Abstract
In this article, we propose multiplerelay selection schemes for multiple source nodes in amplifyandforward wireless relay networks based on the sum capacity maximization criterion. Both optimal and suboptimal relay selection criteria are discussed, considering that suboptimal approaches demonstrate advantages in reduced computational complexity. Using semidefinite programming convex optimization, we present computationally efficient algorithms for multiplesource multiplerelay selection (MSMRS) with both fixed number and varied number of relays. Finally, numerical results are provided to illustrate the comparisons between different relay selection criteria. It is demonstrated that optimal varied number MSMRS outperforms optimal fixed number MSMRS under the same power constraints.
Introduction
Multihop relaying has emerged as a promising approach to achieve highrate coverage in wireless communications [1, 2]. Several amplifyandforward (AF) and decodeandforward (DF) relaying techniques have been introduced such as in [2, 3]. Following those pioneer works, a number of cooperative diversity schemes have been proposed, including, for example, distributed spacetime coding [3–5], adaptive power control for relay networks or relay beamforming [6–9], and relay selection [10–19].
The objective of relay selection is to achieve higher throughput or lower error probability through choosing one or more relays for transmission according to channel conditions. In comparison to relay beamforming, relay selection is attractive due to its deployment of simpler signaling scheme and energy saving. Most currently available relay selection approaches assume only a single source node [10–12, 14–18], and can be classified into two categories:

1.
A majority of relay selection rules are restrictive in the sense that they either always use all the available relays or always use just a single relay, such as in [10–18, 20–29]. In [21], four simple relay selection criteria are described: Two criteria are based on the selection of a single relay according to mean channel gains, while the other two select all available relays. Selecting all available relays are the simplest approach with multiple relays, and this approach may not be allowed when the sum power limit is less than the summation of the power values of all available relays. A singlerelay node is selected based on average channel state information (CSI), e.g., distance or path loss [20, 22, 30], and on the instantaneous fading states of the various links such as in [23].

2.
Multiplerelay selection for a single source has attracted attention as well [31–33]. Jing and Jafarkhani proposed suboptimal twostep optimization approaches for singlesource multiplerelay selection in [31, 33]: In the first step, phase rotation is performed at each relay, and thus only power allocation is considered due to signaltonoise ratio (SNR) consisting of a summation of purely real terms. In the second step, several suboptimal methods were introduced [31, 33]:

(a)
By introducing the idea of relay ordering, several schemes with linear complexity were proposed;

(b)
Based on recursion, a scheme with quadratic complexity was proposed.

(a)
Although both single and multiplerelay selection approaches for a single source node network have been investigated, relay selection approaches for multiple source nodes are rarely addressed in literature. Only the following three existing publications [34–36] have discussed multiplesource relay selection (MSRS) approaches. Elzbieta and Raviraj have proposed MSRS for DF relay networks [34]. Xu et al. have presented MSRS approaches in which only a single source is considered as the desired user over each selected relay per transmission while other sources or users are considered as interferers during the transmission [35]. Guo et al. have analyzed MSRS for opportunistic relays, in other words, only a single source is transmitted over each selected relay per transmission [36]. Further, there have been several recent research works on twoway relay selections [37–40].
In this article, we consider AF relaybased cooperative communication systems for simultaneous multiplesource transmission over each selected relay, and more than one relay is allowed per multiplesource transmission. Each relay is assumed to satisfy practical individual shortterm power constraints, that is, each relay has two power levels: zero and its maximum power. This assumption has been used for singlesource multiplerelay selection in, for example, [31, 33]. The main contributions of this article can be listed as:

1.
Based on the sum capacity criteria, we derive and propose several multiplerelay selection techniques in AF relay networks with multiple source nodes.

