Domain Decomposition Methods for a Complementarity Problem*

  • Haijian YangEmail author
  • Xiao-Chuan Cai
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 78)


We introduce a family of parallel Newton-Krylov-Schwarz methods for solving complementarity problems. The methods are based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is highly scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity that is often associated with obstacle problems. We present numerical results for some large scale calculations obtained on machines with hundreds of processors.


Complementarity Problem Coarse Grid Minimum Function Newton Iteration Obstacle Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B.F. Smith, and H. Zhang. PETSc Users Manual. Argonne National Laboratory, Berkeley, CA, 2009.Google Scholar
  2. 2.
    X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 21:792–797, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Cottle, J.-S. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, Boston, MA, 1992.zbMATHGoogle Scholar
  4. 4.
    M.C. Ferris and J.-S. Pang. Engineering and economic applications of complementarity problems. SIAM Rev., 39:669–713, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Fischer. A special Newton-type optimization method. Optimization, 24:269–284, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    P.T. Harker and J.-S. Pang. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Prog., 48:161–220, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Kanzow. Inexact semismooth Newton methods for large-scale complementarity problems. Optim. Methods Softw., 19:309–325, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    T. De Luca, F. Facchinei, and C. Kanzow. A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Prog., 75:407–439, 1996.CrossRefGoogle Scholar
  9. 9.
    B. Smith, P. Bjørstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Eliptic Partial Differential Equations. Cambridge University Press, New York, NY, 1996.Google Scholar
  10. 10.
    A. Toselli and O. Widlund. Domain Decomposition Methods – Algorithms and Theory. Springer, Berlin, 2005.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.College of Mathematics and Econometrics, Hunan UniversityChangshaP. R. China
  2. 2.Department of Computer ScienceUniversity of ColoradoBoulderUSA

Personalised recommendations