Optimal Beamforming Structure and Efficient Optimization Algorithms for Generalized Multi-Group Multicast Beamforming Optimization

TL;DR

This paper analyzes the optimal structure of beamforming in multi-group multicast systems and introduces efficient algorithms that leverage low-dimensional structures, significantly reducing complexity while maintaining high performance, especially for large-scale MIMO systems.

Contribution

It uncovers the optimal and low-dimensional beamforming structures using KKT conditions and SIT duality, and develops ultra-low-complexity algorithms for large MIMO systems based on these structures.

Findings

Proposed algorithms achieve near-optimal weighted sum rate performance.

Algorithms dramatically reduce computational complexity independent of the number of antennas.

Low-dimensional beamforming structures are asymptotically optimal in various SNR regimes.

Abstract

In this work, we focus on solving non-smooth non-convex maximization problems in multi-group multicast transmission. Leveraging Karush-Kuhn-Tucker (KKT) optimality conditions and successive incumbent transcending (SIT) duality, we thoroughly analyze the optimal beamforming structure for a set of optimization problems characterized by a general utility-based objective function. By exploiting the identified optimal structure, we further unveil inherent low-dimensional beamforming structures within the problems, which are asymptotically optimal in various regimes of transmit signal-to-noise ratios (SNRs) or the number of transmit antennas. Building upon the discovered optimal and low-dimensional beamforming structures, we then propose highly efficient and toolbox-free optimization algorithms to solve a specific multi-group multicast optimization problem based on the weighted sum rate (WSR) utility function. The proposed algorithms first use the cyclic maximization (CM) framework to decompose the problem into multiple subproblems, each has an optimal or low-dimensional closed-form beamforming solution structure. Then, we propose the projected adaptive gradient descent (PAGD) algorithm to compute the optimal Lagrangian dual variables for each subproblem. Numerical results show that the proposed algorithms maintain comparable or improved WSR performance compared to baseline algorithms, while dramatically reducing the computational complexity. Notably, the proposed ultra-low-complexity algorithms based on low-dimensional beamforming structures achieve near optimal WSR performance with extremely low computational complexity. This complexity remains independent of the number of transmit antennas, making them promising and practical for extremely large multiple-input multiple-output (XL-MIMO) applications in 6G.