The finite-basis-set optimized effective potential (OEP) method is presented from an integral equation point of view. It is shown that the projection method for solving the OEP integral equation provides a consistent and convenient approach for including orbital-dependent functionals and potentials in the finite-basis-set implementations of the Kohn-Sham theory. Different finite-basis-set realizations of the OEP method are introduced and tested within the exchange-only approximation. An exact condition involving the local multiplicative exchange potential and the nonlocal Hartree-Fock exchange potential built from Kohn-Sham orbitals is incorporated in our schemes. Numerical results are presented.