Spectral-energy efficiency tradeoff of massive MIMO by a constrained large-scale multi-objective algorithm through decision transfer

To better balance the spectral efficiency (SE) and energy efficiency (EE) in the massive multiple-input multiple output system with a large number of users (MaMIMO-LU), the SE-EE tradeoff is originally constructed as a constrained large-scale multi-objective problem (CLSMOP) for the power allocation of users. To solve this CLSMOP, a constrained large-scale multi-objective evolutionary algorithm (CLSMOEA), considering the dimensionality reduction as well as the balance of objectives and constraints, is explored. The Lagrange multiplier is first used to construct a two-scale optimization model, bridging original large-scale decision space of variables and small-scale decision space of coefficients of Lagrange multiplier. The decision transfer algorithm is then designed to switch between large-scale original decision space and small-scale parametric decision space, while achieving the maximum dimensionality reduction. Finally, the two-scale evolution strategy is proposed for the alternative optimizations in the two decision spaces emphasizing objectives and constraints, respectively. In summary, the optimization in large-scale space pushes the population to unconstrained Pareto front (PF), the optimization in small-scale space helps the population cross the infeasible areas to approach constrained PF, and the GD-based reproduction of offspring further guarantees the solution convergence. Ten representative and state-of-the-art constrained multi-objective evolutionary algorithms (MOEAs) and unconstrained MOEA have been compared to the proposed CLSMOEA to demonstrate its effectiveness through comparative experiments on some well-known benchmark problems (with 1000 variables), and MaMIMO-LU (with 1024 antennas and 256, 512, and 1024 users). Experimental results show that the proposed CLSMOEA can obtain the best SE-EE tradeoff.