An augmented Lagrangean dual algorithm for link capacity side constrained traffic assignment problems

As a means to obtain a more accurate description of traffic flows than that provided by the basic model of traffic assignment, there have been suggestions to impose upper bounds on the link flows. This can be done either by introducing explicit link capacities or by employing travel time functions with asymptotes at the upper bounds. Although the latter alternative has the disadvantage of inherent numerical ill-conditioning, the capacitated assignment model has been studied and applied to a limited extent, the main reason being that the solutions can not be characterized by the classical Wardrop equilibrium conditions; they may, however, be characterized as Wardrop equilibria in terms of a well-defined, natural generalized travel cost. The introduction of link capacity side constraints makes the problem computationally more demanding. The availability of efficient algorithms for the basic model of traffic assignment motivates the use of dualization approaches for handling the capacity constraints. We propose and evaluate an augmented Lagrangean dual method in which the uncapacitated traffic assignment subproblems are solved with the disaggregate simplicial decomposition algorithm. This algorithm fully exploits the subproblem's structure and has very favourable reoptimization capabilities; both these properties are necessary for achieving computational efficiency in iterative dualization schemes. The dual method exhibits a linear rate of convergence under a standard nondegeneracy assumption. The efficiency of the overall algorithm is demonstrated through experiments with capacitated versions of well-known test problems, with the conclusion that the introduction of link capacities increases the computing times with no more than a factor of four. The introduction of capacities and the algorithm suggested can be used to derive tolls for the reduction of flows on overloaded links. The solution strategy can be applied also to other types of traffic assignment models where side constraints have been added in order to refine a descriptive or prescriptive assignment model.