Have any generalizations of maximum weight matching been studied? For example, one way to view maximum weight matching is that each vertex $v$ gets a utility $f_v= w(e_v)$ that equals the weight of the edge it's matched on, and zero otherwise.

accordingly, a maximum weight matching could then be viewed as  maximizing the objective $\sum_v f_v$. 

Have any generalizations of maximum weight matching been studied that consider more general objective functions using  weighted, multivariate or nonlinear $f_v$ ?

Have other variants been studied that are generalizations in a different way?

pleas provide references if applicable!
There are several extensions of the problem to more general structures.  For instance:


Matroid matching (lecture notes, Matroid matching and some applications, Matroid Matching: the Power of Local Search)
Path-matching (Path-matching problems, Matching, matroids, and extensions)
Hypergraph matching (Can't find any good references)


Generally, these extensions are NP-hard.