2.
Using semidefinite programming optimization, we propose computationally efficient algorithms for multiplesource multiplerelay selection (MSMRS) in the presence of both fixed number and varied number of relays.
The following notations are used: ${(\xb7)}^{\mathcal{T}}$ denotes matrix transpose, (·)^{∗}conjugate, ${(\xb7)}^{\mathcal{\mathscr{H}}}$ matrix conjugate transpose, ⊙ Hardmard product operator, ${\left[\mathbf{A}\right]}_{a,b}$ the (a,b)th entry (element) of matrix A, $\mathrm{tr}\left(\xb7\right)$ matrix trace operation, $\mathrm{Re}\left(\xb7\right)$ real part of the object (matrix or variable), $\mathrm{Im}\left(\xb7\right)$ imaginary part of the object (matrix or variable), ${\mathrm{E}}_{\alpha}\left(\xb7\right)$ expectation over random variable or random variable set α, $\text{diag}\left(\mathbf{a}\right)$ denotes a square matrix with allzeros entries except the main diagonal filled with the entries of the vector a, ϕ denotes empty set, and X≽0 denotes that X is a positive semidefinite matrix.
System model and problem formulation
Consider a wireless relay network with M source nodes (transmitters), K relay nodes, and one destination node (receiver). Each node is equipped with a single antenna. Assume no direct channel path between the source nodes and the destination node. The source nodes and the relay nodes are assumed to share the same transmission channel.
Based on twophase halfduplex AF relay assumption, we consider a multiplesource AF relay selection approach. The period of one twophase AF relay procedure is defined as one time channel use. During the t th time channel use, the twophase AF protocol is performed as follows:

1.
In the first phase, the m th source node (transmitter) sends source information symbol ${x}_{m}^{\left(t\right)}$ using power ${P}_{m}^{\left(S\right)}$ to the relay nodes, where m=1,…,M, $\mathrm{E}\left({\left{x}_{m}^{\left(t\right)}\right}^{2}\right)=1$. the information symbols ${x}_{m}^{\left(t\right)}$, m=1,…,M, are selected randomly from M independent codebooks. It is assumed that M source nodes simultaneously send uncorrelated signal streams ${x}_{m}^{\left(t\right)},m=1,\dots ,M$, and the corresponding channel symbols are received at relay k at the same time.

