Skip to main content
SpringerLink
Log in

A modified Perry conjugate gradient method and its global convergence

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

Abstract

In this work, we propose a new conjugate gradient method which consists of a modification of Perry’s method and ensures sufficient descent independent of the accuracy of the line search. An important property of our proposed method is that it achieves a high-order accuracy in approximating the second order curvature information of the objective function by utilizing a new modified secant condition. Moreover, we establish that the proposed method is globally convergent for general functions provided that the line search satisfies the Wolfe conditions. Our numerical experiments indicate that our proposed method is preferable and in general superior to classical conjugate gradient methods in terms of efficiency and robustness.

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
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. http://clas.ufl.edu/users/hager/papers/CG/Archive/CG_DESCENT-C-5.3.tar.gz.

References

  1. Andrei, N.: Scaled conjugate gradient algorithms for unconstrained optimization. Comput. Optim. Appl. 38, 401–416 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrei, N.: Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization. Optim. Methods Softw. 22, 561–571 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrei, N.: A new three-term conjugate gradient algorithm for unconstrained optimization. Numer. Algorithms 22, 1–17 (2014)

    Google Scholar 

  4. Babaie-Kafaki, S.: A modified BFGS algorithm based on a hybrid secant equation. Sci. China Math. 54(9), 2019–2036 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Babaie-Kafaki, S.: Two modified scaled nonlinear conjugate gradient methods. J. Comput. Appl. Math. 261, 172–182 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Babaie-Kafaki, S., Ghanbari, R.: The Dai–Liao nonlinear conjugate gradient method with optimal parameter choices. Eur. J. Oper. Res. 234(3), 625–630 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. Babaie-Kafaki, S., Ghanbari, R., Mahdavi-Amiri, N.: Two new conjugate gradient methods based on modified secant equations. J. Comput. Appl. Math. 234, 1374–1386 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Babaie-Kafaki, S., Ghanbari, R., Mahdavi-Amiri, N.: Two effective hybrid conjugate gradient algorithms based on modified BFGS updates. Numer. Algorithms 58(3), 315–331 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Babaie-Kafaki, S., Mahdavi-Amiri, N.: Two hybrid nonlinear conjugate gradient methods based on a modified secant equation. Optimization, 63(7), 1–16 (2012)

  10. Babaie-Kafaki, S., Mahdavi-Amiri, N.: Two modified hybrid conjugate gradient methods based on a hybrid secant equation. Math. Model. Anal. 18(1), 32–52 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Birgin, E.G., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43, 117–128 (1999)

    Article  Google Scholar 

  12. Bongartz, I., Conn, A., Gould, N., Toint, P.: CUTE: constrained and unconstrained testing environments. ACM Trans. Math. Softw. 21(1), 123–160 (1995)

    Article  MATH  Google Scholar 

  13. Chen, W., Liu, Q.: Sufficient descent nonlinear conjugate gradient methods with conjugacy condition. Numer. Algorithms 53, 113–131 (2010)

    Article  Google Scholar 

  14. Dai, Y.H., Yuan, Y.X.: A nonlinear conjugate gradient with a strong global convergence properties. SIAM J. Optim. 10, 177–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dai, Y.H., Yuan, Y.X.: Nonlinear conjugate gradient methods. Shanghai Scientific and Technical Publishers, Shanghai (2000)

    Google Scholar 

  16. Dai, Z., Wen, F.: A modified CG-DESCENT method for unconstrained optimization. J. Comput. Appl. Math. 235(11), 3332–3341 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dai, Z.F., Tian, B.S.: Global convergence of some modified PRP nonlinear conjugate gradient methods. Optim. Lett. 5(4), 1–16 (2010)

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Du, S.Q., Chen, Y.Y.: Global convergence of a modified spectral FR conjugate gradient method. Appl. Math. Comput. 202(2), 766–770 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Ford, J.A., Narushima, Y., Yabe, H.: Multi-step nonlinear conjugate gradient methods for unconstrained minimization. Comput. Optim. Appl. 40, 191–216 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Hager, W.W., Zhang, H.: The limited memory conjugate gradient method. SIAM J. Optim. 23(4), 2150–2168 (2005)

    Article  MathSciNet  Google Scholar 

  24. 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 

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

    MATH  MathSciNet  Google Scholar 

  26. Hestenes, M.R., Stiefel, E.: Methods for conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  27. Li, D.H., Fukushima, M.: A modified BFGS method and its global convergence in nonconvex minimization. J.Comput. Appl. Math. 129, 15–35 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Li, G., Tang, C., Wei, Z.: New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. J. Comput. Appl. Math. 202, 523–539 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Livieris, I.E., Pintelas, P.: Globally convergent modified Perry conjugate gradient method. Appl. Math. Comput. 218(18), 9197–9207 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Livieris, I.E., Pintelas, P.: A new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization. J. Comput. Appl. Math. 239, 396–405 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lu, A., Liu, H., Zheng, X., Cong, W.: A variant spectral-type FR conjugate gradient method and its global convergence. Appl. Math. Comput. 217(12), 5547–5552 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Narushima, Y., Yabe, H.: Conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization. J. Comput. Appl. Math. 236(17), 4303–4317 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  34. Perry, A.: A modified conjugate gradient algorithm. Oper. Res. 26, 1073–1078 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  35. Polak, E., Ribière, G.: Note sur la convergence de methods de directions conjuguees. Revue Francais d’Informatique et de Recherche Operationnelle 16, 35–43 (1969)

    Google Scholar 

  36. Powell, M.J.D.: Restart procedures for the conjugate gradient method. Math. Program. 12, 241–254 (1977)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. Sugiki, K., Narushima, Y., Yabe, H.: Globally convergent three-term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization. J. Optim. Theory Appl. 153(3), 733–757 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yabe, H., Takano, M.: Global convergence properties of nonlinear conjugate gradient methods with modified secant condition. Comput. Optim. Appl. 28, 203–225 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  40. Yu, G., Guan, L., Chen, W.: Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optim. Methods Softw. 23(2), 275–293 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Yu, G.H.: Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems. PhD thesis, Sun Yat-Sen University (2007)

  42. Zhang, J.Z., Deng, N.Y., Chen, L.H.: New quasi-Newton equation and related methods for unconstrained optimization. J. Optim. Theory Appl. 102, 147–167 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  43. Zhang, J.Z., Xu, C.X.: Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations. J. Comput. Appl. Math. 137, 269–278 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  44. Zhang, L.: Two modified Dai-Yuan nonlinear conjugate gradient methods. Numer. Algorithms 50, 1–16 (2009)

    Article  MathSciNet  Google Scholar 

  45. Zhang, L., Zhou, W.: Two descent hybrid conjugate gradient methods for optimization. J. Comput. Appl. Math. 216, 251–264 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  46. Zhang, L., Zhou, W., Li, D.: A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26(4), 629–640 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Patras, 265-00, Patras, Greece

    Ioannis E. Livieris & Panagiotis Pintelas

Authors
  1. Ioannis E. Livieris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Panagiotis Pintelas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis E. Livieris.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Livieris, I.E., Pintelas, P. A modified Perry conjugate gradient method and its global convergence. Optim Lett 9, 999–1015 (2015). https://doi.org/10.1007/s11590-014-0820-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-014-0820-0

Keywords

Search

Navigation