  By enabling constraint-aware online model adaptation, model predictive
control using Gaussian process (GP) regression has exhibited impressive
performance in real-world applications and received considerable attention in
the learning-based control community. Yet, solving the resulting optimal
control problem in real-time generally remains a major challenge, due to i) the
increased number of augmented states in the optimization problem, as well as
ii) computationally expensive evaluations of the posterior mean and covariance
and their respective derivatives. To tackle these challenges, we employ i) a
tailored Jacobian approximation in a sequential quadratic programming (SQP)
approach, and combine it with ii) a parallelizable GP inference and automatic
differentiation framework. Reducing the numerical complexity with respect to
the state dimension $n_x$ for each SQP iteration from $\mathcal{O}(n_x^6)$ to
$\mathcal{O}(n_x^3)$, and accelerating GP evaluations on a graphical processing
unit, the proposed algorithm computes suboptimal, yet feasible solutions at
drastically reduced computation times and exhibits favorable local convergence
properties. Numerical experiments verify the scaling properties and investigate
the runtime distribution across different parts of the algorithm.