2.
In the second phase, L relays with indices $\left\{{k}_{1},\dots ,{k}_{L}\right\}$ are selected according to some criteria, which will be elaborated later. Here, L, 1≤L≤K, is an integer, which is referred to as “relay selection order” in this article. Then, the k _{ i }th relay, i=1,…,L, scales its received signal power to unity, and, using power ${P}_{{k}_{i}}^{\left(R\right)}$, amplifies and forwards it to the receiver.
Note that, in this twophase AF protocol, multiple source nodes share the same channels. The transmission and reception among the source nodes, the relay nodes and the destination node are assumed to be perfectly synchronized.
In the t th time channel use, the channel from the m th source node (transmitter) to the k th relay is denoted as ${h}_{m}^{(k,t)}$ and the channel from the k th relay to the receiver is denoted as ${g}_{k}^{\left(t\right)}$. The channels are modeled as frequency nonselective Rayleigh fading, and are assumed to independently vary over different time channel uses. Denote ${v}_{k}^{\left(t\right)}$ as the noise component at the k th relay, k=1,…,K, and denote w^{(t)} as the noise component at the destination node, where ${v}_{k}^{\left(t\right)}$ and w^{(t)} are assumed to be independently and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance.
During the t th time channel use, the received signal at the k th relay is
The corresponding scaling factor for the k th relay is given by
During the t th time channel use, the received signal at destination is then obtained as
where ${\alpha}_{k}^{\left(t\right)}$ is the relay selection factor, whose value is equal to 0 or 1, depending upon different relay selection algorithms.
In this article, we choose sum capacity per time channel use as the performance measure for relay selection [41–43]. The system sum capacity is given by
where ρ^{(t)} is the overall system effective SNR, and obtained in our case as
In the above, ${\mathrm{E}}_{\left\{{x}_{m}^{\left(t\right)}\right\}}\left({\left{s}^{\left(t\right)}\right}^{2}\right)$ and ${\mathrm{E}}_{\left\{{v}_{k}^{\left(t\right)},{w}^{\left(t\right)}\right\}}\left({\left{q}^{\left(t\right)}\right}^{2}\right)$ are given by
and
Inserting (6) and (7) into (5) , we have
Relay selection could be expressed using set partition. Define relay index set $\Omega =\left\{1,\dots .,K\right\}$. There exist L distinct relay indices $\left\{{k}_{1},\dots ,{k}_{L}\right\}$, where $1\le \left\{{k}_{1},\dots ,{k}_{L}\right\}\le K$, such that the following hold: $\left\{{\alpha}_{k}^{\left(t\right)}=0,k\notin \left\{{k}_{1},\dots ,{k}_{L}\right\},1\le k\le K\right\}$ and ${\alpha}_{{k}_{1}}^{\left(t\right)}=\cdots ={\alpha}_{{k}_{L}}^{\left(t\right)}=1$. The optimization problem can be now formulated as $\underset{\Omega}{arg}max\left\{{C}^{\left(t\right)}\right\}$. Since $\underset{a}{log}\left(\xb7\right)$, a>1, is a monotonous function, the problem is equivalent to $\underset{\Omega}{arg}max\left\{{\rho}^{\left(t\right)}\right\}$.
Relay selection can be implemented at the destination node (receiver). In this case, the receiver is assumed to know all instantaneous channel state information for sourcerelay paths and relaydestination paths, which may be obtained through channel estimation. After one relay selection algorithm is performed at the destination node, ${\alpha}_{{k}_{1}}^{\left(t\right)}=\cdots ={\alpha}_{{k}_{L}}^{\left(t\right)}$ are obtained, and then the destination node feedbacks onedigit relay selection information to each relay node. The superscripts ^{(t)} used in this section will be omitted in the rest of the article to simplify the notations whenever no ambiguity arises.
Fixed number multiplesource multiplerelay selection
When individual relay power constraints are equal or close, the number of relays may be used as a constraint to stand for sum relay power constraints. In this section, for simultaneous transmission of multiple source signals, the number of relays to be selected is assumed to be a fixed number L, where L>1. This class of approaches are referred to as fixed number multiplesource multiplerelay selection (FNMSMRS), and the corresponding set partition of relay indices for relay selection can be defined as
Optimal FNMSMRS
Using (8) and the values of $\left\{{\alpha}_{k}^{t}\right\}$, the output SNR of optimal FNMSMRS (OFNMSMRS) can be derived as
Thus, the proposed selection criterion is
Fixed number MSMRS based on semidefinite programming optimization
The complexity of OFNMSMRS may be prohibitive particularly when the dimension of the problem becomes larger. Closedform optimization solutions are unfortunately not possible for (10) due to the involvement of multiple sources. Based on semidefinite programming [44], we propose an efficient approach for FNMSMRS. First, note that (8) can be written in a matrix form as
where $\mathbf{p}={\left[{\alpha}_{1},\dots ,{\alpha}_{K}\right]}^{\mathcal{T}}$$\mathbf{P}=\text{diag}\left({P}_{1}^{\left(R\right)},\dots ,{P}_{K}^{\left(R\right)}\right)$${\mathbf{A}}_{s}=\sum _{m=1}^{M}\left({P}_{m}^{\left(S\right)}{\left({\mathbf{a}}_{s}^{\left(m\right)}\right)}^{\ast}{\left({\mathbf{a}}_{s}^{\left(m\right)}\right)}^{\mathcal{T}}\right)$${\mathbf{A}}_{n}=\text{diag}\left({\left({\mathbf{a}}_{n}\right)}^{\ast}\odot {\mathbf{a}}_{n}\right)$${\mathbf{a}}_{s}^{\left(m\right)}={\left[{\beta}_{1}{h}_{m}^{\left(1\right)}{g}_{1},\dots ,{\beta}_{K}{h}_{m}^{\left(K\right)}{g}_{K}\right]}^{\mathcal{T}}$, and ${\mathbf{a}}_{n}^{\left(m\right)}=\left[{\beta}_{1}{g}_{1},\right.$${\left.\dots ,{\beta}_{K}{g}_{K}\right]}^{\mathcal{T}}$. Further, note that p is a real integer vector with $\left\{0,1\right\}$ entries, A_{ s } is a Hermitian matrix, and A_{ n } is a realvalued matrix. It can be readily checked from (6) that
