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A Levenberg–Marquardt Method for Nonlinear Complementarity Problems Based on Nonmonotone Trust Region and Line Search Techniques

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Abstract

Using the FB function, we propose a new Levenberg–Marquardt algorithm for nonlinear complementarity problem. To obtain the global convergence, the algorithm uses the nonmonotone trust region and line search techniques under a convenient boundedness assumption. Furthermore, we get local superlinear/quadratic convergence of the algorithm under a nonsingularity condition. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, B., Harker, P.T.: Smoothing approximations to nonlinear complementarity problems. SIAM J. Optim. 7, 403–423 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Fischer, A.: Solution of monotone complementarity problems with locally Lipschitz functions. Math. Prog. 76, 669–713 (1997)

    Google Scholar 

  5. 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. 67, 519–540 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ma, C., Tang, J., Chen, X.H.: A globally convergent Levenberg-Marquardt method for solving nonlinear complementarity problem. Math. Comput. 192, 370–381 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Xie, Y., Ma, C.: A smoothing Levenberg-Marquardt altorithm for solving a class of stochastic linear complementarity problem. Appl. Math. Comput. 217, 4459–4472 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Wang, X., Ma, C., Li, Meiyan: A smoothing trust region method for NCPs based on the smoothing generalized Fischer-Burmeister function. J. Comput. Math. 29, 261–286 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Chen, B., Ma, C.: Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem. Nonlinear Anal. Real World Appl. 12, 1250–1263 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tang, J., Ma, C.: A smoothing Newton method for solving a class of stochastic linear complementarity problems. Nonlinear Anal. Real World Appl. 12, 3585–3601 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, B., Ma, C.: A new smoothing Broyden-like method for solving nonlinear complementarity problem with a P0-function. J. Glob. Optim. 51, 473–495 (2011)

    Article  MATH  Google Scholar 

  12. Ma, C.: A new smoothing and regularization Newton method for \(P_0\)-NCP. J. Glob. Optim. 48(2), 241–261 (2010)

    Article  MATH  Google Scholar 

  13. Nocedal, J., Yuan, Y.X.: Combining trust region and line search techniques. In: Advances in Nonlinear Programming (Beijing, 1996), Appl. Optim., Vol .14, pp. 153–175. Kluwer Acad. Publ., Dordrecht (1998)

  14. Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Yu, H., Pu, D.: Smoothing Levenberg-Marquardt method for general nonlinear complementarity problems under local error bound. Appl. Math. 35, 1337–1348 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Yang, Y., Qi, L.: Smoothing trust region methods for nonlinear complementarity problems with \(P_0\)-function. Ann. Oper. Res. 133, 99–117 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ahn, B.H.: Iterative methods for linear complementarity problems with upperbounds on primary variables. Math. Prog. 26, 295–315 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kanzow, C.: Global convergence properties of some iterative methods for linear complementarity problems. SIAM J. Optim. 6, 326–341 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kanzow, C.: Some equation-based methods for the nonlinear Complementarity problem. Optim. Method. Softw. 3, 327–340 (1994)

    Article  Google Scholar 

  22. Dirkse, S., Ferris, M.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319–45 (1995)

    Article  Google Scholar 

  23. Levenberg, K.: A method for the solution of certain nonlinear problems in least squares. Quart. Appl. Math. 2, 164–166 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  24. Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear inequalities. SIAM J. Appl. Math. 11, 431–441 (1963)

    Article  MATH  Google Scholar 

  25. Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg-Marquardt method. Computing 15, 239–249 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Mathe. Prog. 58(1–3), 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dan, H., Yamashita, N., Fukushima, M.: Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions. Optimiz. Methods Softw. 17(4), 605–626 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, J., Zhang, X.: A smoothing Levenberg-Marquardt method for NCP. Appl. Math. Comput. 178, 212–228 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Du, S., Gao, Y.: Convergence analysis of nonmonotone Levenberg–Marquardt algorithms for complementarity. Appl. Math. Comput. 216, 1652–1659 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Fan, J.Y.: A modified Levenberg–Marquardt algorithm for singular system of nonlinear equations. Math. Comput. 5, 625–636 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Yamashita, N., Fukushima, M.: Modified Newton methods for solving a semismooth reformulation of monotone complementarity problem. Math. Prog. 76, 469–491 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Gao, D.Y.: Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42, 1161–1193 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Powell, M.J.D.: Convergence properties of a class of minimization algorithms. Nonlinear Prog. 2, 1–27 (1975)

    MATH  Google Scholar 

  35. Dennis, J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  36. Ruggiero, M.A.G., Martínez, J.M., Santos, S.A.: Solving nonsmooth equations by means of quasi-Newton methods with globalization. In: Recent Advances in Nonsmooth Optimization, pp. 121–140. World Scientific, Singapore (1995)

  37. Spedicato, E., Huang, Z.: Numerical experience with Newton-like methods for nonlinear algebraic systems. Computing 58, 69–89 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350007, People’s Republic of China

    Bin Fan & Changfeng Ma

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Bin Fan

  3. College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou, 350007, People’s Republic of China

    Aidi Wu

  4. School of Electronics and Information Engineering, Fuqing Branch of Fujian Normal University, Fuqing, 350300, People’s Republic of China

    Bin Fan & Chao Wu

Authors
  1. Bin Fan
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  2. Changfeng Ma
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  3. Aidi Wu
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  4. Chao Wu
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Corresponding author

Correspondence to Bin Fan.

Additional information

The project is supported by National Natural Science Foundation of China (Grant no. 11071041).

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Fan, B., Ma, C., Wu, A. et al. A Levenberg–Marquardt Method for Nonlinear Complementarity Problems Based on Nonmonotone Trust Region and Line Search Techniques. Mediterr. J. Math. 15, 118 (2018). https://doi.org/10.1007/s00009-018-1168-y

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  • DOI: https://doi.org/10.1007/s00009-018-1168-y

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