Maximal Elements in a Lower Set
Your problem is equivalent to exact learning of monotone DNFs. This task is equivalent to learning all minterms of a monotone function, whereas you are interested in learning all maxterms of a monotone function, but the two problems are equivalent. (Look these terms up if you are not familiar with them.)

If the only queries you allow are membership queries (test whether a given set is feasible or not) then your problem is rather difficult. Theorem 2 in Angluin's Queries and Concept Learning shows that $2^n-1$ queries are required even if there are only $2n$ objects and $n+1$ minterms. See the paper for other models in which your problem can be solved efficiently.