Inverse optimization in minimum cost flow problems on countably infinite networks

Given the costs and a feasible solution for a minimum cost flow problem on a countably infinite network, inverse optimization involves finding new costs that are close to the original ones and that make the given solution optimal. We study this problem using the weighted absolute sum metric to quantify closeness of cost vectors. We provide sufficient conditions under which known results from inverse optimization in minimum cost flow problems on finite networks extend to the countably infinite case. These conditions ensure that recent duality results on countably infinite linear programs can be applied to our setting. Specifically, they enable us to prove that the inverse optimization problem can be reformulated as a capacitated, minimum cost circulation problem on a countably infinite network. Finite-dimensional truncations of this problem can be solved in polynomial time when the weights equal one, which yields an efficient solution method. The circulation problem can also be solved via the shadow simplex method, where each finite-dimensional truncation is tackled using the usual network Simplex algorithm that is empirically known to be computationally efficient. We illustrate these results on an infinite horizon shortest path problem.