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A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations

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Abstract

A fully derivative-free conjugate residual method, using secant condition, is introduced to solve general large-scale nonlinear equations. Under some conditions, global and linear convergence of the proposed method is established by adopting some backtracking type line search. Some numerical results compared with two existing derivative-free methods are reported to show its efficiency.

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References

  1. Yuan, Y.: Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numer. Alge. Cont. Optim. 1, 15–34 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Zhou, W.: A Gauss-Newton-based BFGS method for symmetric nonlinear least squares problems. Pacific J. Optim. 9, 373–389 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Zhou, W., Chen, X.: Global convergence of a new hybrid Gauss-Newton structured BFGS methods for nonlinear least squares problems. SIAM J. Optim. 20, 2422–2441 (2010)

    Article  MathSciNet  Google Scholar 

  5. Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg-Marquardt method. Computing (Supp.) 15, 237–249 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Li, Q., Li, D.: A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31, 1625–1635 (2011)

    Article  MathSciNet  Google Scholar 

  7. Zhou, W., Shen, D.: Convergence properties of an iterative method for solving symmetric nonlinear equations. J. Optim. Theory Appl. 164, 277–289 (2015)

    Article  MathSciNet  Google Scholar 

  8. Zhou, W., Shen, D.: An inexact PRP conjugate gradient method for symmetric nonlinear equations. Numer. Funct. Anal. Optim. 35, 370–388 (2014)

    Article  MathSciNet  Google Scholar 

  9. Zhou, W., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1–7 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Zhou, W., Li, D.: On the Q-linear convergence rate of a class of methods for monotone nonlinear equations. Pacific J. Optim. 14, 723–737 (2018)

    MathSciNet  Google Scholar 

  11. La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)

    Article  MathSciNet  Google Scholar 

  12. La Cruz, W., Martinez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)

    Article  MathSciNet  Google Scholar 

  13. Cheng, W., Xiao, Y., Hu, Q.: A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. J. Comput. Appl. Math. 224, 11–19 (2009)

    Article  MathSciNet  Google Scholar 

  14. Dai, Y., Liao, L.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43, 87–101 (2001)

    Article  MathSciNet  Google Scholar 

  15. Polak, E., Ribière, G.: Note sur la convergence de méthodes de directions conjuguées. Rev. Fr. Inform. Rech. Oper. 16, 35–43 (1969)

    MATH  Google Scholar 

  16. Polyak, B.T.: The conjugate gradient method in extreme problems. USSR Comput. Math. Math. Phys. 9, 94–112 (1969)

    Article  Google Scholar 

  17. Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49, 409–436 (1952)

    Article  MathSciNet  Google Scholar 

  18. Grippo, L., Lucidi, S.: A globally convergent version of the Polak-Ribière conjugate gradient method. Math. Program. 78, 375–391 (1997)

    MATH  Google Scholar 

  19. Dai, Y.: Conjugate gradient methods with Armijo-type line searches. Acta Math. Appl. Sin. -Engl. Ser. 18, 123–130 (2002)

    Article  MathSciNet  Google Scholar 

  20. Zhou, W.: A short note on the global convergence of the unmodified PRP method. Optim. Lett. 7, 1367–1372 (2013)

    Article  MathSciNet  Google Scholar 

  21. Zhou, W., Li, D.: On the convergence properties of the unmodified PRP method with a non-descent line search. Optim. Methods Softw. 29, 484–496 (2014)

    Article  MathSciNet  Google Scholar 

  22. Dai, Y.: Nonlinear Conjugate Gradient Methods. Wiley Encyclopedia of Operations Research and Management Science (2011)

  23. Dennis, J.E., Moré, J.J.: A characterization of superlinear convergence and its applications to quasi-Newton methods. Math. Comput. 28, 549–560 (1974)

    Article  MathSciNet  Google Scholar 

  24. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the editor and the referees whose very helpful suggestions led to much improvement of this paper.

Funding

This work was supported by the SRF (13B137) of Hunan Provincial Education Department, the NSF (14JJ3084) of Hunan Province and Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering.

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Correspondence to Li Zhang.

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Zhang, L. A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations. Numer Algor 83, 1277–1293 (2020). https://doi.org/10.1007/s11075-019-00725-7

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  • DOI: https://doi.org/10.1007/s11075-019-00725-7

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