  We study the correlation between the [O III]$\lambda 5007$ and X-ray
luminosities of local Active Galactic Nuclei (AGNs), using a complete, hard
X-ray ($>10$ keV) selected sample in the Swift/BAT 9-month catalog. From our
optical spectroscopic observations at the South African Astronomical
Observatory and the literature, a catalog of [O III]$\lambda 5007$ line flux
for all 103 AGNs at Galactic latitudes of $|b|>15^\circ$ is complied.
Significant correlations with intrinsic X-ray luminosity ($L_{\rm X}$) are
found both for observed ($L_{\rm [O~III]}$) and extinction-corrected ($L_{\rm
[O~III]}^{\rm cor}$) luminosities, separately for X-ray unabsorbed and absorbed
AGNs. We obtain the regression form of $L_{\rm [O~III]}$ $\propto L_{\rm 2-10\;
keV}^{1.18\pm0.07}$ and $L_{\rm [O~III]}^{\rm cor}$ $\propto L_{\rm 2-10\;
keV}^{1.16\pm0.09}$ from the whole sample. The absorbed AGNs with low
($<$0.5\%) scattering fractions in soft X-rays show on average smaller $L_{\rm
[O~III]}/L_{\rm X}$ and $L_{\rm [O~III]}^{\rm cor}/L_{\rm X}$ ratios than the
other absorbed AGNs, while those in edge-on host galaxies do not. These results
suggest that a significant fraction of this population are buried in tori with
small opening angles. By using these $L_{\rm [O~III]}$ vs. $L_{\rm X}$
correlations, the X-ray luminosity function of local AGNs (including Compton
thick AGNs) in a standard population synthesis model gives much better
agreement with the [O III]$\lambda 5007$ luminosity function derived from the
Sloan Digital Sky Survey than previously reported. This confirms that hard
X-ray observations are a very powerful tool to find AGNs with high
completeness.
