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We study the optimization of the Apodized Pupil Lyot Coronagraph (APLC) inthe context of exoplanet imaging with ground-based telescopes. The APLCcombines an apodization in the pupil plane with a small Lyot mask in the focalplane of the instrument. It has been intensively studied in the literature froma theoretical point of view, and prototypes are currently being manufacturedfor several projects. This analysis is focused on the case of Extremely LargeTelescopes, but is also relevant for other telescope designs.  We define a criterion to optimize the APLC with respect to telescopecharacteristics like central obscuration, pupil shape, low order segmentaberrations and reflectivity as function of the APLC apodizer function and maskdiameter. Specifically, the method was applied to two possible designs of thefuture European-Extremely Large Telescope (E-ELT).  Optimum configurations of the APLC were derived for different telescopecharacteristics. We show that the optimum configuration is a stronger functionof central obscuration size than of other telescope parameters. We also showthat APLC performance is quite insensitive to the central obscuration ratiowhen the APLC is operated in its optimum configuration, and demonstrate thatAPLC optimization based on throughput alone is not appropriate.

Optimization of Apodized Pupil Lyot Coronagraph for ELTs