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Multi-step nonlinear conjugate gradient methods for unconstrained minimization

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Abstract

Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of matrices. Recently, seeking fast convergence of these methods, Dai and Liao (Appl. Math. Optim. 43:87–101, 2001) proposed a conjugate gradient method based on the secant condition of quasi-Newton methods, and later Yabe and Takano (Comput. Optim. Appl. 28:203–225, 2004) proposed another conjugate gradient method based on the modified secant condition. In this paper, we make use of a multi-step secant condition given by Ford and Moghrabi (Optim. Methods Softw. 2:357–370, 1993; J. Comput. Appl. Math. 50:305–323, 1994) and propose two new conjugate gradient methods based on this condition. The methods are shown to be globally convergent under certain assumptions. Numerical results are reported.

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F., Yin, H.X., Yuan, Y.X.: Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10, 345–358 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Ford, J.A., Moghrabi, I.A.: Alternative parameter choices for multi-step quasi-Newton methods. Optim. Methods Softw. 2, 357–370 (1993)

    Article  Google Scholar 

  6. Ford, J.A., Moghrabi, I.A.: Multi-step quasi-Newton methods for optimization. J. Comput. Appl. Math. 50, 305–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone line search for unconstrained optimization. J. Optim. Theory Appl. 60, 401–419 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Moré, J.J., Thuente, D.J.: Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286–307 (1994)

    Article  MATH  Google Scholar 

  11. Moré, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)

    Article  MATH  Google Scholar 

  12. Nocedal, J.: http://www.ece.northwestern.edu/~nocedal/software.html

  13. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Polak, E., Ribière, G.: Note sur la convergence de methods de directions conjuguees. Rev. Fr. Inform. Rech. Oper. 16, 35–43 (1969)

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. 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  MathSciNet  MATH  Google Scholar 

  20. 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  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK

    John A. Ford

  2. Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Yasushi Narushima & Hiroshi Yabe

Authors
  1. John A. Ford
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  2. Yasushi Narushima
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  3. Hiroshi Yabe
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Correspondence to Hiroshi Yabe.

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Ford, J.A., Narushima, Y. & Yabe, H. Multi-step nonlinear conjugate gradient methods for unconstrained minimization. Comput Optim Appl 40, 191–216 (2008). https://doi.org/10.1007/s10589-007-9087-z

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  • DOI: https://doi.org/10.1007/s10589-007-9087-z

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