Variational Inequalities and Complementarity Problems for Stochastic Traffic User Equilibria

 

Autor: Friedemann Sebastian Winkler

Erstkorrektor: Prof. Dr. Joachim Gwinner, Universität der Bundeswehr München, Fakultät für Luft- und Raumfahrttechnik, Institut für Mathematik und Rechneranwendung

Zweitkorrektor: Prof. Dr. Masao Fukushima, Kyoto University, Graduate School of Informatics, Department of Applied Mathematics and Physics

Abgabe: Januar 2011

Abstract

Various models of traffic assignment under stochastic environment have been proposed recently. After recalling the solution theory of optimization, variational inequalities and complementarity problems for the deterministic case, this paper focuses on four different approaches including random influences.

The first approach considered is the Stochastic Nonlinear Complementarity Problem (SNCP) that uses an expected residual formulation and traditional NCP functions such as the (penalized) Fischer-Burmeister function. The second approach is the expected value formulation (EV) that solves the VI for a deterministic problem with an iterative scheme, using the expectation of the defining function. The third model is the Expected Residual Minimization (ERM) in combination with the regularized gap function and the D-gap function that turns the VI into an unconstrained minimization problem. Finally the Random Variational Inequality approach is considered, that divides the intervals of the random variables in partitions and solves the VI for every combination of those, assembling the results in a simple function. To compare this model with the three others, the expectation of the assembled function is used to gain a deterministic solution.

Numerical experiments are carried out to illustrate the characteristics of the proposed models. To compare the results some measures such as fairness, reliability and ratio of delivered demand are used.

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