  We propose a numerical algorithm for computing approximately optimal
solutions of the matching for teams problem. Our algorithm is efficient for
problems involving large number of agent categories and allows for non-discrete
agent type measures. Specifically, we parametrize the so-called transfer
functions and develop a parametric version of the dual formulation, which we
tackle to produce feasible and approximately optimal solutions for the primal
and dual formulations. These solutions yield upper and lower bounds for the
optimal value, and the difference between these bounds provides a direct
sub-optimality estimate of the computed solutions. Moreover, we are able to
control the sub-optimality to be arbitrarily close to 0. We subsequently prove
that the approximate primal and dual solutions converge when the sub-optimality
goes to 0 and their limits constitute a true matching equilibrium. Thus, the
outputs of our algorithm are regarded as an approximate matching equilibrium.
We also analyze the theoretical computational complexity of our parametric
formulation as well as the sparsity of the resulting approximate matching
equilibrium. In the numerical experiments, we study three matching for teams
problems: a problem of business location distribution, the well-known
2-Wasserstein barycenter problem, and a high-dimensional problem involving 100
agent categories. Through the numerical results, we showcase that the proposed
algorithm can produce high-quality approximate matching equilibria in these
settings, provide quantitative insights about the optimal city structure in the
business location distribution problem, and that the sub-optimality estimates
computed by our algorithm are much less conservative than theoretical
estimates.
