  We present distributed subgradient methods for min-max problems with
agreement constraints on a subset of the arguments of both the convex and
concave parts. Applications include constrained minimization problems where
each constraint is a sum of convex functions in the local variables of the
agents. In the latter case, the proposed algorithm reduces to primal-dual
updates using local subgradients and Laplacian averaging on local copies of the
multipliers associated to the global constraints. For the case of general
convex-concave saddle-point problems, our analysis establishes the convergence
of the running time-averages of the local estimates to a saddle point under
periodic connectivity of the communication digraphs. Specifically, choosing the
gradient step-sizes in a suitable way, we show that the evaluation error is
proportional to $1/\sqrt{t}$, where $t$ is the iteration step. We illustrate
our results in simulation for an optimization scenario with nonlinear
constraints coupling the decisions of agents that cannot communicate directly.
