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A smoothing inexact Newton method for P 0 nonlinear complementarity problem

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Abstract

We first propose a new class of smoothing functions for the nonlinear complementarity function which contains the well-known Chen-Harker-Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P 0 nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.

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References

  1. Chen B, Harker P T. A non-interior-point continuation method for linear complementarity problems. SIAM J Matrix Anal Appl, 1993, 14(4): 1168–1190

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen B, Ma C. A new smoothing Broyden-like method for solving nonlinear complementarity problem with a P 0 function. J Global Optim, 2011, 51(3): 473–495

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen X, Qi L, Sun D. Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math Comput, 1998, 67(222): 519–540

    Article  MathSciNet  MATH  Google Scholar 

  4. Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley, 1983

    MATH  Google Scholar 

  5. Ferris M C, Pang J S. Engineering and economic applications of complementarity problems. SIAM Review, 1997, 39(4): 669–713

    Article  MathSciNet  MATH  Google Scholar 

  6. Geiger C, Kanzow C. On the resolution of monotone complementarity problems. Comput Optim Appl, 1996, 5: 155–173

    Article  MathSciNet  MATH  Google Scholar 

  7. Harker P T, Pang J S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math Program, 1990, 48(2): 161–220

    Article  MathSciNet  MATH  Google Scholar 

  8. Hotta K, Yoshise A. Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems. Math Program, 1999, 86: 105–133

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang Z, Han J, Chen Z. Predictor-Corrector Smoothing Newton Method, Based on a New Smoothing Function, for Solving the Nonlinear Complementarity Problem with a P 0 Function. J Optim Theory Appl, 2003, 117(1): 39–68

    Article  MathSciNet  MATH  Google Scholar 

  10. Kanzow C. Some noninterior continuation methods for linear complementarity problems. SIAM J Matrix Anal Appl, 1996, 17(4): 851–868

    Article  MathSciNet  MATH  Google Scholar 

  11. Kanzow C, Kleinmichel H. A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput Optim Appl, 1998, 11: 227–251

    Article  MathSciNet  MATH  Google Scholar 

  12. Luca T D, Facchinei F, Kanzow C. A semismooth equation approach to the solution of nonlinear complementarity problems. Math Program, 1996, 75(3): 407–439

    Article  MATH  Google Scholar 

  13. Ma C, Chen X. The convergence of a one-step smoothing Newton method for P 0-NCP base on a new smoothing NCP-function. J Comput Appl Math, 2008, 216(1): 1–13

    Article  MathSciNet  MATH  Google Scholar 

  14. Mathiesen L. An algorithm based on a sequence of a linear complementarity problems applied to a Walrasian equilibrium model: an example. Math Program, 1987, 37(1): 1–18

    Article  MathSciNet  MATH  Google Scholar 

  15. Mifflin R. Semismooth and semiconvex functions in constrained optimization. SIAM J Control Optim, 1977, 15(6): 957–972

    Article  MathSciNet  Google Scholar 

  16. Natasa K, Sanja R. Globally convergent Jacobian smoothing inexact Newton methods for NCP. Comput Optim Appl, 2008, 41(2): 243–261

    Article  MathSciNet  MATH  Google Scholar 

  17. Pang J S, Gabriel A. NE/SQP: a robust algorithm for nonlinear complementarity problems. Math Program, 1993, 60: 295–337

    Article  MathSciNet  MATH  Google Scholar 

  18. Qi L. Convergence analysis of some algorithms for solving nonsmooth equations. Math Oper Res, 1993, 18(1): 227–244

    Article  MathSciNet  MATH  Google Scholar 

  19. Qi L, Sun D, Zhou G. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math Program, 2000, 87(1): 1–35

    MathSciNet  MATH  Google Scholar 

  20. Qi L, Sun J. A nonsmooth version of Newton,s method. Math Programming, 1993, 58(3): 353–367

    Article  MathSciNet  MATH  Google Scholar 

  21. Sun D, Qi L. On NCP functions. Comput Optim Appl, 1999, 13: 201–220

    Article  MathSciNet  MATH  Google Scholar 

  22. Xie D, Ni Q. An incomplete Hessian Newton minimization method and its application in a chemical database problem. Comput Optim Appl, 2009, 44(3): 467–485

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang X, Jiang H, Wang Y. A smoothing Newton method for generalized nonlinear complementarity problem over a polyhedral cone. J Comput Appl Math, 2008, 212: 75–85

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang J, Zhang K. A variant smoothing Newton method for P 0-NCP based on a new smoothing function. J Comput Appl Math, 2009, 225(1): 1–8

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Weifang University, Weifang, 261061, China

    Haitao Che & Meixia Li

  2. School of Management Science, Qufu Normal University, Rizhao, 276800, China

    Haitao Che & Yiju Wang

Authors
  1. Haitao Che
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  2. Yiju Wang
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  3. Meixia Li
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Correspondence to Haitao Che.

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Che, H., Wang, Y. & Li, M. A smoothing inexact Newton method for P 0 nonlinear complementarity problem. Front. Math. China 7, 1043–1058 (2012). https://doi.org/10.1007/s11464-012-0245-y

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  • DOI: https://doi.org/10.1007/s11464-012-0245-y

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