Convex multicommodity flow problems : a bidual approach

Abstract: The topic of this dissertation, within the subfield of mathematics known as optimization, is the development of a new dual ascent method for convex multicommodity flow problems. Convex multicommodity flow problems arize in many different routing problems such as the design of packet switched computer networks and the computation of traffic network equilibria. The dual problem of a strictly convex twice differentiable convex multicommodity flow problem is an essentially unconstrained maximization problem with a piecewise twice differentiable concave objective. The idea behind this new dual ascent algorithm is to compute Newton-like ascent directions in the current differentiable piece and performing line searches in those directions combined with an active set type treatment of the borders between the differentiable pieces. The first contribution in this dissertation is a detailed investigation of this special structure. The insights gained are then used to explain the proposed algorithm. The algorithm is also tested numerically on well known traffic equilibrium problem instances. The computational results are very promising. The second contribution is a new approach for verifying feasibility in multicommodity flow problems. These feasibility problems arizes naturally in the new dual ascent algorithm proposed. The background of the problem is that if a certain representation of a solution to the dual convex multicommodity flow problem is proved to be feasible for the convex muticommodity flow problem aswell, an optimal solution is found. Hence, it is natural to seek a method for verifying feasibility of a given candidate solution for the multicommodity flow problem. The core of the approach is a distance minimizing method for verifying feasibility and hence for demonstrating optimality. This method is described in detail and some computational results in a traffic assignment setting is also given. Finally, a short note illustrating that many published test problems for traffic assignment algorithms have peculiarities are given. First and foremost, several of them have incorrectly specified attributes, such as the number of nodes. In other testproblems, the network contains subnetworks with constant travel times; subnetworks which to a large extent can be reduced or eliminated. In further test problems, the constant travel time subnetworks imply that the solution has nonunique arc flows.