The covariancematrix adaptation evolution strategy (CMA-ES) is one of themost powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.

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In this paper we study the impact of unmanned aerial vehicles (UAVs) trajectories on terrestrial users' spectral efficiency (SE). Assuming a strong line of sight path to the users, the distance from all users to all UAVs influence the outcome of an online trajectory optimization. The trajectory should be designed in a way that the fairness rate is maximized over time. That means, the UAVs travel in the directions that maximize the minimum of the users' SE. From the free-space path-loss channel model, a data-rate gradient is calculated and used to direct the UAVs in a long-term perspective towards the local optimal solution on the two-dimensional spatial grid. Therefore, a control system implementation is designed. Thereby, the UAVs follow the data-rate gradient direction while having a more smooth trajectory compared with a gradient method. The system can react to changes of the user locations online; this system design captures the interaction between multiple UAV trajectories by joint processing at the central unit, e.g., a ground base station. Because of the wide spread of user locations, the UAVs end up in optimal locations widely apart from each other. Besides, the SE expectancy is enhancing continuously while moving along this trajectory.

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