Model predictive control (MPC) is a promising approach for the lateral and longitudinal control of autonomous vehicles. However, the parameterization of the MPC with respect to high-level requirements such as passenger comfort as well as lateral and longitudinal tracking is a challenging task. Numerous tuning parameters as well as conflicting requirements need to be considered. This contribution formulates the MPC tuning task as a multi-objective optimization problem. Solving it is challenging for two reasons: First, MPC-parameterizations are evaluated on an computationally expensive simulation environment. As a result, the used optimization algorithm needs to be as sampleefficient as possible. Second, for some poor parameterizations the simulation cannot be completed and therefore useful objective function values are not available (learning with crash constraints). In this contribution, we compare the sample efficiency of multi-objective particle swarm optimization (MOPSO), a genetic algorithm (NSGA-II) and multiple versions of Bayesian optimization (BO). We extend BO, by introducing an adaptive batch size to limit the computational overhead and by a method on how to deal with crash constraints. Results show, that BO works best for a small budget, NSGA-II is best for medium budgets and for large budgets none of the evaluated optimizers is superior to random search. Both proposed BO extensions are shown to be beneficial.

Based on Welzl's algorithm for smallest circles and spheres we develop a simple linear time algorithm for finding the smallest circle enclosing a point cloud on a sphere. The algorithm yields correct results as long as the point cloud is contained in a hemisphere, but the hemisphere does not have to be known in advance and the algorithm automatically detects whether the hemisphere assumption is met. For the full-sphere case, that is, if the point cloud is not contained in a hemisphere, we provide hints on how to adapt existing linearithmic time algorithms for spherical Voronoi diagrams to find the smallest enclosing circle.