Relaxation Methods for Network Flow Problems with Convex Arc Costs

We consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs. For the case where all arc costs are strictly convex we study the convergence of a dual Gauss–Seidel type relaxation method that is well suited for parallel computation. We then extend this method to the case where some of the arc costs are linear. As a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in Bertsekas [1] and Bertsekas and Tseng [2].