Distributed abstract programs are a novel class of distributed optimization problems where (i) the number of variables is much smaller than the number of constraints and (ii) each constraint is associated to a network node. Abstract optimization programs are a generalization of linear programs that captures numerous geometric optimization problems. We propose novel constraints consensus algorithms for distributed abstract programs: as each node iteratively identifies locally active constraints and exchanges them with its neighbors, the network computes the active constraints determining the global optimum. The proposed algorithms are appropriate for networks with weak time-dependent connectivity requirements and tight memory constraints. We show how suitable target localization and formation control problems can be tackled via constraints consensus.