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Unconstrained Optimization Reformulations of Variational Inequality Problems

Author

Listed:
  • N. Yamashita

    (Kyoto University)

  • K. Taji

    (Nara Institute of Science and Technology, Ikoma)

  • M. Fukushima

    (Kyoto University)

Abstract

Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.

Suggested Citation

  • N. Yamashita & K. Taji & M. Fukushima, 1997. "Unconstrained Optimization Reformulations of Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 439-456, March.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:3:d:10.1023_a:1022660704427
    DOI: 10.1023/A:1022660704427
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    References listed on IDEAS

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    1. Jong-Shi Pang, 1987. "A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 474-484, August.
    2. Stella Dafermos, 1980. "Traffic Equilibrium and Variational Inequalities," Transportation Science, INFORMS, vol. 14(1), pages 42-54, February.
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    Cited by:

    1. Bui Dinh & Tien-Son Pham, 2022. "Error Bounds of Regularized Gap Functions for Polynomial Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 226-247, January.
    2. H. D. Qi, 1999. "Tikhonov Regularization Methods for Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 193-201, July.
    3. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    4. Giancarlo Bigi & Mauro Passacantando, 2015. "D-gap functions and descent techniques for solving equilibrium problems," Journal of Global Optimization, Springer, vol. 62(1), pages 183-203, May.
    5. L.R. Huang & K. F. Ng, 2005. "Equivalent Optimization Formulations and Error Bounds for Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 299-314, May.
    6. J. M. Peng, 1998. "Derivative-Free Methods for Monotone Variational Inequality and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 235-252, October.
    7. Anna Heusinger & Christian Kanzow, 2009. "Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions," Computational Optimization and Applications, Springer, vol. 43(3), pages 353-377, July.
    8. K. F. Ng & L. L. Tan, 2007. "D-Gap Functions for Nonsmooth Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 77-97, April.
    9. Kwon, Changhyun & Friesz, Terry L. & Mookherjee, Reetabrata & Yao, Tao & Feng, Baichun, 2009. "Non-cooperative competition among revenue maximizing service providers with demand learning," European Journal of Operational Research, Elsevier, vol. 197(3), pages 981-996, September.
    10. B. Qu & C. Y. Wang & J. Z. Zhang, 2003. "Convergence and Error Bound of a Method for Solving Variational Inequality Problems via the Generalized D-Gap Function," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 535-552, December.
    11. Rachana Gupta & Aparna Mehra, 2012. "Gap functions and error bounds for quasi variational inequalities," Journal of Global Optimization, Springer, vol. 53(4), pages 737-748, August.
    12. Massimo Pappalardo & Giandomenico Mastroeni & Mauro Passacantando, 2016. "Merit functions: a bridge between optimization and equilibria," Annals of Operations Research, Springer, vol. 240(1), pages 271-299, May.

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