  We present the first single-stage optimization of islands in finite-$\beta$
stellarator equilibria. Stellarator optimization is traditionally performed as
a two-stage process; in the first stage, an optimal equilibrium is calculated
which balances a set of competing constraints, and in the second stage a set of
coils is found that supports said equilibrium. Stage one is generally performed
using a representation for the equilibrium that assumes nestedness of flux
surfaces, even though this is not warranted and occasionally undesired. The
second stage optimization of coils is never perfect, and the mismatch leads to
worse performing equilibria, and further deteriorates if additional constraints
such as force minimization, coil torsion or port access are included. The
higher fidelity of single-stage optimization is especially important for the
optimization of islands as these are incredibly sensitive to changes in the
field. In this paper we demonstrate an optimization scheme capable of
optimizing islands in finite $\beta$ stellarator equilibria directly from
coils. We furthermore develop and demonstrate a method to reduce the
dimensionality of the single-stage optimization problem to that of the first
stage in the two-stage approach.
