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On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints

Journal article, 2008

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.

stochastic programming

communication networks

network design

sample average approximation

discretization

traffic equilibrium

solution stability

Monte Carlo simulation

penalization

stochastic mathematical program with equilibrium constraints