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Stochastic Nonlinear Complementarity Problem and Applications to Traffic Equilibrium under Uncertainty

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  • C. Zhang

    (Beijing Jiaotong University)

  • X. Chen

    (The Hong Kong Polytechnic University)

Abstract

The expected residual minimization (ERM) formulation for the stochastic nonlinear complementarity problem (SNCP) is studied in this paper. We show that the involved function is a stochastic R 0 function if and only if the objective function in the ERM formulation is coercive under a mild assumption. Moreover, we model the traffic equilibrium problem (TEP) under uncertainty as SNCP and show that the objective function in the ERM formulation is a stochastic R 0 function. Numerical experiments show that the ERM-SNCP model for TEP under uncertainty has various desirable properties.

Suggested Citation

  • C. Zhang & X. Chen, 2008. "Stochastic Nonlinear Complementarity Problem and Applications to Traffic Equilibrium under Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 277-295, May.
    • Handle: RePEc:spr:joptap:v:137:y:2008:i:2:d:10.1007_s10957-008-9358-6
    DOI: 10.1007/s10957-008-9358-6
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    References listed on IDEAS

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    1. Steven A. Gabriel & David Bernstein, 1997. "The Traffic Equilibrium Problem with Nonadditive Path Costs," Transportation Science, INFORMS, vol. 31(4), pages 337-348, November.
    2. B. Chen, 2001. "Error Bounds for R0-Type and Monotone Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 297-316, February.
    3. Xiaojun Chen & Masao Fukushima, 2005. "Expected Residual Minimization Method for Stochastic Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 1022-1038, November.
    4. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
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    Citations

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    Cited by:

    1. Liyan Xu & Bo Yu, 2014. "CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems," Computational Optimization and Applications, Springer, vol. 58(2), pages 483-501, June.
    2. M. J. Luo & G. H. Lin, 2009. "Expected Residual Minimization Method for Stochastic Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 103-116, January.
    3. Zhang, Chao & Chen, Xiaojun & Sumalee, Agachai, 2011. "Robust Wardrop's user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 534-552, March.
    4. Min Li & Chao Zhang, 2020. "Two-Stage Stochastic Variational Inequality Arising from Stochastic Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 324-343, July.
    5. M. J. Luo & G. H. Lin, 2009. "Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 569-581, September.
    6. Fang Lu & Shengjie Li & Jing Yang, 2015. "Convergence analysis of weighted expected residual method for nonlinear stochastic variational inequality problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 229-242, October.
    7. Lu, Fang & Li, Sheng-jie, 2015. "Method of weighted expected residual for solving stochastic variational inequality problems," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 651-663.
    8. M. Wang & M. M. Ali, 2010. "Stochastic Nonlinear Complementarity Problems: Stochastic Programming Reformulation and Penalty-Based Approximation Method," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 597-614, March.
    9. Lin, Gui-Hua & Zhang, Dali & Liang, Yan-Chao, 2013. "Stochastic multiobjective problems with complementarity constraints and applications in healthcare management," European Journal of Operational Research, Elsevier, vol. 226(3), pages 461-470.
    10. Xie, Chi & Liu, Zugang, 2014. "On the stochastic network equilibrium with heterogeneous choice inertia," Transportation Research Part B: Methodological, Elsevier, vol. 66(C), pages 90-109.
    11. Ming-Zheng Wang & M. Ali, 2014. "On the ERM formulation and a stochastic approximation algorithm of the stochastic- $$R_0$$ R 0 EVLCP," Annals of Operations Research, Springer, vol. 217(1), pages 513-534, June.

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