  Nuclear star clusters that surround supermassive black holes (SMBHs) in
galactic nuclei are among the densest systems in the Universe, harbouring
millions of stars and compact objects (COs). Within a few parsecs from the
SMBH, stars can form binaries. In this paper, we model the supernova (SN)
process of massive binaries that are born in proximity of the SMBH and that
produce CO binaries. These binaries can later merge via emission of
gravitational waves as a consequence of the Lidov-Kozai mechanism. We study the
dynamical evolution of these systems by means of high-precision $N$-body
simulations, including post-Newtonian (PN) terms up to 2.5PN order. We adopt
different prescriptions for the natal velocity kicks imparted during the SN
processes and find that larger kicks lead to more compact binaries that merge
closer to the SMBH. We also conclude that most of the mergers enter the LIGO
band with very high eccentricities. Finally, we compute a merger rate of
$0.05$--$0.07\ \mathrm{Gpc}^{-3} \mathrm{yr}^{-1}$, $0.04$--$2\times 10^{-3}\
\mathrm{Gpc}^{-3} \mathrm{yr}^{-1}$, $9.6\times 10^{-6}$--$2.7\times 10^{-3}\
\mathrm{Gpc}^{-3} \mathrm{yr}^{-1}$ for BH-BH, BH-NS, NS-NS, respectively,
smaller than the actual LIGO-Virgo observed rate.
