  Robust controllers that stabilize dynamical systems even under disturbances
and noise are often formulated as solutions of nonsmooth, nonconvex
optimization problems. While methods such as gradient sampling can handle the
nonconvexity and nonsmoothness, the costs of evaluating the objective function
may be substantial, making robust control challenging for dynamical systems
with high-dimensional state spaces. In this work, we introduce multi-fidelity
variants of gradient sampling that leverage low-cost, low-fidelity models with
low-dimensional state spaces for speeding up the optimization process while
nonetheless providing convergence guarantees for a high-fidelity model of the
system of interest, which is primarily accessed in the last phase of the
optimization process. Our first multi-fidelity method initiates gradient
sampling on higher fidelity models with starting points obtained from cheaper,
lower fidelity models. Our second multi-fidelity method relies on ensembles of
gradients that are computed from low- and high-fidelity models. Numerical
experiments with controlling the cooling of a steel rail profile and laminar
flow in a cylinder wake demonstrate that our new multi-fidelity gradient
sampling methods achieve up to two orders of magnitude speedup compared to the
single-fidelity gradient sampling method that relies on the high-fidelity model
alone.
