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A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations

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Abstract

In this paper, a modified nonmonotone BFGS algorithm is developed for solving a smooth system of nonlinear equations. Different from the existent techniques of nonmonotone line search, the value of an algorithmic parameter controlling the magnitude of nonmonotonicity is updated at each iteration by the known information of the system of nonlinear equations such that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence of the algorithm is established for solving a generic nonlinear system of equations. Implementing the algorithm to solve some benchmark test problems, the obtained numerical results demonstrate that it is more effective than some similar algorithms available in the literature.

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References

  1. Gu, G.Z., Li, D.H., Qi, L.Q., Zhou, S.Z.: Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 40, 1763–1774 (2003)

    Article  MathSciNet  Google Scholar 

  2. Yuan, G.L., Liu, X.W.: A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput. Math. Appl. 55, 116–129 (2009)

    Article  Google Scholar 

  3. Guo, Q., Liu, J.G., Wang, D.H.: A modified BFGS method and its superlinear convergence in nonconvex minimization with general line search rule. J. Appl. Math. Comput. 28, 435–446 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Guo, Q., Liu, J.G.: Global convergence of a modified BFGS-type method for unconstrained nonconvex minimization. J. Appl. Math. Comput. 24(1–2), 325–331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Li, D.H., Fukushima, M.: On the global convergence of the BFGS method for nonconvex unconstrained optimization problems. SIAM J. Optim. 11(4), 1054–1064 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Li, D.H., Fukushima, M.: A globally and superlinearly convergent Gauss-Newton-based BFGS methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li, D.H., Cheng, W.Y.: Recent progress in global convergence of quasi-Newton methods for nonlinear equations. Hokkaido Math. J. 36, 729–743 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu, J.G., Guo, Q.: Global convergence properties of the modified BFGS method associating with general line search model. J. Appl. Math. Comput. 18(1–2), 195–205 (2004)

    Article  Google Scholar 

  9. Zhou, W.J., Zhang, L.: Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization. J. Comput. Appl. Math. 223, 40–47 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Birgin, E.G., Krejic, N., Martínez, J.M.: Globally convergent inexact quasi-Newton methods for solving nonlinear systems. Numer. Algorithms 32, 249–260 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  12. Martínez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97–122 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grippo, L., Lampariello, F., Lucidi, S.: Newton-type algorithms with nonmonotone line search for large-scale unconstrained optimization. Syst. Model. Optim. 113, 187–196 (1988)

    Article  MathSciNet  Google Scholar 

  14. Grippo, L., Lampariello, F., Lucidi, S.: A class of nonmonotone stabilization methods in unconstrained optimization. Numerische Mathematik, vol 59, pp. 779–805 (1991)

  15. Xiao, Y.H., Sun, H.J., Wang, Z.G.: A globally convergent BFGS method with nonmonotone line search for non-convex minimization. J. Comput. Appl. Math. 230, 95–106 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Yuan, G.L., Wei, Z.X.: The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions. Acta Mathematica Sinica (English Series) 24(1), 35–42 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhu, D.T.: Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations. Appl. Math. Comput. 161, 875–895 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Shi, Z.J., Shen, J.: Convergence of nonmonotone line search method. J. Comput. Appl. Math. 193, 397–412 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Luksan, L.: Inexact trust region method for large sparse systems of nonlinear equations. Comput. Optim. Appl. 81(3), 569–590 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to express their thanks to the three anonymous referees for their constructive comments on the paper, which have greatly improved its presentation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, People’s Republic of China

    Zhong Wan, Yu Chen, Shuai Huang & Dongdong Feng

Authors
  1. Zhong Wan
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  2. Yu Chen
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  3. Shuai Huang
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  4. Dongdong Feng
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Corresponding author

Correspondence to Zhong Wan.

Additional information

This research is supported by the National Natural Science Foundation of China (Grant No. 71071162, 71221061) and Natural Science Foundation of Hunan Province (Grant No. 13JJ3002).

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Wan, Z., Chen, Y., Huang, S. et al. A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations. Optim Lett 8, 1845–1860 (2014). https://doi.org/10.1007/s11590-013-0678-6

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  • DOI: https://doi.org/10.1007/s11590-013-0678-6

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