Convexification of Generalized Network Flow Problem

This paper is concerned with the minimum-cost flow problem over an arbitrary flow network. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. This relaxation may fail to find optimal flows because the mapping from injections to flows may not be unique. However, it will be shown that it yields correct flows if the optimal injection vector is a Pareto point. More generally, if the optimal injection vector is on the boundary of its feasible region, then the graph can be decomposed into two subgraphs, where the lines between the subgraphs are congested at optimality and convexified GNF finds correct optimal flows over the lines of one of these subgraphs. Finally, 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. A primary application of this work is in optimization over power networks. Recent work on the optimal Somayeh Sojoudi Department of Computing and Mathematical Sciences, California Institute of Technology Tel.: E-mail: [email protected] Salar Fattahi Department of Electrical Engineering, Columbia University Tel.: +1-631-417-2564 E-mail: [email protected] Javad Lavaei Department of Electrical Engineering, Columbia University Tel.: E-mail: [email protected] 2 Somayeh Sojoudi et al. power flow (OPF) problem has shown that this non-convex problem can be solved efficiently using semidefinite programming (SDP) after two approximations: relaxing angle constraints (by adding virtual phase shifters) and relaxing power balance equations to