Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design

We provide a sensitivity analysis of separable traffic equilibrium models with travel cost and demand parameters. We establish that while equilibrium link flows may not always be directionally differentiable (even when the link travel costs are strictly increasing), travel demands and link costs are; this improves the general results of Patriksson [Patriksson, M., 2004. Sensitivity analysis of traffic equilibria. Transportation Science 37, 258-281]. The new results contradict common belief that equilibrium cost and demand sensitivities hinge on that of equilibrium flows. The paper by Tobin and Friesz [Tobin, R.L., Friesz, T.L., 1988. Sensitivity analysis for equilibrium network flow. Transportation Science 22, 242-250] brought the classic non-linear programming subject of sensitivity analysis to transportation science. Theirs is still the most widely used device by which "gradients" of traffic equilibrium solutions are calculated, for use in bilevel transportation planning applications such as network design, origin-destination (OD) matrix estimation and problems where link tolls are imposed on the users in order to reach a traffic management objective. However, it is not widely understood that the regularity conditions proposed by them are stronger than necessary. Also, users of their method sometimes misunderstand its limitations and are not aware of the computational advantages offered by more recent methods. In fact, a more often applicable formula was proposed already by Qiu and Magnanti [Qiu, Y., Magnanti, T.L., 1989. Sensitivity analysis for variational inequalities defined on polyhedral sets. Mathematics of Operations Research 14, 410-432], and Bell and Iida [Bell, M.G.H., Iida, Y., 1997. Transportation Network Analysis. John Wiley & Sons, Chichester, UK] describe one of the cases in practice in which the formula by Tobin and Friesz would not be able to generate sensitivity information, because one of their regularity conditions fails to hold. This paper provides an overview of this formula, and illustrates by means of examples that there are several cases where it is not applicable. Our findings are illustrated with small numerical examples, as are our own analysis. The findings of this paper are hoped to motivate replacing the previous approach with the more often applicable one, not only because of this fact but equally importantly because it is intuitive and also can be much more efficiently utilized: the sensitivity problem that provides the directional derivative is a linearized traffic equilibrium problem, and the sensitivity information can be generated efficiently by only slightly modifying a state-of-the-art traffic equilibrium solver. This is essential for bringing the use of sensitivity analysis in transportation planning beyond the solution of only toy problems. We finally utilize a new sensitivity solver in the preliminary testing of a simple heuristic for bilevel optimization in continuous traffic network design, and compare it favourably to previous heuristics on known small-scale problems.