  Many stellarator coil design problems are plagued by multiple minima, where
the locally optimal coil sets can sometimes vary substantially in performance.
As a result, solving a coil design problem a single time with a local
optimization algorithm is usually insufficient and better optima likely do
exist. To address this problem, we propose a global optimization algorithm for
the design of stellarator coils and outline how to apply box constraints to the
physical positions of the coils. The algorithm has a global exploration phase
that searches for interesting regions of design space and is followed by three
local optimization algorithms that search in these interesting regions (a
``global-to-local" approach). The first local algorithm (phase I), following
the globalization phase, is based on near-axis expansions and finds stellarator
coils that optimize for quasisymmetry in the neighborhood of a magnetic axis.
The second local algorithm (phase II) takes these coil sets and optimizes them
for nested flux surfaces and quasisymmetry on a toroidal volume. The final
local algorithm (phase III) polishes these configurations for an accurate
approximation of quasisymmetry. Using our global algorithm, we study the
trade-off between coil length, aspect ratio, rotational transform, and quality
of quasi-axisymmetry. The database of stellarators, which comprises
approximately 200,000 coil sets, is available online and is called QUASR, for
`QUAsi-symmetric Stellarator Repository'.
