User grouping and admission control for multi-group multicast beamforming in MIMO systems

Abstract

In this paper, we consider the multi-group multicasting problem in a MIMO wireless cellular system, whereby the base station can generate multiple beampatterns to serve multiple groups simultaneously. Each user is interested in multiple multicast groups, but is only allowed to be assigned to one of them eventually during each transmission interval. We formulate it as a joint optimization problem of multicast beamformer design as well as user grouping, and admission control is also considered to account for the quality of service demands of individual users. To deal with this NP-hard problem, we adopt the big-M formulation method and the semidefinite relaxation scheme to reformulate it to a mixed-integer semidefinite programming problem. Three algorithms are developed to solve the obtained approximate problem with different computational costs. Simulation results are presented to demonstrate the performance of the proposed algorithms in terms of the number of admitted users, power consumption and computation time.

Appendix

1.1 Proof of Lemma 1

In this section, we prove that under the condition $0<\rho <\frac{1}{P_{\text{max}}}$, the single-stage problem in (7) is equivalent to the two-stage problems in (5) and (6) in terms of achieving the maximum subset of admitted users. In addition, it can automatically obtain the maximum subset with the minimum transmit power.

Assume the optimal solution of problem (7) is $\{\mathcal{U}_l^\star ,\mathbf {w}_l^\star \}_{l \in \mathcal{L}}$ and define the maximum subset of admitted users is U. Then the optimal objective of problem (5) is U if only if the optimal objective of problem (7) is lies in $(U-1,U)$. In addition, if the maximum admissible set of the system is $\{\mathcal{U}_l^\star \}_{l \in \mathcal{L}}$, the minimum transmit power for the admitted users is $P_{\text{cost}}\triangleq \sum _{l \in \mathcal{L}} \Vert \mathbf {w}_l^*\Vert _2^2$.

Firstly, we demonstrate that when the optimal objective of problem (7) is lies in $(U-1,U)$, then the optimal objective of problem (5) is U. According to the total power constraint (7b) and the parameter constraint (9), we have:

$$\begin{aligned} \rho \sum \limits _{l \in \mathcal{L}} \Vert \mathbf {w}_l\Vert _2^2 <1, \end{aligned}$$

which means that the second term of the objective function in (7) cannot exceed 1, in spite of the beamforming vectors. Therefore, we can see that if the optimal objective of the problem (7) is lies in $(U-1,U)$, then the optimal objective of problem (5) is U.

Then, we show that if the optimal objective of problem (5) is U, the one-stage problem (7) is capable of achieving the maximum admissible set, i.e., the optimal objective of problem (7) is lies in $(U-1,U)$. The proof is executed by contradiction. Suppose $\{\widehat{\mathcal{U}}_l,\widehat{\mathbf {w}}_l\}_{l \in \mathcal{L}}$ is a feasible solution of problem (7), where $\sum _{l \in \mathcal{L}}|\widehat{\mathcal{U}}_l|=U$. Assume the optimal solution of problem (7) is $\sum _{l \in \mathcal{L}} |\mathcal{U}_l^*|<U$, we have:

$$\begin{aligned} \quad \sum \limits _{l \in \mathcal{L}} |\mathcal{U}_l^*| -\rho \sum \limits _{l \in \mathcal{L}} \Vert \mathbf {w}_l^*\Vert _2^2< U-1 < \sum \limits _{l \in \mathcal{L}} |\widehat{\mathcal{U}}_l| -\rho \sum \limits _{l \in \mathcal{L}} \Vert \widehat{\mathbf {w}}_l\Vert _2^2, \end{aligned}$$

wherein the first inequality is obtained by the fact that the number of admitted users is discontinuous with an increment of 1. It can be seen that this result is contradict to the assumption that $\{\mathcal{U}_l^*,\mathbf {w}_l^*\}_{l \in \mathcal{L}}$ is the global optimum of problem (7). Therefore, when the optimal objective of problem (5) is U, the solution of problem (7) can be achieved with the suitable beamforming vectors such that U users are served, that is, the objective of problem (7) lies between $U-1$ and U.

Finally, we show the fact that if the maximum subset of admitted users has multiple solutions, the achieved solution of problem (7) requires the minimum transmit power. The reason is that when the first term of objective function in (7) is a constant, the second term possess the effect of selecting the one of admitted set from the candidate multiple maximum admission sets with the minimum transmit power. Therefore, the proof of Lemma 1 is completed.

1.2 Proof of Lemma 2

Equation (13a) contains multiple-choice constraint corresponding to the binary variables $\eta _l^k,\forall l,k$, so we proof this lemma from the following conditions.

Firstly, if $\eta _l^k=1$, the constraint (13a) is equivalent to

$$\begin{aligned} |\mathbf {h}_k^H \mathbf {w}_l|^2 \ge \gamma _k \sum \limits _{m \ne l, m \in \mathcal{L}} | \mathbf {h}_k^H \mathbf {w}_m|^2 + \gamma _k \sigma _k^2, \end{aligned}$$

which is the same as Eq. (10a) in this case.

Secondly, if $\eta _l^k=0$, the constraint (13a) is given as

$$\begin{aligned} \gamma _k \sum \limits _{m \ne l, m \in \mathcal{L}} |\mathbf {h}_k^H \mathbf {w}_m|^2 + \gamma _k \sigma _k^2 - |\mathbf {h}_k^H \mathbf {w}_l|^2 \le M_k. \end{aligned}$$

(20)

The left-hand side of (20) can be rewritten as:

$$\begin{aligned} \begin{aligned}&~ \gamma _k \sum \limits _{m \ne l, m \in \mathcal{L}} |\mathbf {h}_k^H \mathbf {w}_m|^2 + \gamma _k \sigma _k^2 - |\mathbf {h}_k^H \mathbf {w}_l|^2 \\&\quad \le \gamma _k \sum \limits _{l \in \mathcal{L}} |\mathbf {h}_k^H \mathbf {w}_l|^2 + \gamma _k \sigma _k^2 \\&\quad \le \gamma _k P |\mathbf {h}_k|^2+\gamma _k \sigma _k^2\\&\quad = M_k, \end{aligned} \end{aligned}$$

where the second inequality is obtained according to the Cauchy–Schwarz inequality. Therefore, Eq. (13a) is satisfied automatically when we set the values of $M_k$s according to Lemma 2 in this case.

Based on the above analysis, the constraint in (13a) is equivalent to the constraint in (10a) if only if $M_k$ satisfies the condition (14).