Convex analysis of generalized flow networks

This paper is concerned with the generalized network flow (GNF) problem, which aims to find a minimum-cost solution for a generalized flow network. The objective is to determine the optimal injections at the nodes as well as optimal flows over the lines of the network. In this problem, each line is associated with two flows in opposite directions that are related to each other via a given nonlinear function. Under some monotonicity and convexity assumptions, we have shown in our recent work that a convexified generalized network flow (CGNF) problem always finds optimal injections for GNF, but may fail to find optimal flows. In this paper, we develop three results to explore the possibility of obtaining optimal flows. First, we show that CGNF yields optimal flows for GNF if the optimal injection vector is a Pareto point. Second, we show that if CGNF fails to find an optimal flow vector, then the graph can be decomposed into two subgraphs, where the lines between the subgraphs are congested at optimality and CGNF finds correct optimal flows over the lines of one of these subgraphs. Third, we fully characterize the set of all optimal flow vectors. In particular, we show that this non-convex set is a subset of the boundary of a convex set, and may include an exponential number of disconnected components.