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

In this work, we focus on solving non-smooth non-convex maximization problems in multi-group multicast transmission. By leveraging Karush-Kuhn-Tucker (KKT) optimality conditions, 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 optimization algorithms to solve a specific multi-group multicast optimization problem based on the weighted power mean (WPM) utility function. The proposed algorithms first use the successive convex approximation (SCA) framework to decompose the problem into a sequence of convex subproblems, each with an optimal closed-form beamforming solution structure. Then, we propose a hyperplane fixed point iteration (HFPI) algorithm to compute the optimal Lagrangian dual variables for each subproblem. Numerical results show that the proposed algorithms maintain comparable or improved utility 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 utility 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.