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Norm descent conjugate gradient methods for solving symmetric nonlinear equations

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Abstract

Nonlinear conjugate gradient method is very popular in solving large-scale unconstrained minimization problems due to its simple iterative form and lower storage requirement. In the recent years, it was successfully extended to solve higher-dimension monotone nonlinear equations. Nevertheless, the research activities on conjugate gradient method in symmetric equations are just beginning. This study aims to developing, analyzing, and validating a family of nonlinear conjugate gradient methods for symmetric equations. The proposed algorithms are based on the latest, and state-of-the-art descent conjugate gradient methods for unconstrained minimization. The series of proposed methods are derivative-free, where the Jacobian information is needless at the full iteration process. We prove that the proposed methods converge globally under some appropriate conditions. Numerical results with differentiable parameter’s values and performance comparisons with another solver CGD to demonstrate the superiority and effectiveness of the proposed algorithms are reported.

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References

  1. An, X.M., Li, D.H., Xiao, Y.: Sufficient descent directions in unconstrained optimization. Comput. Optim. Appl. 48, 515–532 (2011)

    Article  MathSciNet  Google Scholar 

  2. Barzilai, J., Borwein, J.M.: Two point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  Google Scholar 

  3. Cheng, W., Chen, Z.: Nonmonotone spectral method for large-scale symmetric nonlinear equations. Numer. Algorithms 62, 149–162 (2013)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. Conn, A.R., Gould, N.I.M., Toint, PhL: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21, 123–160 (1995)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dai, Y.-H., Kou, C.-X.: A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search. SIAM J. Optim. 23, 296–320 (2013)

    Article  MathSciNet  Google Scholar 

  8. Gu, G.-Z., Li, D.-H., Qi, L., Zhou, S.-Z.: Desecent direction of quasi-Newton methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 40, 1763–1774 (2002)

    Article  MathSciNet  Google Scholar 

  9. Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)

    Article  MathSciNet  Google Scholar 

  10. Hager, W.W., Zhang, H.: Algorithm 851: \(\text{ CG }\_\text{ DESCENT }\), a conjugate gradient method with guaranteed descent. ACM Trans. Math. Soft. 32, 113–137 (2006)

    Article  MathSciNet  Google Scholar 

  11. 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  MathSciNet  Google Scholar 

  12. Li, D.-H., Wang, X.: A modified Fletcher–Reeves-type derivative-free method for symmetric nonlinear equations. Numer. Algebra Control Optim. 1, 71–82 (2011)

    Article  MathSciNet  Google Scholar 

  13. Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405, 310–319 (2013)

    Article  MathSciNet  Google Scholar 

  14. Yuan, G., Lu, X.: A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput. Math. Appl. 55, 11–129 (2008)

    MathSciNet  Google Scholar 

  15. Yuan, G., Lu, X., Wei, Z.: BFGS trust-region method for symmetric nonlinear equations. J. Comput. Appl. Math. 230, 44–58 (2009)

    Article  MathSciNet  Google Scholar 

  16. Zhang, L., Zhou, W., Li, D.: A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)

    Article  MathSciNet  Google Scholar 

  17. Zhang, L., Zhou, W., Li, D.-H.: Global convergence of a modied Fletcher–Reeves conjugate gradient method with Armijo-type line search. Numer. Math. 104, 561–572 (2006)

    Article  MathSciNet  Google Scholar 

  18. Zhou, W., Shen, D.: Convergence properties of an iterative method for solving symmetric nonlinear equations. J. Optim. Theory Appl. doi:10.1007/s10957-014-0547-1

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

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank two anonymous referees for their constructive suggestions which improved the paper greatly. This work is supported by Natural Science Foundation of Henan Province Grant 13HASTIT050 and 2011GGJS030.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Henan University, Kaifeng , 475000, China

    Yunhai Xiao & Chunjie Wu

  2. National Center for Theoretical Sciences (South), National Cheng Kung University, Tainan , 700, Taiwan

    Soon-Yi Wu

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  1. Yunhai Xiao
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  2. Chunjie Wu
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  3. Soon-Yi Wu
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Correspondence to Yunhai Xiao.

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Xiao, Y., Wu, C. & Wu, SY. Norm descent conjugate gradient methods for solving symmetric nonlinear equations. J Glob Optim 62, 751–762 (2015). https://doi.org/10.1007/s10898-014-0218-7

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  • DOI: https://doi.org/10.1007/s10898-014-0218-7

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