Building on this notion, for configs. IIIa and IIIb, we recast the Lyot coronagraph optimization as a nonlinear program in which both the apodizer and the Lyot stop transmission profile are free vectors, optimized simultaneously. See Appendix A1 for further description of this procedure. The program seeks to maximize the sum of the transmission of both apodizer and Lyot stop, given the same contrast and 10% bandwidth goal as before. The results are illustrated in Figs. 67 and 8 and listed in Table 1. As in the case of config. IIa (not plotted), the mismatch between the apodizer and the Babinet subtrahend profiles leads to high amplitude, sharp residual features in the Lyot plane. This time, however, a subtle rearrangement of Lyot stop obstructions is enough to enable an apodizer with far more open area. Most of the sharp residual Lyot plane features are not obstructed by the freely varying stop, contrary to what one might expect. Apparently, not even a modest level of field cancellation in the Lyot plane is required to create deep, broadband destructive interference in the image plane. The FWHM throughput of this solution is 0.33, more than triple that of the comparable clear Lyot stop configuration (IIa). The coronagraph PSF also sharpens, giving an FWHM area only 25% larger than the Airy disk. We also tested the effect of decreasing the focal occulting spot radius from to and arrived at a similar design with a throughput of 17%. The contrast curve of this design is plotted in Fig. 7, showing the intensity pattern at three wavelengths, as well as the average over five wavelength samples spanning the 10% passband.