Nonsmooth optimization methods for parallel decomposition of multicommodity flow problems

The authors develop an iterative algorithm based on right-hand side decomposition for the solution of multicommodity network flow problems. At each step of the proposed iterative procedure the coupling constraints are eliminated by subdividing the shared capacity resource among the different commodities and a master problem is constructed which attempts to improve sharing of the resources at each iteration. As the objective function of the master problem is nonsmooth, the authors apply to it a new optimization technique which does not require the exact solutions of the single commodity flow subproblems. This technique is based on the notion of ∈-subgradients instead of subgradients and is suitable for parallel implementation. Extensions to the nonlinear, convex separable case are also discussed.