Thus, (11) can be further simplified in a realvalued matrix form as
Denote
where $\underline{\mathbf{c}}$ is an integer vector with $\left\{1,1\right\}$ entries, and 1_{ K }is an allone column vector of length K. For an arbitrary matrix M of size K×K, the following relationship always holds,
where
and matrix $\underline{\mathbf{M}}$ is related to M by a function f defined as
Now (11) can be rewritten as
where $\mathbf{S}=f\left({\mathbf{P}}^{1/2}\mathrm{Re}\left\{{\mathbf{A}}_{s}\right\}{\mathbf{P}}^{1/2}\right)$ and $\mathbf{N}=f\left({\mathbf{P}}^{1/2}\right.$$\left.{\mathbf{A}}_{n}{\mathbf{P}}^{1/2}\right)$.
The optimization problem can be now formulated as
where G_{ k }, k=1,…,K, are allzero matrices except ${\left[{\mathbf{G}}_{k}\right]}_{k,k}=1$, k=1,…,K. The fixed relay selection order is quantified in (18c). Note that, it is necessary to include individual relay selection factor constraints, such as (18d), which are actually related to individual relay power constraints, otherwise individual relay selection factors can be arbitrary in the optimization process. Only using vector p, it is hard to quantify individual relay selection factor constraints. However, based on the vector transformation in (13) and (15), individual relay selection factor constraints can be advantageously written as the form shown in (18d).
Denote $\mathbf{B}=\mathbf{c}{\mathbf{c}}^{\mathcal{T}}$. Note that ${\mathbf{c}}^{\mathcal{T}}\mathbf{Xc}=\mathrm{tr}\left(\mathbf{Xc}{\mathbf{c}}^{\mathcal{T}}\right)=\mathrm{tr}\left(\mathbf{XB}\right)$. Thus, the optimization problem (18) now becomes
subject to:
The optimization problem is not convex due to rank constraint (19e) and fractional constraint (19b). We can perform semidefinite relaxation through removing rank constraint (19e) [45]. Choosing a positive variable u, where $\frac{\mathrm{tr}\left(\mathbf{S}\mathbf{B}\right)}{\mathrm{tr}\left(\mathbf{N}\mathbf{B}\right)+1}\ge u$, the above optimization problem can be written as
The optimization problem (20) is still nonconvex. However, using the bisection Algorithm 1 as shown in Appendix, with the aids of convex programming tools, such as CVX[46, 47] which we have used in the simulations, the problem (20) can be solved iteratively, since it is quasiconvex in each loop within the bisection Algorithm 1, where u acts as a constant. This problem (20) can now be efficiently solved by standard interior point algorithms based on semidefinite programming (SDP) [48]. Denote the optimal estimation of B through the proposed bisection Algorithm 1 as $\hat{\mathbf{B}}$.
The above SDP procedure requires bisection Algorithm 1, which might introduce higher complexity when the number of iterative loops is high for convergence. Now we may consider another approach without the requirement of an bisection algorithm. Denote
Using (20c), we have
Denote
where λ>0 is chosen to make sure
Denote $\mathbf{W}={\mathbf{ww}}^{\mathcal{T}}$, and thus
Through removing rank constraint (19e), the problem (19) is now relaxed to
The problem (26) now could be solved using semidefinite programming without the requirement of a bisection algorithm. Note that the above method could be considered as the extension of Charnes–Cooper algorithm [49] from linear fractional programming to linear quadratic programming.
Note that the above solutions are obtained through removing the rank1 constraint (19e), which may lead to an increased problem dimension. Thus it is required to convert the semidefinite relaxation solution to some Boolean solution. In [45, 50, 51], a randomization method has been introduced to achieve this conversion. Note that in those works, the randomization approach is implemented without additional constraint. Here, we extend such randomization approach to support extra constraints, such as (20c). Based on the randomization procedure as proposed in the Appendix, the decision of $\underline{\mathbf{c}}$$\hat{\underline{\mathbf{c}}}$, can be obtained, where $\hat{\underline{\mathbf{c}}}={\left[{\left[\hat{\mathbf{c}}\right]}_{1,2},\dots ,{\left[\hat{\mathbf{c}}\right]}_{1,K+1}\right]}^{\mathcal{T}}$ and ${\left[\hat{\mathbf{c}}\right]}_{1,k}$ is the k th entry of $\hat{\mathbf{c}}$. It should be noted that Steps 9) and 10) of Algorithm 2 are introduced to satisfy constraint (20c). In [45, 50, 51], only $\mathbf{c}=\text{sign}\left({\mathbf{V}}^{\mathcal{T}}\mathbf{u}\right)$ is used in the randomization process. However, it has been further proved that $\pm \text{sign}\left(\mathbf{c}\right)=\text{sign}\left({\mathbf{V}}^{\mathcal{T}}\mathbf{u}\right)$ holds with probability 1 in Property 2 of [45]^{a}. Thus it may be meaningful to perform both “+” and “−” of “sign” operations in the randomization process as we have proposed in Steps 9) and 10) of Algorithm 2.
The above proposed MSMRS based on semidefinite programming is termed as SDPFNMSMRS:

