  We discuss fundamental properties of the fastest apparent convergence (FAC)
condition which is used as a variational criterion in optimized perturbation
theory (OPT). We examine an integral representation of the FAC condition and a
distribution of the zeros of the integral in a complex artificial parameter
space on the basis of theory of Lefschetz thimbles. We find that the zeros
accumulate on a certain line segment so-called anti-Stokes line in the limit $K
\to \infty$, where $K$ is a truncation order of a perturbation series. This
phenomenon gives an underlying mechanism that physical quantities calculated by
OPT can be insensitive to the choice of the artificial parameter.
