  Designing experiments for generalized linear models is difficult because
optimal designs depend on unknown parameters. Here we investigate local
optimality. We propose to study for a given design its region of optimality in
parameter space. Often these regions are semi-algebraic and feature interesting
symmetries. We demonstrate this with the Rasch Poisson counts model. For any
given interaction order between the explanatory variables we give a
characterization of the regions of optimality of a special saturated design.
This extends known results from the case of no interaction. We also give an
algebraic and geometric perspective on optimality of experimental designs for
the Rasch Poisson counts model using polyhedral and spectrahedral geometry.