1.
In the case of solving Problem 20 and using randomization procedure Algorithm 2: SDPFNMSMRS B1,

2.
In the case of solving Problem 20 and using randomization procedure Algorithm 2 without step 10: SDPFNMSMRS A1,

3.
In the case of solving Problem 26 and using randomization procedure Algorithm 2: SDPFNMSMRS B2,

4.
In the case of solving Problem 26 and using randomization procedure Algorithm 2 without step 10: SDPFNMSMRS A2.
Note that both the solutions of SDPFNMSMRS B1 and SDPFNMSMRS A1 require bisection algorithms to solve SDP problems iteratively, while both the solutions of SDPFNMSMRS B2 and SDPFNMSMRS A2 do not.
Best worse FNMSMRS and random FNMSMRS
Note that (9) cannot be further simplified without additional approximations. Intuitively, it can be questioned whether bestworse single source singlerelay selection [33] can be extended to this case. To address this concern, calculate all
where k=1,…,K. Then, permutate a_{ k } in descending order such that a_{σ(1)}≥⋯≥a_{σ(K)}, where $\sigma \left(\xb7\right)$denotes the permutation function. This yields $\left\{\sigma \left(1\right),\dots ,\sigma \left(L\right)\right\}$, and such a selection criterion is termed as best worse FN MSMRS (BWFNMSMRS).
For comparison purpose, we also define random fixed number MSMRS (RANDFNMSMRS), which randomly selects L relays, as a baseline benchmark FNMSMRS scheme.
Varied number multiplesource multiplerelay selection (MSMRS)
When individual relay power constraints are diverse, sum power constraints can no longer be described using a fixed number of relays. Unlike the previous section, we assume in this section that the number of relays to be selected is not predetermined but rather a varied number which is optimized depending on both individual and sum relay power constraints. This class of proposed approaches are abbreviated as VNMSMRS, and the corresponding set partition of relay indices for relay selection can be defined as
For comparison purpose, a baseline benchmark VNMSMRS scheme using predetermined relay selection, PVNMSMRS, is also defined. In this scheme, a feasible relay selection is chosen, assuming that this selection satisfies given relay power constraints, and no more relays can be added, otherwise the given sum power constraint is violated.
Optimal VNMSMRS (OVNMSMRS)
For VNMSMRS, the overall effective system SNR is still given by (9). However, L is no longer a fixed number but a variable to be chosen from a set $L\in \left\{1,\dots ,K\right\}$. The proposed optimal selection criterion, OVNMSMRS, becomes
Varied number MSMRS based on semidefinite programming optimization
For VNMSMRS, the formulation of optimization problem is the same as (18) except that (18c) is replaced by a sum power constraint
The corresponding semidefinite relaxation formulation is written as
The optimization problem (29) can be solved using the bisection procedure similar to the proposed Algorithm 1 as depicted in Appendix. The difference is that Step 4) of the bisection procedure for VNMSMRS is changed into “solve the SDP optimization problem (29).” To obtain the estimation of $\underline{\mathbf{c}}$, $\hat{\underline{\mathbf{c}}}$, the randomization procedure Algorithm 3 for VNMSMRS is proposed in Appendix.
The above proposed MSMRS based on semidefinite programming is defined as SDPVNMSMRS: the SDPVNMSMRS using randomization procedure Algorithm 3 is termed as SDPVNMSMRS B, while the SDPVNMSMRS using randomization procedure Algorithm 3 without steps 9) and 10) is called as SDPVNMSMRS A.
Numerical results
In this section, we present the performance of the sum capacity per time channel use for the relay selection approaches under considerations. In all figures, the horizontal axis indicates unit power P, and ${P}_{k}^{\left(R\right)},k=1,\dots ,K,$ and ${P}_{m}^{\left(S\right)},m=1\dots ,M$, are scaled values of P. In this section, the number of sources is set to M=2. We further assume that channels ${h}_{m}^{(k,t)}$ and ${g}_{k}^{\left(t\right)}$, m=1,…,M and k=1,…,K, are Rayleigh fading channel gains (modeled as complex Gaussian with zero mean and unit variance), and they change independently over different time channel uses.
FNMSMRS results
In Figures 1 and 2, we assume K=8, L=4, ${P}_{k}^{\left(R\right)}=\mathrm{PM},k=1,\dots ,K$, and ${P}_{m}^{\left(S\right)}=P,m=1,\dots ,M$. The settings of randomization procedure in SDPFNMSMRS A1, SDPFNMSMRS A2, SDPFNMSMRS B1, and SDPFNMSMRS B2 are N_{ c }=2 and N_{ l }=14.
In Figure 1, we observe that, to achieve the same average sum capacity per time channel use,

