Skip to main content
SpringerLink
Log in

Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems

  • Original paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

It is well known that the sufficient descent condition is very important to the global convergence of the nonlinear conjugate gradient method. In this paper, some modified conjugate gradient methods which possess this property are presented. The global convergence of these proposed methods with the weak Wolfe–Powell (WWP) line search rule is established for nonconvex function under suitable conditions. Numerical results are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dai Y.: A nonmonotone conjugate gradient algorithm for unconstrained optimization. J. Syst. Sci. Complex. 15, 139–145 (2002)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Dai Y., Yuan Y.: A nonlinear conjugate gradient with a strong global convergence properties. SIAM J. Optim. 10, 177–182 (2000)

    Article  Google Scholar 

  4. Dai, Y., Yuan, Y.: Nonlinear conjugate gradient Methods. Shanghai Scientific and Technical Publishers (1998)

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Fletcher R.: Practical method of optimization, vol I: unconstrained optimization, 2nd edn. Wiley, New York (1997)

    Google Scholar 

  7. Fletcher R., Reeves C.: Function minimization bu conjugate gradients. Comput. J. 7, 149–154 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gibert J.C., Nocedal J.: Global convergence properties of conugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992)

    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  MATH  MathSciNet  Google Scholar 

  10. Hager W.W., Zhang H.: Algorithm 851: CG D ESCENT, A conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32, 113–137 (2006)

    Article  MathSciNet  Google Scholar 

  11. Hager W.W., Zhang H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35–58 (2006)

    MATH  Google Scholar 

  12. Hestenes M.R., Stiefel E.: Method of conjugate gradient for solving linear equations. J, Res. Nat. Bur. Stand. 49, 409–436 (1952)

    MATH  MathSciNet  Google Scholar 

  13. Li G., Tang C., Wei Z.: New conjugacy condition and related new conjugate gradient methods for unconstrained optimization problems. J. Comput. Appl. Math. 202, 532–539 (2007)

    Article  MathSciNet  Google Scholar 

  14. Liu Y., Storey C.: Effcient generalized conjugate gradient algorithms part 1: theory. J. Optim. Theory Appl. 69, 17–41 (1992)

    Google Scholar 

  15. Polak E., Ribiere G.: Note sur la xonvergence de directions conjugees. Rev. Francaise informat Recherche Operatinelle, 3e Annee 16, 35–43 (1969)

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  17. Powell, M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. Lecture Notes in Mathematics, vol. 1066, pp. 122–141. Spinger, Berlin (1984)

  18. Powell M.J.D.: Convergence properties of algorithm for nonlinear optimization. SIAM Rev. 28, 487–500 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wei Z., Li G., Qi L.: New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Appl. Math. Comput. 179, 407–430 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wei Z., Yao S., Liu L.: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341–1350 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yu, G.H.: Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems. Thesis of Doctor’s Degree, Sun Yat-Sen University (2007)

  22. Yu, G., Guan, L.: Chen Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optimization methods and software (in press, 2007)

  23. Yuan Y., Sun W.: Theory and methods of optimization. Science Press of China, Beijing (1999)

    Google Scholar 

  24. Zhang L., Zhou W., Li D.: A descent modified Polak-Ribière-Polyak conjugate method and its global convergence. IMA J. Numer. Anal. 26, 629–649 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zoutendijk G.: Nonlinear programming computational methods. In: Abadie, J.(eds) Integer and nonlinear programming, pp. 37–86. North-Holland, Amsterdam (1970)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, People’s Republic of China

    Gonglin Yuan

Authors
  1. Gonglin Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gonglin Yuan.

Additional information

This work is supported by Guangxi University SF grands X061041 and China NSF grands 10761001.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yuan, G. Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optim Lett 3, 11–21 (2009). https://doi.org/10.1007/s11590-008-0086-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-008-0086-5

Keywords

Search

Navigation