  In this paper, we propose an extension of the forward improvement iteration
algorithm, originally introduced in Irle (2006) and recently reconsidered in
Miclo and Villeneuve (2021). The main new ingredient is a flexible window
parameter describing the look-ahead distance in the improvement step. We
consider the framework of a Markovian optimal stopping problem in discrete time
with random discounting and infinite time horizon. We prove convergence and
show that the additional flexibility may significantly reduce the runtime.