1.
SDPFNMSMRS A1, SDPFNMSMRS A2, SDPFNMSMRS B1, and SDPFNMSMRS B2 use less unit power P than BWFNMSMRS by 1.6 and 1.3 dB, respectively;

2.
BWFNMSMRS use less unit power P than RANDFNMSMRS by only 2.2 dB;

3.
With the advantage of lower complexity, SDPFNMSMRS A1, SDPFNMSMRS A2, SDPFNMSMRS B1, and SDPFNMSMRS B2 require more unit power P than OFNMSMRS by 2.2 and 2.5 dB, respectively.
It is observed that both SDPFNMSMRS B1 and SDPFNMSMRS B2 achieve notably higher average sum capacity over both SDPFNMSMRS A1 and SDPFNMSMRS A2 for the same unit power P. This also verifies the importance of step 10 of Algorithm 2. With very close performance to SDPFNMSMRS A1 and SDPFNMSMRS B1, respectively, SDPFNMSMRS A2 and SDPFNMSMRS B2 are quite computationally effective due to avoiding the needs of additional bisection algorithms.
VNMSMRS results
In this section, the settings of randomization procedure in SDPVNMSMRS B and SDPVNMSMRS A are N_{ c }=2 and N_{ l }=14.
In Figures 3 and 4, we assume K=8, M=2, P^{(Sum)}=4PM, ${P}_{k}^{\left(R\right)}=\mathrm{PM},k=1,\dots ,K$, ${P}_{m}^{\left(S\right)}=P,m=1,\dots ,M$. From Figure 3, we observe that, to achieve the same average sum capacity per time channel use,

