  We use the parabolic wave equation to study the propagation of x-rays in
tapered waveguides by numercial simulation and optimization. The goal of the
study is to elucidate how beam concentration can be best achieved in x-ray
optical nanostructures. Such optimized waveguides can e.g. be used to
investigate single biomolecules. Here, we compare tapering geometries, which
can be parametrized by linear and third-order (Bezier-type) functions and can
be fabricated using standard e-beam litography units. These geometries can be
described in two and four-dimensional parameter spaces, respectively. In both
geometries, we observe a rugged structure of the optimization problem's ``gain
landscape''. Thus, the optimization of x-ray nanostructures in general will be
a highly nontrivial optimization problem.
