Ultra-Low-Complexity Algorithms with Structurally Optimal Multi-Group Multicast Beamforming in Large-Scale Systems

TL;DR

This paper introduces ultra-low-complexity algorithms for multi-group multicast beamforming in large-scale systems, leveraging optimal structure and first-order methods to achieve near-optimal performance with reduced computational effort.

Contribution

It develops two fast first-order algorithms based on SCA, dual reformulation, and ADMM for efficient multi-group multicast beamforming, including a simple scaling scheme for max-min fairness.

Findings

Algorithms achieve near-optimal performance.

Significantly lower computational complexity.

Effective initialization methods for fast convergence.

Abstract

In this work, we propose ultra-low-complexity design solutions for multi-group multicast beamforming in large-scale systems. For the quality-of-service (QoS) problem, by utilizing the optimal multicast beamforming structure obtained recently in [2], we convert the original problem into a non-convex weight optimization problem of a lower dimension and propose two fast first-order algorithms to solve it. Both algorithms are based on successive convex approximation (SCA) and provide fast iterative updates to solve each SCA subproblem. The first algorithm uses a saddle point reformulation in the dual domain and applies the extragradient method with an adaptive step-size procedure to find the saddle point with simple closed-form updates. The second algorithm adopts the alternating direction method of multipliers (ADMM) method by converting each SCA subproblem into a favorable ADMM structure. The structure leads to simple closed-form ADMM updates, where the problem in each update block can be further decomposed into parallel subproblems of small sizes, for which closed-form solutions are obtained. We also propose efficient initialization methods to obtain favorable initial points that facilitate fast convergence. Furthermore, taking advantage of the proposed fast algorithms, for the max-min fair (MMF) problem, we propose a simple closed-form scaling scheme that directly uses the solution obtained from the QoS problem, avoiding the conventional computationally expensive method that iteratively solves the inverse QoS problem. We further develop lower and upper bounds on the performance of this scaling scheme. Simulation results show that the proposed algorithms offer near-optimal performance with substantially lower computational complexity than the state-of-the-art algorithms for large-scale systems.