1.
SDPVNMSMRS B and SDPVNMSMRS A use less unit power P than PVNMSMRS by 4.2 and 3.9 dB, respectively,

2.
With the advantage of lower complexity, SDPVNMSMRS B and SDPVNMSMRS A require more unit power P than OVNMSMRS by 1.4 and 1.7 dB, respectively.
Unlike in Figures 3 and 4, relay powers in Figures 5 and 6 are not uniformly distributed, and we assume K=8, M=2, P^{(Sum)}=3.62PM, $\left\{{P}_{k}^{\left(R\right)}=\mathrm{PM},k=1,2\right\}$, $\left\{{P}_{k}^{\left(R\right)}=0.65\mathrm{PM},k=3,4\right\}$, $\left\{{P}_{k}^{\left(R\right)}=0.4\mathrm{PM},k=5,\dots ,8\right\}$, $\left\{{P}_{m}^{\left(S\right)}=P,m=1,\dots ,M\right\}$. In Figure 5, similar conclusions can be drawn except for different gains as shown in Figure 3. For example, in Figure 5, SDPVNMSMRS B uses 3.55 dB less unit power P than PVNMSMRS. The above results verify the importance of steps 9 and 10 of Algorithm 3.
Comparison between OFNMSMRS and OVNMSMRS
In Figures 7 and 8, we compare OFNMSMRS with OVNMSMRS under the same power constraints, and we assume K=8, M=2, P^{(Sum)}=4PM, $\left\{{P}_{k}^{\left(R\right)}=\mathrm{PM},k=1,\dots ,K\right\}$, $\left\{{P}_{m}^{\left(S\right)}=P,m=1,\dots ,M\right\}$. Note that, for OFNMSMRS, P^{(Sum)}=4PM is equivalent to set L=4. It is evident that OVNMSMRS outperforms OFNMSMRS under the same power constraints. This implies that the best selection solution for some channel realizations may not necessarily always reach full sum power constraints.
Note that the complexity of optimal MSMRS significantly increases when K becomes larger. In simulations, we choose a small number of K=8, for reduced simulation time. For such low K values, the complexity advantage for the proposed approaches may not be that significant. However, with the increase of K, complexity advantage for proposed approaches in Sections ‘Fixed number multiplesource multiplerelay selection’ and ‘Varied number multiplesource multiplerelay selection’ will become more pronounced.
Conclusion
Based on the sum capacity maximization criterion, we have proposed a number of multiplerelay selection approaches for simultaneously transmitting multiple source nodes with fixed power relays in an amplifyandforward cooperative relay network. We propose computationally efficient algorithms based on semidefinite programming for MSMRS with both fixed number and varied number of relays. We have demonstrated that optimal varied number MSMRS outperforms fixed number MSMRS under the same sum power constraints. Although we have discussed the convex relaxation approaches in this article, as the future research directions, it may be deserved to investigate other nonconvexrelaxation approaches with better performance, such as in [52–54].
Endnote
^{a} In [45], the authors express sign operation using notation “σ” instead of “sign”
Appendix
Algorithms
Algorithm 1 Bisection procedure

1.
Initialize the upper and lower limits of u, u ^{(U)}and u ^{(L)};

2.
If $\left(\left{u}^{\left(U\right)}{u}^{\left(L\right)}\right<\epsilon \right)$, go to step 7), otherwise go to step 3);

3.
$u:=\frac{1}{2}\left({u}^{\left(U\right)}+{u}^{\left(L\right)}\right)$;

4.
Perform the SDP optimization procedure for problem (20);

5.
If the optimization problem (20) is infeasible or unbounded, {u ^{(U)}:=u; }
else {
u^{(L)}:=u;
$\hat{\mathbf{B}}=\mathbf{B}$;
}

6.
Go to step 2);

7.
The optimization procedure ends.
Algorithm 2 Randomization procedure for FNMSMRS

1.
Compute V such that $\hat{\mathbf{B}}={\mathbf{V}}^{\mathcal{T}}\mathbf{V}$, where $\mathbf{V}=\left[{\mathbf{v}}_{1},\dots ,{\mathbf{v}}_{K}\right]$;

2.
Set a _{ c }=0, a _{ s }=0, and ${\rho}^{(max)}=0$;

3.
If a _{ s }≠0, go to step 13), otherwise go to step 4);

4.
If a _{ c }≥N _{ c }, {
choose $\hat{\underline{\mathbf{c}}}$ using BWFNMSMRS such that L entries of $\hat{\underline{\mathbf{c}}}$ equal to 1 and the rest equal to −1, then go to step 13);
}
else { go to step 5); }

