  This paper provides a framework for the policy relevant treatment effects
using instrumental variables. In the framework, a treatment selection may or
may not satisfy the classical monotonicity condition and can accommodate
multidimensional unobserved heterogeneity. We can bound the target parameter by
extracting information from identifiable estimands. We also provide a more
conservative yet computationally simpler bound by applying a convex relaxation
method. Linear shape restrictions can be easily incorporated to further improve
the bounds. Numerical and simulation results illustrate the informativeness of
our convex-relaxation bounds, i.e., that our bounds are sufficiently tight.
