Relaxation methods for minimum cost network flow problems

We view the optimal single commodity network flow problem with linear arc costs and its dual as a pair of monotropic programming problems, i.e. problems of minimizing the separable sum of scalar extended real-valued convex functions over a subspace. For such problems directions of cost improvement can be selected from among a finite set of directions--the elementary vectors of the constraint subspace. The classical primal simplex, dual simplex, and primal-dual methods turn out to be particular implementations of this idea. This paper considers alternate implementations leading to new dual descent algorithms which are conceptually related to coordinate descent and Gauss-Seidel relaxation methods for unconstrained optimization or solution of equations. Contrary to primal simplex and primal-dual methods, these algorithms admit a natural extension to network problems with strictly convex arc costs. Our first coded implementation of relaxation methods is compared with mature state-of-the-art primal simplex and primal-dual codes and is found to be substantially faster on most types of network flow problems of practical interest.