Skip to main content
SpringerLink
Log in

Applying Powell’s symmetrical technique to conjugate gradient methods

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A new conjugate gradient method is proposed by applying Powell’s symmetrical technique to conjugate gradient methods in this paper. Using Wolfe line searches, the global convergence of the method is analyzed by using the spectral analysis of the conjugate gradient iteration matrix and Zoutendijk’s condition. Based on this, some concrete descent algorithms are developed. 200s numerical experiments are presented to verify their performance and the numerical results show that these algorithms are competitive compared with the PRP+ algorithm. Finally, a brief discussion of the new proposed method is given.

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. 5, 121–124 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrei, N.: Scaled conjugate gradient algorithms for unconstrained optimization. Comput. Optim. Appl. 38, 401–416 (2007). doi:10.1007/s10589-007-9055-7

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrei, N.: Another hybrid conjugate gradient algorithm for unconstrained optimization. Numer. Algorithms 47, 143–156 (2008). doi10.1007/s11075-007-9152-9

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Birgin, E.G., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43, 117–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buckley, A.: Extending the relationship between the conjugate gradient and BFGS algorithms. Math. Program. 15, 343–348 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2(1), 21–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49(6), 409–439 (1952)

    MathSciNet  MATH  Google Scholar 

  11. Liu, Y., Storey, C.: Efficient generalized conjugate gradient algorithms, part 1: theory. J. Optim. Theory Appl. 69, 129–137 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  13. Perry, A.: A modified conjugate gradient algorithm. Oper. Res. Tech. Notes 26(6), 1073–1078 (1978)

    MathSciNet  MATH  Google Scholar 

  14. Powell, M.J.D.: A new algorithm for unconstrained optimization. In: Rosen, J.B., Mangasarian, O.L., Ritter, K. (eds.) Nonlinear Programming, pp. 31–66. Academic Press, New York (1970)

    Google Scholar 

  15. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shanno, D.F.: Conjugate gradient methods with inexact searches. Math. Oper. Res. 3, 244–256 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shanno, D.F., Phua, K.H.: Algorithm 500, minimization of unconstrained multivariate functions. ACM Trans. Math. Softw. 2, 87–94 (1976)

    Article  MATH  Google Scholar 

  18. Touati-Ahmed, D., Storey, C.: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64(2), 379–397 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu, G.H., Zhao, Y.L., Wei, Z.X.: A descent nonlinear conjugate gradient method for large-scale unconstrained optimization. Appl. Math. Comput. 187, 636–643 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tianjin University, Tianjin, 300072, P.R. China

    Liu Dongyi & Xu Genqi

Authors
  1. Liu Dongyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xu Genqi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liu Dongyi.

Additional information

This research is supported by the Natural Science Foundation of China grant NSFC-60874034.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dongyi, L., Genqi, X. Applying Powell’s symmetrical technique to conjugate gradient methods. Comput Optim Appl 49, 319–334 (2011). https://doi.org/10.1007/s10589-009-9302-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9302-1

Search

Navigation