  Combined plasma-coil optimization approaches for designing stellarators are
discussed and a new method for calculating free-boundary equilibria is
proposed. Four distinct categories of stellarator optimization, two of which
are novel approaches, are the fixed-boundary optimization, the generalized
fixed-boundary optimization, the quasi free-boundary optimization, and the
free-boundary (coil) optimization. These are described using the multi-region
relaxed magnetohydrodynmics (MRxMHD) energy functional, the Biot-Savart
integral, the coil-penalty functional and the virtual casing integral, and
their derivatives. The proposed free-boundary equilibrium calculation differs
from existing methods in how the boundary-value problem is posed, and for the
new approach it seems that there is not an associated energy minimization
principle because a non-symmetric functional arises. We propose to solve the
weak formulation of this problem using a spectral-Galerkin method, and this
will reduce the free-boundary equilibrium calculation to something comparable
to a fixed-boundary calculation. In our discussion of combined plasma-coil
optimization algorithms, we emphasize the importance of the stability matrix.
