Skip to main content
SpringerLink
Log in

Some three-term conjugate gradient methods with the inexact line search condition

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

The three-term conjugate gradient methods solving large-scale optimization problems are favored by many researchers because of their nice descent and convergent properties. In this paper, we extend some new conjugate gradient methods, and construct some three-term conjugate gradient methods. An remarkable property of the proposed methods is that the search direction always satisfies the sufficient descent condition without any line search. Under the standard Wolfe line search, the global convergence properties of the proposed methods are proved merely by assuming that the objective function is Lipschitz continuous. Preliminary numerical results and comparisons show that the proposed methods are efficient and promising.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 5, 409–432 (1952)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Polak, E., Ribire, G.: Note surla convergence des methodse de directions conjugees. Rev Francaise Imformmat Recherche Opertionelle 16, 35–43 (1969)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (1987)

    MATH  Google Scholar 

  6. Liu, Y., Story, C.: Efficient generalized conjugate gradient algorithms. Part 1: theory. J. Optim. Theory Appl. 69, 129–137 (1992)

    Article  Google Scholar 

  7. Dai, Y.H., Yuan, Y.X.: Nonlinear conjugate gradient with a strong global convergence property. SIAM J. Optim. 10, 177–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rivaie, M., Mamat, M., June, L.W., Mohd, I.: A new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 218, 11323–11332 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Rivaie, M., Mamat, M., Abashar, A.: A new class of nonlinear conjugate gradient coefficients with exact and inexact line searches. Appl. Math. Comput. 268, 1152–1163 (2015)

    MathSciNet  Google Scholar 

  10. Nocedal, J.: Updating quasi-Newton matrixes with limited storage. Math. Comput. 35, 773–782 (1980)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, L., Zhou, W.J., Li, D.H.: Some descent three-term conjugate gradient methods and their global convergence. Optim. Methods. Softw. 22, 697–711 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Narushima, Y., Yabe, H., Ford, J.A.: A three-term conjugate gradient method with sufficient descent property for unconstrained optimization. SIAM J. Optim. 21, 212–230 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, J.K., Li, S.J.: New three-term conjugate gradient method with guaranteed global convergence. Int. J. Comput. Math. 91, 1744–1754 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, J.K., Wu, X.S.: New three-term conjugate gradient method for solving unconstrained optimization problems. ScienceAsia 40, 295–300 (2014)

    Article  Google Scholar 

  16. Andrei, N.: A simple three-term conjugate gradient algorithm for unconstrained optimization. J. Comput. Appl. Math. 241, 19–29 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Andrei, N.: On three-term conjugate gradient algorithm for unconstrained optimization. Appl. Math. Comput. 219, 6316–6327 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Andrei, N.: An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization. Numer. Algorithms 65, 859–874 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Al-Bayati, A.Y., Sharif, W.H.: A new three-term conjugate gradient method for unconstrained optimization. Can. J. Sci. Eng. Math. 1, 108–124 (2010)

    Google Scholar 

  20. Nazareth, L.: A conjugate direction algorithm without line search. J. Optim. Theory Appl. 23, 373–387 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wolfe, P.: Convergence conditions for ascent methods. SIAM Rev. 11, 226–235 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zoutendijk, G.: Nonlinear Programming, Computational Methods. In: Abadie, J. (ed.) Integer and Nonlinear Programming, pp. 38–86. North-Holland, Amsterdam (1970)

    Google Scholar 

  23. Bongartz, I., Conn, A.R., Gould, N.I.M., Toint., P.L.: CUTE: constrained and unconstrained testing environments. ACM Trans. Math. Softw. 21, 123–160 (1995)

    Article  MATH  Google Scholar 

  24. Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10, 147–161 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China

    J. K. Liu, Y. M. Feng & L. M. Zou

Authors
  1. J. K. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. M. Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. M. Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. K. Liu.

Additional information

This research is funded by Chongqing Research Program of Basic Research and Frontier Technology (Grant No.: cstc2017jcyjAX0318), the fund of Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant Nos.: KJ1710251, KJ1501003), Program for Innovation Team Building at Institutions of Higher Education in Chongqing (Grant number: CXTDX201601035) and Project Supported by Chongqing Municipal Key Laboratory of Institutions of Higher Education (Grant No. [2017]3).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J.K., Feng, Y.M. & Zou, L.M. Some three-term conjugate gradient methods with the inexact line search condition. Calcolo 55, 16 (2018). https://doi.org/10.1007/s10092-018-0258-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-018-0258-3

Keywords

Mathematics Subject Classification

Search

Navigation