Distributed Combined Space Partitioning and Network Flow Optimization: an Optimal Transport Approach

This paper studies a combined space partitioning and network flow optimization problem, with applications to large-scale electric power, transportation, or communication systems. In dense wireless networks for instance, one may want to simultaneously optimize the assignment of many spatially distributed users to base stations and route the resulting communication traffic through the backbone network. We formulate the overall problem by coupling a semi-discrete optimal transport (SDOT) problem, capturing the space partitioning component, with a minimum-cost flow problem on a discrete network. This formulation jointly optimizes the assignment of a continuous demand distribution to certain endpoint network nodes and the routing of flows over the network to serve the demand, under capacity constraints. As for SDOT problems, we establish that the formulation of our problem admits a tight relaxation taking the form of an infinite-dimensional linear program, derive its finite-dimensional dual, and prove that strong duality holds. We leverage this theory to design a distributed dual (super)gradient ascent algorithm solving the problem, where nodes in the graph perform computations based solely on locally available information. Simulation results illustrate the algorithm’s performance and its applicability to an electric power distribution network reconfiguration problem.