  Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility
functionals and convex risk measures. Explicit solutions are found for
quantile-based coherent risk measures and related utility functionals.
Typically, these solutions exhibit a critical phenomenon: If the capital
constraint is below some critical value, then the solution will coincide with a
classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For
general risk measures and utility functionals, it is shown that there exists a
solution that can be written as a deterministic increasing function of the
price density.
