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Solving a class of constrained 'black-box' inverse variational inequalities

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  • He, Bingsheng
  • He, Xiao-Zheng
  • Liu, Henry X.

Abstract

It is well known that a general network economic equilibrium problem can be formulated as a variational inequality (VI) and solving the VI will result in a description of network equilibrium state. In this paper, however, we discuss a class of normative control problem that requires the network equilibrium state to be in a linearly constrained set. We formulate the problem as an inverse variational inequality (IVI) because the variables and the mappings in the IVI are in the opposite positions of a classical VI. In addition, the mappings in IVI usually do not have any explicit forms and only implicit information on the functional value is available through exogenous evaluation or direct observation. For such class of network equilibrium control problem, we present a linearly constrained implicit IVI formulation and a solution method based on proximal point algorithm (PPA) that only needs functional values for given variables in the solution process.

Suggested Citation

  • He, Bingsheng & He, Xiao-Zheng & Liu, Henry X., 2010. "Solving a class of constrained 'black-box' inverse variational inequalities," European Journal of Operational Research, Elsevier, vol. 204(3), pages 391-401, August.
    • Handle: RePEc:eee:ejores:v:204:y:2010:i:3:p:391-401
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    References listed on IDEAS

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    1. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    2. Anna Nagurney & Sten Thore & Jie Pan, 1996. "Spatial Market Policy Modeling with Goal Targets," Operations Research, INFORMS, vol. 44(2), pages 393-406, April.
    3. Florian, Michael & Los, Marc, 1982. "A new look at static spatial price equilibrium models," Regional Science and Urban Economics, Elsevier, vol. 12(4), pages 579-597, November.
    4. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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    Cited by:

    1. He, Xiaozheng & Liu, Henry X., 2011. "Inverse variational inequalities with projection-based solution methods," European Journal of Operational Research, Elsevier, vol. 208(1), pages 12-18, January.
    2. Jiawei Chen & Elisabeth Köbis & Markus Köbis & Jen-Chih Yao, 2018. "Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 816-834, June.
    3. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2019. "Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 105-121, October.
    4. Phan Tu Vuong & Xiaozheng He & Duong Viet Thong, 2021. "Global Exponential Stability of a Neural Network for Inverse Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 915-930, September.

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