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Decomposition Method for a Class of Monotone Variational Inequality Problems

Author

Listed:
  • B. S. He

    (Nanjing University)

  • L. Z. Liao

    (Hong Kong Baptist University)

  • H. Yang

    (Hong Kong University of Science and Technology, Clear Water Bay)

Abstract

In the solution of the monotone variational inequality problem VI(Ω, F), with $$u = \left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right],Fu = \left[ {\begin{array}{*{20}c} {fx - ATy} \\ {Ax - b} \\ \end{array} } \right],\Omega = \mathcal{X} \times \mathcal{Y},$$ the augmented Lagrangian method (a decomposition method) is advantageous and effective when $$\mathcal{X} = \mathcal{R}^m$$ . For some problems of interest, where both the constraint sets $$\mathcal{X}$$ and $$\mathcal{Y}$$ are proper subsets in $$\mathcal{R}^n$$ and $$\mathcal{R}^m$$ , the original augmented Lagrangian method is no longer applicable. For this class of variational inequality problems, we introduce a decomposition method and prove its convergence. Promising numerical results are presented, indicating the effectiveness of the proposed method.

Suggested Citation

  • B. S. He & L. Z. Liao & H. Yang, 1999. "Decomposition Method for a Class of Monotone Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 103(3), pages 603-622, December.
    • Handle: RePEc:spr:joptap:v:103:y:1999:i:3:d:10.1023_a:1021736008175
    DOI: 10.1023/A:1021736008175
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    References listed on IDEAS

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    1. Anna Nagurney & Sten Thore & Jie Pan, 1996. "Spatial Market Policy Modeling with Goal Targets," Operations Research, INFORMS, vol. 44(2), pages 393-406, April.
    2. Anna Nagurney & Padma Ramanujam, 1996. "Transportation Network Policy Modeling with Goal Targets and Generalized Penalty Functions," Transportation Science, INFORMS, vol. 30(1), pages 3-13, February.
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    Cited by:

    1. J. Fuller & William Chung, 2005. "Dantzig—Wolfe Decomposition of Variational Inequalities," Computational Economics, Springer;Society for Computational Economics, vol. 25(4), pages 303-326, June.
    2. Fuller, J. David & Chung, William, 2008. "Benders decomposition for a class of variational inequalities," European Journal of Operational Research, Elsevier, vol. 185(1), pages 76-91, February.
    3. S. L. Wang & L. Z. Liao, 2001. "Decomposition Method with a Variable Parameter for a Class of Monotone Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 415-429, May.

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