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The inclusion of nucleonic exchange energy has been a long-standing challenge for the relativistic density functional theory (RDFT) in nuclear physics. We propose an orbital-dependent relativistic Kohn-Sham density functional theory to incorporate the exchange energy with local Lorentz scalar and vector potentials. The relativistic optimized effective potential equations for the local exchange potentials are derived and solved efficiently. The obtained binding energies and charge radii for nuclei are benchmarked with the results given by the traditional relativistic Hartree-Fock approach, which involves complicated nonlocal potentials. It demonstrates that the present framework is not only accurate but also efficient.