We study some generalizations to subsets of the unit circle of Adamjan-Arov-Kr- ein type problems and mainly the one of extending a given function to the missing part of the boundary so as to make it as close to meromorphic with $N$ poles as possible in the $sup$ norm while meeting some gauge constraint. To make our analysis computationally effective, a generic non--multipleness result of the singular values of Hankel operators is established which allows us to provide a convergent resolution algorithm in separable Hölder-Zygmund classes.