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In this work, we focus on solving non-smooth non-convex maximization problemsin multi-group multicast transmission. Leveraging Karush-Kuhn-Tucker (KKT)optimality conditions and successive incumbent transcending (SIT) duality, wethoroughly analyze the optimal beamforming structure for a set of optimizationproblems characterized by a general utility-based objective function. Byexploiting the identified optimal structure, we further unveil inherentlow-dimensional beamforming structures within the problems, which areasymptotically optimal in various regimes of transmit signal-to-noise ratios(SNRs) or the number of transmit antennas. Building upon the discovered optimaland low-dimensional beamforming structures, we then propose highly efficientand toolbox-free optimization algorithms to solve a specific multi-groupmulticast optimization problem based on the weighted sum rate (WSR) utilityfunction. The proposed algorithms first use the cyclic maximization (CM)framework to decompose the problem into multiple subproblems, each has anoptimal or low-dimensional closed-form beamforming solution structure. Then, wepropose the projected adaptive gradient descent (PAGD) algorithm to compute theoptimal Lagrangian dual variables for each subproblem. Numerical results showthat the proposed algorithms maintain comparable or improved WSR performancecompared to baseline algorithms, while dramatically reducing the computationalcomplexity. Notably, the proposed ultra-low-complexity algorithms based onlow-dimensional beamforming structures achieve near optimal WSR performancewith extremely low computational complexity. This complexity remainsindependent of the number of transmit antennas, making them promising andpractical for extremely large multiple-input multiple-output (XL-MIMO)applications in 6G.

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