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Nonlinear Optimization and Related Topics pp 463–487Cite as

A New Derivative-Free Descent Method for the Nonlinear Complementarity Problem

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Part of the book series: Applied Optimization ((APOP,volume 36))

Abstract

Recently, much effort has been made in solving and analyzing the nonlinear complementarity problem (NCP) by means of a reformulation of the problem as an equivalent unconstrained optimization problem involving a merit function. In this paper, we propose a new merit function for the NCP and show several favorable properties of the proposed function. In particular, we give conditions under which the function provides a global error bound for the NCP and conditions under which its level sets are bounded. Moreover, we propose a new derivative-free descent algorithm for solving the NCP based on this function. We show that any accumulation point generated by the algorithm is a solution of the NCP under the monotonicity assumption on the problem. Also, we prove that the sequence generated by the algorithm converges linearly to the solution under the strong monotonicity assumption.

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References

  1. B. Chen, X. Chen and C. Kanzow (1997), A penalized Fischer-Burmeister NCPfunction: Theoretical investigation and numerical results, Working paper, School of Mathematics, University of New South Wales.

    Google Scholar 

  2. R.W. Cottle, J.-S. Pang and R.E. Stone (1992), The Linear Complementarity Problem, Academic Press, New York.

    MATH  Google Scholar 

  3. S. Dafermos (1983), An iterative scheme for variational inequalities, Mathematical Programming, vol. 26, pp. 40–47.

    MathSciNet  MATH  Google Scholar 

  4. F. Facchinei and C. Kanzow (1997), On unconstrained and constrained stationary points of the implicit Lagrangian, Journal of Optimization Theory and Applications, vol. 92, pp. 99–115.

    MathSciNet  MATH  Google Scholar 

  5. F. Facchinei and J. Soares (1997), A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal on Optimization, vol. 7, pp. 225–247.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Fischer (1992), A special Newton-type optimization method, Optimization, vol. 24, pp. 269–284.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Fischer (1995), An NCP-Function and its use for the solution of complementarity problems, Recent Advances in Nonsmooth Optimization, D.Z. Du, L. Qi, and R.S. Womersley (eds.), World Scientific Publishers, Singapore, pp. 88–105.

    Google Scholar 

  8. M. Fukushima (1992), Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, vol. 53, pp. 99–110.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Fukushima (1996), Merit functions for variational inequality and complementarity problems, Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi (eds.), Plenum Press, New York, pp. 155–170.

    Google Scholar 

  10. C. Geiger and C. Kanzow (1996), On the resolution of monotone complementarity problems, Computational Optimization and Applications, vol. 5, pp. 155–173.

    Article  MathSciNet  MATH  Google Scholar 

  11. P.T. Harker and J.-S. Pang (1990), Finite dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, vol. 48, pp. 161–220.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Jiang (1996), Unconstrained minimization approaches to nonlinear complementarity problems, Journal of Global Optimization, vol. 9, pp. 169–181.

    Article  MATH  Google Scholar 

  13. C. Kanzow (1994), An unconstrained optimization technique for large-scale linearly constrained convex minimization problems, Computing, vol. 53, pp. 101117.

    Google Scholar 

  14. C. Kanzow (1996), Nonlinear complementarity as unconstrained optimization, Journal of Optimization Theory and Applications, vol. 88, pp. 139–155.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Kanzow, N. Yamashita and M. Fukushima (1997), New NCP-functions and their properties, Journal of Optimization Theory and Applications, vol. 94, pp. 115–135.

    Article  MathSciNet  MATH  Google Scholar 

  16. O.L. Mangasarian (1976), Equivalence of the complementarity problem to a system of nonlinear equations, SIAM Journal on Applied Mathematics, vol. 31, pp. 89–92.

    Article  MathSciNet  MATH  Google Scholar 

  17. O.L. Mangasarian and M.V. Solodov (1993), Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming, vol. 62, pp. 277–297.

    Article  MathSciNet  MATH  Google Scholar 

  18. O.L. Mangasarian and M.V. Solodov (forthcoming), A linearly convergent descent method for strongly monotone complementarity problems, Computational Optimization and Applications.

    Google Scholar 

  19. J.-S. Pang (1987), A posteriori error bounds for the linearly-constrained variational inequality problem, Mathematics of Operations Research, vol. 12, pp. 474–484.

    Article  MathSciNet  Google Scholar 

  20. J.-S. Pang (1994), Complementarity problems, Handbook of Global Optimization, R. Horst and P. Pardalos (eds.), Kluwer Academic Publishers, Boston, Massachusetts, pp. 271–338.

    Google Scholar 

  21. J.-S. Pang (1990), Newton’s method for B-differentiable equations, Mathematics of Operations Research, vol. 15, pp. 311–341.

    Article  MathSciNet  MATH  Google Scholar 

  22. J.-S. Pang and D. Chan (1982), Iterative methods for variational and complementarity problems, Mathematics Programming, vol. 24, pp. 284–313.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Tseng (1996), Growth behaviour of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, vol. 89, pp. 17–37.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Yamashita and M. Fukushima (1995), On stationary points of the implicit Lagrangian for nonlinear complementarity problems, Journal of Optimization Theory and Applications, vol. 84, pp. 653–663.

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Yamashita and M. Fukushima (1997), Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming, vol. 76, pp. 469–491.

    MathSciNet  MATH  Google Scholar 

  26. N. Yamashita and M. Fukushima (1998), A new merit function and a descent method for semidefinite complementarity problems, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (eds.), Kluwer Academic Publishers, Boston, Massachusetts, pp. 401–416.

    Google Scholar 

  27. N. Yamashita, K. Taji and M. Fukushima (1997), Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications, vol. 92, pp. 439–456.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Nihonbashi Hakozaki-cho, IBM Japan, Ltd., Chuo-ku, Tokyo, 103-8510, Japan

    Kenjiro Yamada

  2. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Nobuo Yamashita & Masao Fukushima

Authors
  1. Kenjiro Yamada
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  2. Nobuo Yamashita
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  3. Masao Fukushima
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Editor information

Editors and Affiliations

  1. University Di Roma La Sapienza, Italy

    Gianni Di Pillo

  2. University of Pisa, Italy

    Franco Giannessi

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© 2000 Springer Science+Business Media Dordrecht

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Yamada, K., Yamashita, N., Fukushima, M. (2000). A New Derivative-Free Descent Method for the Nonlinear Complementarity Problem. In: Pillo, G.D., Giannessi, F. (eds) Nonlinear Optimization and Related Topics. Applied Optimization, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3226-9_25

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  • DOI: https://doi.org/10.1007/978-1-4757-3226-9_25

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4823-6

  • Online ISBN: 978-1-4757-3226-9

  • eBook Packages: Springer Book Archive

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