  We investigate weighted settings of popular matching problems with matroid
constraints. The concept of popularity was originally defined for matchings in
bipartite graphs, where vertices have preferences over the incident edges.
There are two standard models depending on whether vertices on one or both
sides have preferences. A matching $M$ is popular if it does not lose a
head-to-head election against any other matching. In our generalized models,
one or both sides have matroid constraints, and a weight function is defined on
the ground set. Our objective is to find a popular optimal matching, i.e., a
maximum-weight matching that is popular among all maximum-weight matchings
satisfying the matroid constraints. For both one- and two-sided preferences
models, we provide efficient algorithms to find such solutions, combining
algorithms for unweighted models with fundamental techniques from combinatorial
optimization. The algorithm for the one-sided preferences model is further
extended to a model where the weight function is generalized to an
M$^\natural$-concave utility function. Finally, we complement these
tractability results by providing hardness results for the problems of finding
a popular near-optimal matching. These hardness results hold even without
matroid constraints and with very restricted weight functions.
