  CMB-S4---the next-generation ground-based cosmic microwave background (CMB)
experiment---is set to significantly advance the sensitivity of CMB
measurements and enhance our understanding of the origin and evolution of the
Universe, from the highest energies at the dawn of time through the growth of
structure to the present day. Among the science cases pursued with CMB-S4, the
quest for detecting primordial gravitational waves is a central driver of the
experimental design. This work details the development of a forecasting
framework that includes a power-spectrum-based semi-analytic projection tool,
targeted explicitly towards optimizing constraints on the tensor-to-scalar
ratio, $r$, in the presence of Galactic foregrounds and gravitational lensing
of the CMB. This framework is unique in its direct use of information from the
achieved performance of current Stage 2--3 CMB experiments to robustly forecast
the science reach of upcoming CMB-polarization endeavors. The methodology
allows for rapid iteration over experimental configurations and offers a
flexible way to optimize the design of future experiments given a desired
scientific goal. To form a closed-loop process, we couple this semi-analytic
tool with map-based validation studies, which allow for the injection of
additional complexity and verification of our forecasts with several
independent analysis methods. We document multiple rounds of forecasts for
CMB-S4 using this process and the resulting establishment of the current
reference design of the primordial gravitational-wave component of the Stage-4
experiment, optimized to achieve our science goals of detecting primordial
gravitational waves for $r > 0.003$ at greater than $5\sigma$, or, in the
absence of a detection, of reaching an upper limit of $r < 0.001$ at $95\%$ CL.
