Maximal Elements in a Lower Set

Question

I have a collection of objects, and a feasibility property for sets of objects which is slow to compute. If a set is feasible then so is any subset. For example, it could be whether the set of things will fit into a certain-sized packing crate.

I want to compute all feasible sets (equivalently - all maximal feasible sets), minimizing the number of evaluations of the feasibility property (worst-case or average). Does anyone know of any theoretical work on this?

Bonus question - what if the cost to compute the property is linear (or superlinear) in the size of the set being evaluated?