  Modern coronagraph design relies on advanced, large-scale optimization
processes that require an ever increasing amount of computational resources. In
this paper, we restrict ourselves to the design of Apodized Pupil Lyot
Coronagraphs (APLCs). To produce APLC designs for future giant space
telescopes, we require a fine sampling for the apodizer to resolve all small
features, such as segment gaps, in the telescope pupil. Additionally, we
require the coronagraph to operate in broadband light and be insensitive to
small misalignments of the Lyot stop. For future designs we want to include
passive suppression of low-order aberrations and finite stellar diameters. The
memory requirements for such an optimization would exceed multiple terabytes
for the problem matrix alone.
  We therefore want to reduce the number of variables and constraints to
minimize the size of the problem matrix. We show how symmetries in the pupil
and Lyot stop are expressed in the complete optimization problem, and allow
removal of both variables and constraints. Each mirror symmetry reduces the
problem size by a factor of four. Secondly, we introduce progressive
refinement, which uses low-resolution optimizations as a prior for higher
resolutions. This lets us remove the majority of variables from the
high-resolution optimization. Together these two improvements require up to
256x less computer memory, with a corresponding speed increase. This allows for
greater exploration of the phase space of the focal-plane mask and Lyot-stop
geometry, and easier simulation of sensitivity to Lyot-stop misalignments.
Moreover, apodizers can now be optimized at their native manufactured
resolution.
