Journal of Optimization Theory and Applications

, Volume 102, Issue 1, pp 193–201

Tikhonov Regularization Methods for Variational Inequality Problems

  • H. D. Qi
Article

Abstract

Motivated by the work of Facchinei and Kanzow (Ref. 1) on regularization methods for the nonlinear complementarity problem and the work of Ravindran and Gowda (Ref. 2) for the box variational inequality problem, we study regularization methods for the general variational inequality problem. A sufficient condition is given which guarantees that the union of the solution sets of the regularized problems is nonempty and bounded. It is shown that solutions of the regularized problems form a minimizing sequence of the D-gap function under a mild condition.

Variational inequality problems regularization methods minimizing sequences 

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References

  1. 1.
    Facchinei, F., and Kanzow, C., Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems, SIAM Journal on Optimization, Vol. 37, pp. 1150–1161, 1999.Google Scholar
  2. 2.
    Ravindran, G., and Gowda, M. S., Regularization of P 0-Functions in Box Variational Inequality Problems, SIAM Journal on Control and Optimization, to appear.Google Scholar
  3. 3.
    Sznajder, R., and Gowda, M. S., On the Limiting Behavior of the Trajectory of the Regularized Solution of a P 0-Camplementarity Problem, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Edited by M. Fukushima and L. Qi, Kluwer Academic Publications, pp. 371–379, 1998.Google Scholar
  4. 4.
    Kanzow, C., and Fukushima, M., Theoretical and Numerical Investigation of the D-Gap Function for Box Constrained Variational Inequalities, Mathematical Programming, Vol. 83, pp. 55–87, 1998.Google Scholar
  5. 5.
    Harker, P. T., and Pang, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 9, 624–645, 1999.Google Scholar
  6. 6.
    MorÉ, J. J., and Rheinboldt, W. C., On P-and S-Functions and Related Classes of n-Dimensional Nonlinear Mappings, Linear Algebra and Applications, Vol. 6, pp. 45–68, 1973.Google Scholar
  7. 7.
    Karamardian, S., and Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.Google Scholar
  8. 8.
    Chen, B., and Chen, X., A Global and Local Superlinear Continuation-Smoothing Method for P 0 + R 0 and Monotone NCP, SIAM Journal on Optimization, Vol. xx, pp. xxx-xxx, xxxx.Google Scholar
  9. 9.
    Subramanian, P. K., Note on Least Two Norm Solution of Monotone Complementarity Problems, Applied Mathematics Letters, Vol. 1, pp. 395–397, 1988.Google Scholar
  10. 10.
    Peng, J. M., Equivalence of Variational Inequality Problems to Unconstrained Minimization, Mathematical Programming, Vol. 78, pp. 347–356, 1997.Google Scholar
  11. 11.
    Yamashita, N., Taji, K., and Fukushima, M., Unconstrained Optimization Reformulations of Variational Inequality Problems, Journal of Optimization Theory and Applications, Vol. 92, pp. 439–456, 1997.Google Scholar
  12. 12.
    Mangasarian, O. L., and Solodov, M. V., Nonlinear Complementarity as Unconstrained and Constrained Minimization, Mathematical Programming, Vol. 62, pp. 277–297, 1993.Google Scholar
  13. 13.
    Fukushima, M., and Pang, J. S., Minimizing and Stationary Sequences of Merit Functions for Complementarity Problems and Variational Inequalities, Complementarity and Variational Problems: State of the Art, Edited by M. C. Ferris and J. S. Pang, SIAM Philadelphia, Pennsylvania, pp. 91–104, 1997.Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • H. D. Qi
    • 1
  1. 1.Institute of Computational Mathematics and Scientific/Engineering ComputingChinese Academy of SciencesBeijingChina

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