Modification of the Armijo line search to satisfy the convergence properties of HS method

Authors

  • Mohammed Belloufi
  • Rachid Benzine
  • Laskri Yamina

DOI:

https://doi.org/10.11121/ijocta.01.2013.00141

Keywords:

unconstrained optimization, conjugate gradient method, line search algorithm

Abstract

The Hestenes-Stiefel (HS) conjugate gradient algorithm is a useful tool of unconstrainednumerical optimization, which has good numerical performance but no global convergence result under traditional line searches. This paper proposes a line search technique that guarantee the globalconvergence of the Hestenes-Stiefel (HS) conjugate gradient method. Numerical tests are presented tovalidate the different approaches.

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References

Armijo, L., Minimization of functions having Lipschits continuous partial derivatives, Pacific J. Math., 16, (1966) 1–3.

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). Crossref

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

Chen, X.D., Sun, J., Global convergence of a two-parameter family of conjugate gradient methods without line search, Journal of Computational and Applied Mathematics, 146, 37–45 (2002). Crossref

Dai, Y.H., Yuan, Y., Convergence properties of the conjugate descent method, Advances in Mathematics, 25, 552–562 (1996).

Dai, Y.H., Yuan, Y., A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal of Optimization, 10, 177–182 (1999). Crossref

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 Journal of Optimization, 10, 345–358 (1999). Crossref

Fletcher, R., Practical Method of Optimization, second ed., vol. I: Unconstrained Optimization, Wiley, New York, (1997).

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

Fasano, G., Planar conjugate gradient algorithm for large scale unconstrained optimization. Part 1: Theory, Journal of Optimization Theory and Applications, 125, 523–541 (2005). Crossref

Fasano, G., Planar conjugate gradient algorithm for large-scale unconstrained optimization. Part 1: Applications, Journal of Optimization Theory and Applications, 125, 543–558 (2005). Crossref

Grippo, L., Lucidi, S., A globally convergent version of the Polak–Ribi`ere conjugate gradient method, Mathematical Programming, 78, 375–391 (1997). Crossref

Grippo, L., Lucidi, S., Convergence conditions, line search algorithms and trust region implementations for the Polak Ribi`ere conjugate gradient method, Optimization Methods and Software, 20(1), 71–98 (2005). Crossref

Goldstein, A.A., On steepest descent, SIAM J. Control, 3, 147–151 (1965).

Hestenes, M.R., and Stiefel, E.L., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standars Sect., 5(49), 409-436 (1952). Crossref

Hu, Y.F., Storey, C., Global convergence result for conjugate gradient methods, Journal of Optimization Theory and Applications, 71, 399–405 (1991). Crossref

Liu, Y., Storey, C., Efficient generalized conjugate gradient algorithms. Part 1: Theory, Journal of Optimization Theory and Applications, 69, 129–137 (1991). Crossref

Mor´e, J.J., Garbow, B.S., Hillstrom, K.E., Testing unconstrained optimization software, ACM Trans. Math. Softw., 7, 17–41 (1981). Crossref

Polak, E., Ribi`ere, G., Note Sur la convergence de directions conjug`ees, Rev. Francaise Informat Recherche Operationelle, 3eAnn`ee16, 35–43 (1969).

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

Powell, M.J.D., Nonconvex minimization calculations and the conjugate gradient method, Lecture Notes in Mathematics, 1066, Springer-Verlag, Berlin, 122–141 (1984).

Sun, J., Zhang, J.P., Convergence of conjugate gradient methods without line search, Annals of Operations Research, 103, 161–173 (2001). Crossref

Sun, J., Yang, X.Q., Chen, X.D., Quadratic cost flow and the conjugate gradient method, European Journal of Operational Research, 164, 104–114 (2005). Crossref

Shi, Z.J., Shen, J., Convergence of descent method without line search, Appl. Math. Comput., 167, 94–107 (2005). Crossref

Shi, Z.J., Restricted PR conjugate gradient method and its global convergence, Advances in Mathematics, 31(1), 47–55 (2002)(in Chinese).

Shi, Z.J., Shen, J., Convergence of Polak Rebi`ere Polyak conjugate gradient method, Nonlinear Analysis, 66, 1428–1441 (2007). Crossref

Shi, Z.J., Shen, J., Convergence of LiuStorey conjugate gradient method, European Journal of Operational Research,182(2), 552–560 (2007). Crossref

Oktrik.com. (2020). Oktrik.com. [online] Available at: https://www.oktrik.com/ [Accessed 18 Jan. 2014].

Yuan, Y. Numerical Methods for Nonlinear Programming, Shanghai Scientiï¬c & Technical Publishers, (1993).

Yuan, Y. Analysis on the conjugate gradient method, Optimization Methods and Software, 2, 19–29 (1993). Crossref

Zoutendijk, G. Nonlinear programming computational methods, in: J. Abadie (Ed.), Integer and Nonlinear Programming, NorthHolland, Amsterdam, 37–86 (1970).

Published

2013-05-29
CITATION
DOI: 10.11121/ijocta.01.2013.00141
Published: 2013-05-29

How to Cite

Belloufi, M., Benzine, R., & Yamina, L. (2013). Modification of the Armijo line search to satisfy the convergence properties of HS method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 3(2), 145–152. https://doi.org/10.11121/ijocta.01.2013.00141

Issue

Section

Optimization & Applications