to combat correlated channel uncertainties, where the matrixform conjugate gradient (MCG) method is used for optimization. Moreover, the robust joint optimization of the relay weights and the input covariance matrix of jamming signals for secrecy rate maximization is addressed in [18]. In [19], the authors investigated the worstcase robust relay precoder optimization for multiantenna AF relaying in the presence of either deterministic additive or multiplicative channel uncertainties. Specifically, the authors in [7, 20] have considered a twoway relay network consisting of multiple pairs of singleantenna users and distributed, multiple relays.
Furthermore, in [21–26], the generalization of SLemma to complexvalued variables and multiple semiinfinite (SI) constraints are presented. They have investigated iterative algorithms based on alternating convex search (ACS) to devise practical algorithms for solving the semiinfinite problems. In addition, in [26], the authors investigated a robust transceiver design for downlink multiuser MIMO AFrelay systems with normbounded channel uncertainties presenting that the cuttingset method proposed improves the performance of the worstcase MSE.
Since that, for the multiple relays scheme, not only the performance of the capacity outperforms that of the single relay one, but also the multiple relays scheme is more practical and challenging for the wireless communication scenarios. The downlink network considered here consists of a base station (BS), multiple relays, and user equipment (UE). We have named this particular AF system base BSRelaysUE beamforming. The main contributions of this paper are summarized in the following:(i)We comprehensively investigate a generalized beamforming design for a cooperative MIMO multirelay networks with imperfect CSI. In the proposed scheme, the joint optimal design considers the precoding matrix at the BS, the beamforming matrix at the relays and the receiving beamforming matrix at the UE with multiple relays, which is general and practical.(ii)Since the considered worstcase MSE optimization problem is not only nonconvex but also subject to the semiinfinite relay power constraints, a generalized optimal beamforming structure of the relay amplifying matrix is investigated. In addition, the SI relay power constraints are converted into linear matrix inequalities (LMIs). By this way, the objective problem recasts as a decoupled biconvex semidefinite programming (SDP) one, which can be efficiently solved by our proposed alternating algorithm.(iii)By means of the numerical results, the proposed beamforming design significantly reduces the computational cost and improves the performance in terms of the worstcase MSE compared with the nonrobust and cuttingset cases.
In the following, a system model for the proposed cooperative MIMO multirelay networks is presented in Section 2 and a generalized BSRelayUE beamforming design is described in Section 3 (see Algorithm 1). In Section 4, the performance of the proposed system is evaluated using qualitative analysis and the numerical results are provided. Finally, the conclusions are made in Section 5.

Notations. , , , , and denote the transpose, conjugate transpose, trace, Euclidean norm, and Frobenius norm of a matrix , respectively. and are operators for taking the real part and imaginary part of a complexvalued matrix, respectively. The field of dimensional complexvalued matrix is denoted by . is an identity matrix. and are the operators for computing Hadamard and Kronecker products.
2. System Model
In this section, an amplifyandforward relaying system model is described. In the proposed system, users are away from a base station and they are isolated and out of the BS coverage. In other words, the direct link between BS and UE does not exist so that UE can get a connection to the BS only via relays in order to achieve any signal receptions. To be more specific, the proposed multirelay network consists of one BS, relays, and one UE as shown in Figure 1. In this network, communications are performed using a halfduplex mode. It is further assumed that the BS and UE are equipped with antennas while each relay node is with antennas.
Now, concerning the communication channels related to the proposed network, there are two of them. The one is the group of estimated communication channels , which are from the BS to relays , for ; the other one is another group of communication channels , which are from relays to UE. We have also considered to satisfy the decodeable condition.
It is assumed that the BS transmits the signals to the UE using two consecutive time slots. At the first time slot, after being linearly processed by the matrix , which is constrained by , the information data with unit power is transmitted to the relay nodes. Therefore, the received signal at can be expressed aswhere represents the additive white Gaussian noise (AWGN) vector with zero mean and variance at the relay node .
At the second time slot, the relay node linearly amplifies with an beamforming matrix and then broadcasts the amplified signal vector to the UE, which results inTherefore, the transmission power used by the relay node is given byAt the UE, the received data is linearly utilized by using the receiving beamforming matrix with , where ≤ is a slack value. The received signal vectors at the UE can be finally denoted aswhere is the received noise vector at the UE with mean zero and variance .
By taking into account the estimation error, the CSI is considered partially known at each node. With this consideration, the actual channel coefficients of the links follow thatwith as the channel uncertainties. For simplicity, the channel uncertainties are assumed to be normbounded errors (NBEs) [24] aswhere .
With this observation, the worstcase MSE at the UE node can be obtained asSince , after introducing the slack value , the objective problem is to minimize the worstcase MSE subjects to the BS and the relay transmission power constraints which can be formulated as follows:where denotes the power constraint at the relay node. In problem , it is clear that not only the objective problem is nonconvex, but also the semiinfinite expressions of the optimal , , and are intractable. In particular, the globally optimal solution is difficult to be obtained; in this paper, to efficiently solve , a biconvex SDP is proposed to obtain the suboptimal solution of the worstcase MSE for our proposed scheme.
