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Parallel Newton methods for the nonlinear complementarity problem

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Abstract

In this paper, we discuss how the basic Newton method for solving the nonlinear complementarity problem can be implemented in a parallel computation environment. We propose some synchronized and asynchronous Newton methods and establish their convergence.

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References

  1. G.M. Baudet, “Asynchronous iterative methods for multiprocessors,”Journal of the ACM 2 (1978) 226–244.

    Google Scholar 

  2. A. Berman and R.J. Plemmons,Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).

    Google Scholar 

  3. D.P. Bertsekas, “Distributed asynchronous computation of fixed points,”Mathematical Programming 27 (1983) 107–120.

    Google Scholar 

  4. B.C. Eaves, “Where solving for stationary points by LCPs is mixing Newton iterates,” in B.C. Eaves, F.J. Gould, H.O. Peitgen and M.J. Todd eds.,Homotopy Methods and Global Convergence (Plenum Press, New York, 1983) pp. 63–78.

    Google Scholar 

  5. N.H. Josephy, “Newton's method for generalized equations,” Technical Summary Report No. 1965, Mathematics Research Center, University of Wisconsin (Madison, Wisconsin, 1979).

    Google Scholar 

  6. N.H. Josephy, “Quasi-Newton methods for generalized equations,” Technical Summary Report No. 1966, Mathematics Research Center, University of Wisconsin (Madison, Wisconsin, 1979).

    Google Scholar 

  7. L. Kronsjo, Computational Complexity of Sequential and Parallel Algorithms (John Wiley and Sons, Great Britain, 1985).

    Google Scholar 

  8. H.T. Kung, “Synchronized and asynchronous parallel algorithms for multiprocessors,” in J.F. Traub, ed.,Algorithms and Complexity: New Directions and Recent Results (Academic Press, New York, 1976) 153–200.

    Google Scholar 

  9. O.L. Mangasarian and R. De Leone, “Parallel successive overrelaxation methods for symmetric linear complementarity problems and linear programs,” Computer Sciences Technical Report #647, University of Wisconsin (Madison, Wisconsin, 1986).

    Google Scholar 

  10. O.L. Mangasarian and R. De Leone, “Parallel gradient projection successive overrelaxation methods for symmetric linear complementarity problems and linear programs,” Computer Sciences Technical Report #659, University of Wisconsin (Madison, Wisconsin, 1986).

    Google Scholar 

  11. H. Mukai, “Parallel algorithms for solving systems of nonlinear equations,”Proceedings of the 17th Annual Allerton Conference on Communications, Control and Computations (October 10–12, 1979) pp. 37–46.

  12. J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).

    Google Scholar 

  13. J.S. Pang, “On the convergence of a basic iterative scheme for the implicit complementarity problem,”Journal of Optimization Theory and Applications 37 (1982) 149–162.

    Google Scholar 

  14. J.S. Pang, “Necessary and Sufficient Conditions for the convergence of iterative methods for the linear complementarity problem,”Journal of Optimization Theory and Applications 42 (1984) 1–18.

    Google Scholar 

  15. J.S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming 36 (1986) 54–71.

    Google Scholar 

  16. J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.

    Google Scholar 

  17. J.S. Pang and J.M. Yang, “Two-stage parallel iterative methods for the symmetric linear complementarity problem,” to appear inAnnals of Operations Research (1988).

  18. S.M. Robinson, “Strongly Regular Generalized Equations,”Mathematics of Operations Research 5 (1980) 43–62.

    Google Scholar 

  19. V.E. Shamanskii, “On a modification of the Newton method,”Ukrainskiî Matematiceskiî Žurnal 19 (1967) 133–138.

    Google Scholar 

  20. K.M. Thompson, “Parallel algorithms for solving the linear complementarity problem,” paper presented at the TIMS/ORSA meeting (New Orleans, May 4–6, 1987).

  21. K.M. Thompson, “A parallel asynchronous successive overrelaxation algorithm for solving the linear complementarity problem,” Computer Sciences Technical Report #705, University of Wisconsin (Madison, Wisconsin, 1987).

    Google Scholar 

  22. J.F. Traub,Iterative Methods for the Solution of Equations (Prentice-Hall, New Jersey, 1964).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, The Johns Hopkins University, Charles & 34th Street, 21218, Baltimore, Maryland, USA

    Jong-Shi Pang

  2. Department of Management Information Systems, National Cheng-Chi University, Taipei, Taiwan, Republic of China

    Jiann-Min Yang

Authors
  1. Jong-Shi Pang
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  2. Jiann-Min Yang
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Additional information

This work was based on research supported by the National Science Foundation under grant ECS-8407240 and by a University Research and Development grant from Cray Research Inc. The research was initiated when the authors were with the University of Texas at Dallas.

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Pang, JS., Yang, JM. Parallel Newton methods for the nonlinear complementarity problem. Mathematical Programming 42, 407–420 (1988). https://doi.org/10.1007/BF01589414

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  • DOI: https://doi.org/10.1007/BF01589414

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