Solving the Convex Flow Problem

In this paper, we introduce the solver ConvexFlows for the convex flow problem first defined in the authors' previous work. In this problem, we aim to optimize a concave utility function depending on the flows over a graph. However, unlike the classic network flows literature, we also allow for a concave relationship between the input and output flows of edges. This nonlinear gain describes many physical phenomena, including losses in power network transmission lines. We outline an efficient algorithm for solving this problem which parallelizes over the graph edges. We provide an open source implementation of this algorithm in the Julia programming language package ConvexFlows.jl. This package includes an interface to easily specify these flow problems. We conclude by walking through an example of solving for an optimal power flow using ConvexFlows.