Before revision:  We  demonstrate how various functional form, independence, and monotonicity assumptions can be imposed as constraints in our optimization procedure to tighten the identified set, and we show how these assumptions have meaningful interpretations in terms of restrictions on latent types. In the special case when the index function is linear in the latent variables, we leverage results in computational geometry to provide a tractable means of constructing the sharp set of constraints for our optimization problems .
After revision:  We demonstrate how various functional form, independence, and monotonicity assumptions can be imposed as constraints in our optimization procedure to tighten the identified set, and we show how these assumptions have meaningful interpretations in terms of restrictions on latent response types .