  We discuss the sign problem in the Polyakov loop extended Nambu--Jona-Lasinio
model with repulsive vector-type interaction by using the path optimization
method. In this model, both of the Polyakov loop and the vector-type
interaction cause the model sign problem, and several prescriptions have been
utilized even in the mean field treatment. In the path optimization method,
integration variables are complexified and the integration path (manifold) is
optimized to evade the sign problem, or equivalently to enhance the average
phase factor. Within the homogeneous field ansatz, the path is optimized by
using the feedforward neural network. We find that the assumptions adopted in
previous works, $\mathrm{Re}\,A_8 \simeq 0$ and $\mathrm{Re}\,\omega \simeq 0$,
can be justified from the Monte-Carlo configurations sampled on the optimized
path. We also derive the Euler-Lagrange equation for the optimal path to
satisfy. The two optimized paths, the solution of the Euler-Lagrange equation
and the variationally optimized path, agree with each other in the region with
large statistical weight.
