Distributed Constraint-Coupled Optimization over Lossy Networks

This paper considers distributed resource allocation and sum-preserving constrained optimization over lossy networks, where the links are unreliable and subject to packet drops. We define the conditions to ensure convergence under packet drops and link removal by focusing on two main properties of our allocation algorithm: (i) The weight-stochastic condition in typical consensus schemes is reduced to balanced weights, with no need for readjusting the weights to satisfy stochasticity. (ii) The algorithm does not require all-time connectivity but instead uniform connectivity over some non-overlapping finite time intervals. First, we prove that our algorithm provides primal-feasible allocation at every iteration step and converges under the conditions (i)-(ii) and some other mild conditions on the nonlinear iterative dynamics. These nonlinearities address possible practical constraints in real applications due to, for example, saturation or quantization among others. Then, using (i)-(ii) and the notion of bond-percolation theory, we relate the packet drop rate and the network percolation threshold to the (finite) number of iterations ensuring uniform connectivity and, thus, convergence towards the optimum value.