In ,  a theory for degree-constrained analytic interpolation was developed in terms of the minimizers of certain convex entropy functionals. In the present paper, we introduce and study relevant inverse problems. More specifically, we answer the following two questions. First, given a function f which satisfies specified interpolation conditions, when is it that f can be obtained as the minimizer of a suitably chosen entropy functional? Second, given a function g, when does there exist a suitably entropy functional so that the unique minimizer f which is subject to interpolation constraints also satisfies |f| = |g| on the unit circle. The theory and answers to these questions suggest an approach to identifying interpolants of a given degree and of a given approximate shape.