5.
Set a _{ l }=0;

6.
Choose random vector u from the uniform distribution on the unit sphere;

7.
Compute $\mathbf{c}=\text{sign}\left({\mathbf{V}}^{\mathcal{T}}\mathbf{u}\right)$, and thus obtain $\underline{\mathbf{c}}$ as in (15);

8.
Compute $\mathbf{p}=\frac{1}{2}\left(\underline{\mathbf{c}}+1\right)$;

9.
If $\left(\sum _{k=1}^{K}{\left[\mathbf{p}\right]}_{1,k}\right)==L$, {
Compute ρ based on (18b);
If $\rho >{\rho}^{(max)}$, ${\rho}^{(max)}=\rho $, $\hat{\underline{\mathbf{c}}}=\underline{\mathbf{c}}$;
a_{ s }=1;
}

10.
If $\left(K\left(\sum _{k=1}^{K}{\left[\mathbf{p}\right]}_{1,k}\right)\right)==L$, {
Compute c=−c,${\left[\mathbf{c}\right]}_{1,1}=1$, and thus obtain $\underline{\mathbf{c}}$;
Compute ρ based on (18b);
If $\rho >{\rho}^{(max)}$, ${\rho}^{(max)}=\rho $ ,$\hat{\underline{\mathbf{c}}}=\underline{\mathbf{c}}$;
a_{ s }=1;
}

11.
a _{ l }=a _{ l } + 1;

12.
If a _{ l }≥N _{ l }, {
a_{ c }=a_{ c } + 1;
go to 3);
}
else { go to 6); }

13.
The randomization procedure ends.
Algorithm 3 Randomization procedure for VNMSMRS

1.
Compute V such that $\hat{\mathbf{B}}={\mathbf{V}}^{\mathcal{T}}\mathbf{V}$, where $\mathbf{V}=\left[{\mathbf{v}}_{1},\dots ,{\mathbf{v}}_{K}\right]$, v _{ k }is the k th column vector of V;

2.
Set a _{ c }=0, a _{ s }=0, and ${\rho}^{(max)}=0$;

3.
If a _{ s }≠0, go to step 13), otherwise go to step 4);

4.
If a _{ c }≥N _{ c }, {
Choose $\hat{\underline{\mathbf{c}}}$ using optimal multiplesource singlerelay selection such that the sum power constraint (28) is satisfied, then go to step 13);
}
else { go to step 5); }

5.
Set a _{ l }=0;

6.
Choose random vector u from the uniform distribution on the unit sphere;

7.
Compute $\mathbf{c}=\text{sign}\left({\mathbf{V}}^{\mathcal{T}}\mathbf{u}\right)$, and thus obtain $\underline{\mathbf{c}}$;

8.
If the sum power constraint (28) is satisfied, {
Compute ρ based on (18b);
If $\rho >{\rho}^{(max)}$, ${\rho}^{(max)}=\rho $, $\hat{\underline{\mathbf{c}}}=\underline{\mathbf{c}}$;
a_{ s }=1;
}

9.
Compute c=−c, ${\left[\mathbf{c}\right]}_{1,1}=1$, and thus obtain $\underline{\mathbf{c}}$;

10.
If the sum power constraint (28) is satisfied, {
Using c, compute ρ based on (18b);
If $\rho >{\rho}^{(max)}$, ${\rho}^{(max)}=\rho $ , $\hat{\underline{\mathbf{c}}}=\underline{\mathbf{c}}$;
a_{ s }=1;
}

11.
a _{ l }=a _{ l } + 1;

12.
if a _{ l }≥N _{ l }, {
a_{ c }=a_{ c } + 1;
go to 3);
}
else { go to 6); }

13.
The randomization procedure ends.
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Acknowledgements
The study was performed when J. Wu was with the Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA
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Wu, J., Zhang, Y.D., Amin, M.G. et al. Multiplerelay selection in amplifyandforward cooperative wireless networks with multiple source nodes. J Wireless Com Network 2012, 256 (2012). https://doi.org/10.1186/168714992012256
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Keywords
 Channel State Information
 Relay Node
 Power Constraint
 Relay Selection
 Relay Network