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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.