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Some equilibrium problems under uncertainty and random variational inequalities

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  • Joachim Gwinner
  • Fabio Raciti

Abstract

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems. Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loève expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples. Copyright Springer Science+Business Media, LLC 2012

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  • Joachim Gwinner & Fabio Raciti, 2012. "Some equilibrium problems under uncertainty and random variational inequalities," Annals of Operations Research, Springer, vol. 200(1), pages 299-319, November.
    • Handle: RePEc:spr:annopr:v:200:y:2012:i:1:p:299-319:10.1007/s10479-012-1109-2
    DOI: 10.1007/s10479-012-1109-2
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    References listed on IDEAS

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    1. Stella Dafermos, 1980. "Traffic Equilibrium and Variational Inequalities," Transportation Science, INFORMS, vol. 14(1), pages 42-54, February.
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    5. Huifu Xu, 2010. "Sample Average Approximation Methods For A Class Of Stochastic Variational Inequality Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 27(01), pages 103-119.
    6. P. Daniele & A. Maugeri & W. Oettli, 1999. "Time-Dependent Traffic Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 103(3), pages 543-555, December.
    7. Arkadii V. Kryazhimskii & Andrzej Ruszczyński, 2001. "Constraint Aggregation in Infinite-Dimensional Spaces and Applications," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 769-795, November.
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    Cited by:

    1. B. Jadamba & F. Raciti, 2015. "Variational Inequality Approach to Stochastic Nash Equilibrium Problems with an Application to Cournot Oligopoly," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 1050-1070, June.
    2. Baasansuren Jadamba & Massimo Pappalardo & Fabio Raciti, 2018. "Efficiency and Vulnerability Analysis for Congested Networks with Random Data," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 563-583, May.
    3. Georgia Fargetta & Antonino Maugeri & Laura Scrimali, 2022. "A Stochastic Nash Equilibrium Problem for Medical Supply Competition," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 354-380, June.
    4. Francesca Faraci & Baasansuren Jadamba & Fabio Raciti, 2016. "On Stochastic Variational Inequalities with Mean Value Constraints," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 675-693, November.
    5. A. Causa & B. Jadamba & F. Raciti, 2017. "A migration equilibrium model with uncertain data and movement costs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 159-175, November.
    6. Jie Jiang & Shengjie Li, 2021. "Regularized Sample Average Approximation Approach for Two-Stage Stochastic Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 650-671, August.
    7. Patrizia Daniele & Sofia Giuffrè, 2015. "Random Variational Inequalities and the Random Traffic Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 363-381, October.
    8. Baasansuren Jadamba & Fabio Raciti, 2015. "On the Modeling of Some Environmental Games with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 959-968, December.

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