  A complete formalism for constructing initial data representing black-hole
binaries in quasi-equilibrium is developed. Radiation reaction prohibits, in
general, true equilibrium binary configurations. However, when the timescale
for orbital decay is much longer than the orbital period, a binary can be
considered to be in quasi-equilibrium. If each black hole is assumed to be in
quasi-equilibrium, then a complete set of boundary conditions for all initial
data variables can be developed. These boundary conditions are applied on the
apparent horizon of each black hole, and in fact force a specified surface to
be an apparent horizon. A global assumption of quasi-equilibrium is also used
to fix some of the freely specifiable pieces of the initial data and to
uniquely fix the asymptotic boundary conditions. This formalism should allow
for the construction of completely general quasi-equilibrium black hole binary
initial data.
