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Bilevel road pricing: theoretical analysis and optimality conditions

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Abstract

We consider the bilevel road pricing problem. In contrary to the Karush-Kuhn-Tucker (one level) reformulation, the optimal value reformulation is globally and locally equivalent to the initial problem. Moreover, in the process of deriving optimality conditions, the optimal value reformulation helps to preserve some essential data involved in the traffic assignment problem that may disappear with the Karush-Kuhn-Tucker (KKT) one. Hence, we consider in this work the optimal value reformulation of the bilevel road pricing problem; using some recent developments in nonsmooth analysis, we derive implementable KKT type optimality conditions for the problem containing all the necessary information. The issue of estimating the (fixed) demand required for the road pricing problem is a quite difficult problem which has been also addressed in recent years using bilevel programming. We also show how the ideas used in designing KKT type optimality conditions for the road pricing problem can be applied to derive optimality conditions for the origin-destination (O-D) matrix estimation problem. Many other theoretical aspects of the bilevel road pricing and O-D matrix estimation problems are also studied in this paper.

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Acknowledgements

We would like to thank the two anonymous referees for carefully reading our manuscript. Their comments and suggestions have been of a great help for improvements in the paper. The work of A.B. Zemkoho was supported by the Deutscher Akademischer Austausch Dienst (DAAD).

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  1. Institut für Numerische Mathematik und Optimierung, Technische Universität Bergakademie Freiberg, Akademie Straße 6, 09596, Freiberg, Germany

    S. Dempe & A. B. Zemkoho

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  1. S. Dempe
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  2. A. B. Zemkoho
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Correspondence to A. B. Zemkoho.

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Dempe, S., Zemkoho, A.B. Bilevel road pricing: theoretical analysis and optimality conditions. Ann Oper Res 196, 223–240 (2012). https://doi.org/10.1007/s10479-011-1023-z

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