The Apodized Pupil Lyot Coronagraph (APLC) is a di_raction suppression system adopted in the baseline of several new and recent high-contrast imaging instruments (Palomar P1640, Gemini Planet Imager, VLT/SPHERE) to enable direct imaging of exoplanets at small angular separations (> 0.2 arcsec) from their host star. This coronagraph combines an entrance pupil apodizer, a hard-edge focal plane mask (FPM) and a Lyot stop in a relayed pupil plane to form the coronagraphic image of an observed star onto a camera located in the image plane. The APLC designs underlying these instruments take advantage of the eigen-properties between entrance pupil and Lyot stop, and rely on prolate apodization to reach a contrast performance of 107 over a 20% spectral bandwidth and with a moderate inner working angle (IWA, ~ 5 λ/=D) in the presence of central obstruction and support structures. In this communication we propose novel designs relying on the linearity between the coronagraphic electric field at the science camera and the apodization function. We use this relationship to devise a numerical optimization scheme that extends the APLC performance (contrast, IWA, apodizer through- put, chromaticity) beyond the intrinsic properties of prolate apodizations. We explore the parameter space by considering different aperture geometries, contrast levels, dark-zone size, spectral bandwidth and FPM size. We present the application of these new solutions to the case of the High-Contrast Imager for Complex Aperture Telescopes (HiCAT).