3. Generalized BSRelayUE Beamforming Design
3.1. The Optimal Relay Beamforming Design
With the fixed BS and UE beamforming matrices and , suppose the singular value decomposition (SVD) of and as follows:where , , , and are unitary matrices.
Theorem 1. Using the SVDs in (9), the optimal relay beamforming matrix as the solution to the objective problem can be obtained aswhere is a matrix to be determined.
Proof. The proof is similar to [8, 25].
Without loss of generality, can be further partitioned as follows:From (9), (11) and Theorem 1, one can haveIn this manner, we havewhere and . It is worth noting that the term becomes , because if we retain this term in , when calculating the MSE, it will result in some terms involving and order of channel uncertainties which are negligible.
Using and , from (13), we haveSince the channel uncertainty can be further partitioned,upon substituting (10), (11), and (15) into the relay power constraint (3), we havewhere . Synthesizing (14) and (16), it is clear that, for any feasible with , one can always findwhich can achieve the smaller relay power constraint and MSE. Thus, we conclude that if and only if , the objective problem has the optimal solution. By this way, the optimal expression of can be denoted as
3.2. The Joint Source and Relay Beamforming Design
From (14), it is clear that, with fixed , the objective problem is equivalent to the problem:Therefore, similar to [22], by introducing the auxiliary optimization variable , for , problem (19) can be recast in the epigraph form [27] as which subjects to , . In this manner, the objective problem can be equivalently converted into the problem as follows:Using the identities and , for any given matrix , and , for the constraint , we haveFurther assuming , the constraint can be represented in the term of the following LMI form asBy employing SLemma [22], (22) can be recast aswhere , for , is a slack variable, andwhereOn the other hand, similarly, letting , the constraint can be equivalently converted into the LMI version as follows:where is a slack variable. In this observation, (23) and (26) represented the equivalent SDP formulation of the constraints and which are affine in the channel uncertainties.
For the relay power constraint, after introducing the slack variable , it can be decoupled into two parts as follows:where , , and . Defining the constraint (28) can be reformulated aswhich can be converted to the following LMI:where is a slack variable.
Thus, the objective problem is recast as
It is clear that the problem is a biconvex SDP which can be efficiently solved by the following alternating algorithm by using CVX [27].
By initializing the small and setting the limitation of the number of iterations , the suboptimal solution can be obtained when .
4. Numerical Results
In this section, we examine the performance of the proposed robust scheme in terms of the worstcase MSE compared with cuttingset method in [26], the nonrobust case and the perfect design. All results are averaged over 5,000 channel realizations and the initial BS and the UE beamforming matrices are chosen at random. We set the number of the relay nodes , the number of antennas and , respectively. The transmission power at the BS and the relay node is . In the iterative algorithm, and are set to and , respectively.
Figure 2 depicts the convergence of the proposed algorithm with the fixed NBEs as for the transmission dB and dB. Clearly, our proposed scheme has satisfactory convergent performance compared with the cuttingset method. It is because, in our proposed scheme, the objective problem converts the worstcase MSE into the one with which significantly reduces the computational complexity. In addition, the optimal (local) relay beamforming matrix can be efficiently derived which supports the practical utility of our design. It is also observed that, for the higher transmission SNR, the convergence rate of our proposed scheme outperforms the cuttingset one while both these two schemes provide the optimal (local) solution with more iterations.
Figure 3 compares the worstcase MSE in cases of our proposed scheme, the cuttingset one and the perfect one versus the transmission SNR with and , where the perfect one is served as the upper bound with the NBEs as while the robust solutions are obtained by using the corresponding algorithms. Results reveal that our proposal has a relative advantage over the other two schemes, especially for the larger NBE case. This is reasonable, since employing the approximation may lose the performance, in our proposed scheme, the objective problem is equivalent to the original target one without using the approximations (except the terms involving the highorder channel uncertainties) which is different to the work in [26].
5. Conclusions
So far, we have presented an efficient way to achieve signal transmission from a BS to an isolated, outofcoverage user by realizing a generalized BSRUE beamforming. In our proposed cooperative MIMO multirelay networks with imperfect CSI; first, the optimal relay beamforming is derived as a means to improve the efficiency. Then, the semiinfinite objective problem is converted into a biconvex problem, which is subject to the LMI constraints. Furthermore, the converted, biconvex problem is efficiently solved by our alternating algorithm. The system performance in terms of worstcase MSE is verified using a set of qualitative analysis; the results show us that the proposed beamforming method outperforms the conventional schemes and can also effectively reduce the computational complexity when it is compared to the cuttingset schemes and also to the nonrobust ones. Our future concerns will be the generalized optimization problems on the twoway relay networks and cooperative relaying nonorthogonal multiple access (NOMA) systems.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would gratefully acknowledge the grants from the National Natural Science Foundation of China (61371113, 61401241, 61401240, and 61771264), Nantong UniversityNantong Joint Research Center for Intelligent Information Technology (KFKT2016B01 and KFKT2017B01), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (no. 2015D02), and the Brain Korea BK21 plus.
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Copyright © 2018 Haiyang Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.