  In this article, we apply the path optimization method to handle the
complexified parameters in the 1+1 dimensional pure $U(1)$ gauge theory on the
lattice. Complexified parameters make it possible to explore the Lee-Yang zeros
which helps us to understand the phase structure and thus we consider the
complex coupling constant with the path optimization method in the theory. We
clarify the gauge fixing issue in the path optimization method; the gauge
fixing helps to optimize the integration path effectively. With the gauge
fixing, the path optimization method can treat the complex parameter and
control the sign problem. It is the first step to directly tackle the Lee-Yang
zero analysis of the gauge theory by using the path optimization method.
