  We present a general numerical approach to construct local Kohn-Sham
potentials from orbital-dependent functionals within the all-electron
full-potential linearized augmented-plane-wave (FLAPW) method, in which core
and valence electrons are treated on an equal footing. As a practical example,
we present a treatment of the orbital-dependent exact-exchange (EXX) energy and
potential. A formulation in terms of a mixed product basis, which is
constructed from products of LAPW basis functions, enables a solution of the
optimized-effective-potential (OEP) equation with standard numerical algebraic
tools and without shape approximations for the resulting potential. We find
that the mixed product and LAPW basis sets must be properly balanced to obtain
smooth and converged EXX potentials without spurious oscillations. The
construction and convergence of the exchange potential is analyzed in detail
for diamond. Our all-electron results for C, Si, SiC, Ge, GaAs semiconductors
as well as Ne and Ar noble-gas solids are in very favorable agreement with
plane-wave pseudopotential calculations. This confirms the adequacy of the
pseudopotential approximation in the context of the EXX-OEP formalism and
clarifies a previous contradiction between FLAPW and pseudopotential results.
