  We study the problem of designing optimal auctions under restrictions on the
set of permissible allocations. In addition to allowing us to restrict to
deterministic mechanisms, we can also indirectly model non-additive valuations.
We prove a strong duality result, extending a result due to Daskalakis et al.
[2015], that guarantees the existence of a certificate of optimality for
optimal restricted mechanisms. As a corollary of our result, we provide a new
characterization of the set of allocations that the optimal mechanism may
actually use. To illustrate our result we find and certify optimal mechanisms
for four settings where previous frameworks do not apply, and provide new
economic intuition about some of the tools that have previously been used to
find optimal mechanisms.
