Duality in convex minimum cost flow problems in infinite networks

Minimum cost flow problems in infinite networks arise, for example, in infinite-horizon sequential decision problems such as production planning. Strong duality for these problems was recently established for the special case of linear costs using an infinite-dimensional simplex algorithm. Here, we use a different approach to derive duality results when the costs are convex. We formulate the primal and dual problems in appropriately paired sequence spaces such that weak duality and complementary slackness can be established using finite-dimensional proof techniques. We then prove, using a planning horizon proof technique, that the absence of a duality gap between carefully constructed finite-dimensional truncations of the primal problem and their duals is preserved in the limit. We then establish that strong duality holds when optimal solutions to the finite-dimensional duals are bounded. These theoretical results are illustrated via an infinite-horizon shortest path problem.