ADMM-Based Fast Algorithm for Multi-Group Multicast Beamforming in Large-Scale Wireless Systems

Abstract

Multi-group multicast beamforming in wireless systems with large antenna arrays and massive audience is investigated in this paper. Multicast beamforming design is a well-known non-convex quadratically constrained quadratic programming (QCQP) problem. A conventional method to tackle this problem is to approximate it as a semi-definite programming problem via semi-definite relaxation, whose performance, however, deteriorates considerably as the number of per-group users goes large. A recent attempt is to apply convex-concave procedure (CCP) to find a stationary solution by treating it as a difference of convex programming problem, whose complexity, however, increases dramatically as the problem size increases. In this paper, we propose a low-complexity highperformance algorithm for multi-group multicast beamforming design in large-scale wireless systems by leveraging the alternating direction method of multipliers (ADMM) together with CCP. In specific, the original non-convex QCQP problem is first approximated as a sequence of convex subproblems via CCP. Each convex subproblem is then reformulated as a novel ADMM form. Our ADMM reformulation enables that each updating step is performed by solving multiple small-size subproblems with closed-form solutions in parallel. Numerical results show that our fast algorithm maintains the same favorable performance as state-of-the-art algorithms but reduces the complexity by orders of magnitude. Index TermsPhysical layer multicasting, large-scale optimization, non-convex quadratically constrained quadratic programming (QCQP), convex-concave procedure (CCP), alternating direction method of multipliers (ADMM). I. INTRODUCTIONMulticasting is a promising approach to deliver a common message to multiple receivers by exploiting the broadcast nature of wireless medium. It has great potential in many applications such as live video streaming, venue casting, mobile application updates, advertisements, and public group communications [2]- [4]. It can also be used in heterogeneous networks (HetNets) with wireless backhaul to push common information from a macro base station (BS) to multiple small BSs [5]. Recently, multicasting is shown to be useful for content delivery even when user demands are different in wireless cache networks [6], [7].Physical layer multicasting via beamforming further boosts the energy and spectrum efficiencies by exploiting channel state information at the transmitter. Multicast beamforming is first considered in [8] for a single group of users. The similar problem for multiple co-channel groups is later studied in [9]. In [10], coordinated multicast beamforming among BSs in multi-cell networks is investigated. The design of multicast beamforming in cellular networks with massive multiple-input multiple-output (MIMO) is studied in [11]. In general, the design of multicast beamforming is a non-convex quadratically constrained quadratic programming (QCQP) problem and its global optimal solution is difficult to obtain. A prevailing method to tackle this p...

Over-the-Air Computation via Intelligent Reflecting Surfaces

“…On the other hand, if the optimal solution of the SDP problem fails to be rank-one, additional steps such as Gaussian randomization [16] need to be applied to extract a suboptimal solution for the original problem. However, it was observed that for the high-dimensional optimization problems (e.g., the number of antennas N increases), the probability of returning a rank-one solution for the SDR approach becomes low, which yields significant performance deterioration [6], [17]. To overcome the limitations of the SDR methods, we instead propose a novel DC framework in the following section to solve problem P 1 and problem P 2 .…”

“…A popular way to convexify the nonconvex QCQP problem is to reformulate it as a rank-one constrained matrix optimization problem via matrix lifting, followed by the semidefinite relaxation (SDR) technique to drop the nonconvex rank-one constraint [16]. However, it was observed that the performance of SDR approach degenerates in the scenarios with large number of antennas due to its low probability of returning rank-one solutions [2], [6], [17]. To address the limitations of the popular SDR technique, in this paper, we develop a general framework to solve the rank-one constrained matrix optimization problem via difference-of-convex (DC) programming.…”

Fast Algorithms for Joint Multicast Beamforming and Antenna Selection in Massive MIMO

“…In [15], the authors used a successive convex approximation (SCA) approach [16]- [18] to tackle the max-min fair multi-group multicasting problem under PAPC, where each SCA sub-problem was formulated as a second order cone program. A subsequent work [19] used an alter-native reformulation which entails solving a sequence of perantenna minimization problems within a bisection framework. Then, an ADMM algorithm is proposed for (approximately) solving these per-antenna minimization problems.…”

“…A high performance, low complexity successive convex approximation (SCA) based approach is proposed for solving the joint problem. Although SCA has not been considered before for joint multicast beamforming and antenna selection, our SCA framework compared to that of [15], [19]-which studied the max-min fair multicasting without antenna selection-features the following; (i) it approximates the objective function of the problem directly without introducing any additional variables or constraints, (ii) exploits the structure inherent in each SCA subproblem by using specially tailored low-complexity firstorder methods, and (iii) guarantees convergence to a stationary point.…”

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“…In [26], a primal decomposition-based algorithm and an alternating direction method of multipliers (ADMM)-based algorithm have been proposed for the SDR version of the original problem. In [27], instead of directly dealing with the relaxed problem, the authors proposed to apply ADMM for each of the convexified SCA problems, obtaining a doule-loop scheme. In [24], an ADMM-based algorithm is proposed for a distributed solution of the problem with imperfect CSI after relaxing the original problem with S-procedure.…”