On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of solutions to SMPEC models

Preprint, 2007

Bilevel optimization models, and more generally MPEC (mathematical program with equilibrium constraints) models, constitute important modelling tools in transportation science and network games, as they place the classic ``what-if'' analysis in a proper mathematical framework. The MPEC model is also becoming a standard for the computation of optimal design solutions, where ``design'' may include either or both of network infrastructure investments and various types of tolls. At the same time, it does normally not sufficiently well take into account possible uncertainties and/or perturbations in problem data (travel costs and demands), and thus may not a priori guarantee robust designs under varying conditions. We consider natural stochastic extensions to a class of MPEC traffic models which explicitly incorporate data uncertainty. In stochastic programming terminology, we consider ``here-and-now'' models where decisions on the design must be made before observing the uncertain parameter values and the responses of the network users, and the design is chosen to minimize the expectation of the upper-level objective function. Such a model could, for example, be used to derive a fixed link pricing scheme that provides the best revenue for a given network over a given time period, where the varying traffic conditions are described by distributions of parameters in the link travel time and OD demand functions. For a general such SMPEC network model we establish not only the existence of optimal solutions, but in particular their stability to perturbations in the probability distribution. We also provide convergence results for general algorithmic schemes based on the penalization of the equilibrium conditions or possible joint upper-level constraints, as well as for algorithms based on the discretization of the probability distribution, the latter enabling the utilization of standard MPEC algorithms. Especially the latter part utilizes relations between the traffic application of SMPEC and stochastic structural topology optimization problems.