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Error bounds for complementarity problems with tridiagonal nonlinear functions

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Abstract

In this paper we consider the complementarity problem NCP(f) with f(x) = Mx + φ(x), where M ∈ R n×n is a real matrix and φ is a so-called tridiagonal (nonlinear) mapping. This problem occurs, for example, if certain classes of free boundary problems are discretized. We compute error bounds for approximations \({\hat x}\) to a solution x* of the discretized problems. The error bounds are improved by an iterative method and can be made arbitrarily small. The ideas are illustrated by numerical experiments.

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References

  1. Alefeld GE, Chen X, Potra FA (1999) Validation of solution to linear complementarity problems. Numer Math 83: 1–23

    Article  MATH  MathSciNet  Google Scholar 

  2. Alefeld GE, Mayer G (2000) Interval analysis: theory and applications. J Comp Appl Math 121: 421–464

    Article  MATH  MathSciNet  Google Scholar 

  3. Alefeld GE, Chen X, Potra FA (2002) Numerical validation of solutions of complementarity problems: the nonlinear case. Numer Math 92: 1–16

    Article  MATH  MathSciNet  Google Scholar 

  4. Alefeld GE, Schäfer U (2003) Iterative methods for linear complementarity problems with interval data. Computing 70: 235–259

    MATH  MathSciNet  Google Scholar 

  5. Alefeld GE, Wang Z, Shen Z (2004) Enclosing solutions of linear complementarity problems for H-matrices. Reliab Comput 10: 423–435

    Article  MATH  MathSciNet  Google Scholar 

  6. Alefeld GE, Wang Z (2005) Verification of solutions for almost linear complementarity problems. Ann Eur Acad Sci 211–231

  7. Alefeld GE, Wang Z (2008) Error estimation for nonlinear complementarity problems via linear systems with interval data. Numer Funct Anal Optim 29(3–4): 243–267

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen B (2001) Error bounds for R0-type and monotone nonlinear complementarity problems. J Optim Theory Appl 108: 297–316

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen X, Xiang S (2006) Computation of error bounds for P-matrix linear complementarity problems. Math Prog Ser A 106: 513–525

    Article  MATH  MathSciNet  Google Scholar 

  10. Collatz L (1966) The numerical treatment of differential equations. 2nd Printing of the 3rd edn. Springer, Berlin

    Google Scholar 

  11. Facchinei F, Pang JS (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, New York

    Google Scholar 

  12. Ferris MC, Mangasarian OL (1993) Error bounds and strong upper semicontinuity for monotone affine variational inequalities. Ann Oper Res 47: 293–305

    Article  MathSciNet  Google Scholar 

  13. Ferris MC, Pang JS (1997) Engineering and economic applications of complementarity problems. SIAM Rev 39(4): 669–713

    Article  MATH  MathSciNet  Google Scholar 

  14. Mangasarian OL, Ren J (1994) New improved error bounds for the linear complementarity problem. Math Prog 66: 241–257

    Article  MathSciNet  Google Scholar 

  15. Mathias R, Pang JS (1990) Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra Appl 132: 123–136

    Article  MATH  MathSciNet  Google Scholar 

  16. Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  17. Pang JS (1997) Error bounds in mathematical programming. Math Prog 79: 199–332

    Google Scholar 

  18. Rump SM (1999) INTLAB-INTerval LABoratory. In: Csendes T (eds) Developments in reliable computing. Kluwer, Dordrecht, pp 77–104

    Google Scholar 

  19. Varga RS (1962) Matrix iterative analysis. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  20. Wang Z (2007) Validation and enclosure of solutions of linear complementarity problems. Computing 79: 61–77

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied and Numerical Mathematics, University of Karlsruhe (Karlsruhe Institute of Technology KIT), 76131, Karlsruhe, Germany

    G. Alefeld

  2. Department of Mathematics, Nanjing University, 210093, Nanjing, China

    Z. Wang

Authors
  1. G. Alefeld
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  2. Z. Wang
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Corresponding author

Correspondence to G. Alefeld.

Additional information

The work of Z. Wang was supported in part by a grant from the State of Baden-Württemberg and by grant-10771099 from the National Natural Science Foundation of China. Z. Wang would also like to appreciate Universität Karlsruhe for the kind invitation.

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Alefeld, G., Wang, Z. Error bounds for complementarity problems with tridiagonal nonlinear functions. Computing 83, 175–192 (2008). https://doi.org/10.1007/s00607-008-0021-8

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  • DOI: https://doi.org/10.1007/s00607-008-0021-8

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