  We study an one dimensional model where an interface is the stationary
solution of a mesoscopic non local evolution equation which has been derived by
a microscopic stochastic spin system. Deviations from this evolution equation
can be quantified by obtaining the large deviations cost functional from the
underlying stochastic process. For such a functional, derived in a companion
paper, we investigate the optimal way for a macroscopic interface to move from
an initial to a final position distant by R within fixed time T. We find that
for small values of R/T the interface moves with a constant speed, while for
larger values there appear nucleations of the other phase ahead of the front.
