Skip to main content
SpringerLink
Log in

Conjugate Gradient Methods with Armijo-type Line Searches

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica Aims and scope Submit manuscript

Abstract

Two Armijo-type line searches are proposed in this paper for nonlinear conjugate gradient methods. Under these line searches, global convergence results are established for several famous conjugate gradient methods, including the Fletcher-Reeves method, the Polak-Ribiére-Polyak method, and the conjugate descent method.

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. Al-Baali, M. Descent Property and Global Convergence of the Fletcher-Reeves Method with Inexact Line Search. IMA J. Numer. Anal., 1985, 5: 121–124

    Article  MATH  MathSciNet  Google Scholar 

  2. Armijo, L. Minimization of Functions having Lipschitz Continuous First Partial Derivatives. Pacific Journal of Mathematics, 1966, 16(1): 1–3

    MATH  MathSciNet  Google Scholar 

  3. Dai, Y.H. Further Insight Into the Convergence of the Fletcher-Reeves Method. Sciences in China (Series A), 1999, 42(9): 905–916

    MATH  Google Scholar 

  4. Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F., Yin, H.X., Yuan, Y. Convergence Properties of Nonlinear Conjugate Gradient Methods. SIAM Journal on Optimization, 1999, 10(2): 345–358

    Article  MATH  Google Scholar 

  5. Dai, Y.H., Yuan, Y. A Nonlinear Conjugate Gradient Method with A Strong Global Convergence Property. SIAM Journal on Optimization, 1999, 10(1): 177–182

    Article  MATH  MathSciNet  Google Scholar 

  6. Dai, Y.H., Yuan, Y. Convergence of the Fletcher-Reeves Method under A Generalized Wolfe Search. Journal Computational Mathematics of Chinese Universities, 1996, 2: 142–148

    Google Scholar 

  7. Dai, Y.H., Yuan, Y. Convergence Properties of the Conjugate Descent Method. Mathematical Advances, 1996, 6: 552–562

    Google Scholar 

  8. Dai, Y.H., Yuan, Y. Convergence Properties of the Fletcher-Reeves Method. IMA J. Numer. Anal., 1996, 16(2): 155–164

    Article  MATH  MathSciNet  Google Scholar 

  9. Fletcher, R. Practical Methods of Optimization, Vol.1: Unconstrained Optimization. John Wiley & Sons, New York, 1987

  10. Fletcher, R., Reeves, C. Function Minimization by Conjugate Gradients. Comput. J., 1964, 7: 149–154

    Article  MATH  MathSciNet  Google Scholar 

  11. Gilbert, J.C., Nocedal, J. Global Convergence Properties of Conjugate Gradient Methods for Optimization. SIAM Journal on Optimization, 1992, 2(1): 21–42

    Article  MATH  MathSciNet  Google Scholar 

  12. Grippo, L., Lucidi, S. A globally convergent version of the Polak-Ribière Conjugate Gradient method. Math. Prog., 1997, 78: 375–391

    Article  MathSciNet  Google Scholar 

  13. Hestenes, M.R., Stiefel E.L. Methods of Conjugate Gradients for Solving Linear Systems. J. Res. Nat. Bur. Standards Sect., 1952, 5(49): 409–436

    MathSciNet  Google Scholar 

  14. Liu, D.C., Nocedal, J. On the Limited Memory Method for Large Scale Optimization. Math. Prog., 1989, 45: 503–528

    Article  MATH  Google Scholar 

  15. Liu, G.H., Han, J.Y., Yin, H.X. Global Convergence of the Fletcher-Reeves Algorithm with an Inexact Line Search. Appl. Math. J. Chinese Univ. (Series B), 1995, 10: 75–82

    MATH  Google Scholar 

  16. Nocedal, J. Large Scale Unconstrained Optimization. Research Report, Northwestern University, 1996

  17. Polak, E., Ribière, G. Note sur la Convergence de Directions Conjugées. Rev. Francaise Informat Recherche Opertionelle, 3e Année, 1969, 16: 35–43

    MathSciNet  Google Scholar 

  18. Polyak, B.T. The Conjugate Gradient Method in Extremem Problems. USSR Comp. Math. and Math. Phys., 1969, 9: 94–112

    Article  Google Scholar 

  19. Powell, M.J.D. Convergence Properties of Algorithms for Nonlinear Optimization. SIAM Review, 1986, 28: 487–500

    Article  MATH  MathSciNet  Google Scholar 

  20. Powell, M.J.D. Nonconvex Minimization Calculations and the Conjugate Gradient Method. In: D.F. Griffths ed., Numerical Analysis, Lecture Notes in Mathematics, 1066, Springer-Verlag, Berlin, 1984, 122–141

  21. Touati-Ahmed, D., Storey, C. Efficient Hybrid Conjugate Gradient Techniques. J. Optimization Theory Appl., 1990, 64: 379–397

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang, C.Y., Zhang, Y.Z. Global Convergence Properties of s-related Conjugate Gradient Methods. Chinese Science Bulletin, 1998, 43(23): 1959–1965

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 10080, China

    Yu-Hong Dai

Authors
  1. Yu-Hong Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-Hong Dai.

Additional information

Supported by the National Natural Science Foundation of China (No.19801033 and 10171104).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dai, YH. Conjugate Gradient Methods with Armijo-type Line Searches. Acta Mathematicae Applicatae Sinica, English Series 18, 123–130 (2002). https://doi.org/10.1007/s102550200010

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s102550200010

Keywords

2000 MR Subject Classification

Search

Navigation