The requirements for a coronagraph instrument to image and obtain spectra of rocky planets around bright stars from space are tight. Indeed, the goal of imaging an Earth-like planet requires a starlight suppression system that cancels light to a level of ^{10  −  10} with sufficient stability and robustness to errors. Furthermore, the key science questions necessitate an adequate sample size; consequently, the throughput of the coronagraph drives the achievable yield of a given mission. The trade among achievable raw contrast, sensitivity to wavefront errors, and throughput poses a challenging problem in coronagraph design. The complexity of this problem drives us toward the simultaneous solving of all optical elements. We present a set of methods to optimize the design of a coronagraph. We implement these for the case of the hybrid Lyot coronagraph in the context of the Nancy Grace Roman Space Telescope Coronagraph Instrument. We discuss our findings in terms of coronagraph instrument design, and optical subsystems, and performance interplay.