A New Nonlinear Conjugate Gradient Method Based on the Scaled Matrix


  • Basim A. Hassan Department of Mathematics, College of Computers Sciences and Mathematics, Baghdad
  • Haneen A. Alashoor Department of Mathematics, College of Computers Sciences and Mathematics




Conjugate gradient, Descent condition, global convergent, Numerical results.


In this paper, a new type nonlinear conjugate gradient method based on the Scale
Matrix is derived. The new method has the decent and globally convergent
properties under some assumptions. Numerical results indicate the efficiency of
this method to solve the given test problems.


Andrie N. (2008) ' An Unconstrained Optimization Test functions collection ' Advanced Modeling and optimization. 10, pp.147-161.

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

https://doi.org/10.1093/imanum/8.1.141 DOI: https://doi.org/10.1093/imanum/8.1.141

Cao W. and Wang K. (2010) ' Global convergence of a new conjugate gradient method for modified Liu-Storey formula' Journal of East China Normal University (Natural Science), 1, pp.44-51.

Dai Y. and Liao Z. (2001). New conjugate conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43, pp. 87-101.

https://doi.org/10.1007/s002450010019 DOI: https://doi.org/10.1007/s002450010019

Dai, Y. and Yuan Y. (1999) ' A non-linear conjugate gradients method with a strong global convergence property' SIAM J. Optimization,10, pp.177-182.

https://doi.org/10.1137/S1052623497318992 DOI: https://doi.org/10.1137/S1052623497318992

Fletcher, R. (1987) ' Practical Methods of Optimization (second edition). John Wiley and Sons, New York.

Fletcher, R. and Reeves C. (1964) ' Function minimization by conjugate gradients ' Computer J. 7, pp.149-154.

https://doi.org/10.1093/comjnl/7.2.149 DOI: https://doi.org/10.1093/comjnl/7.2.149

Hestenes, M. R. and Stiefel E. L. (1952) ' Method of conjugate gradients for solving linear systems' J. Research Nat. Standards 49, pp.409-436.

https://doi.org/10.6028/jres.049.044 DOI: https://doi.org/10.6028/jres.049.044

Liu Y. and Storey C. (1991) ' Efficient generalized conjugate gradients algorithms ' Part 1 : Theory. J. Optimization Theory and Applications 69, pp.129-137.

https://doi.org/10.1007/BF00940464 DOI: https://doi.org/10.1007/BF00940464

Nocedal J. and Wright S. J. (1999). Numerical Optimization. Springer. New York.

https://doi.org/10.1007/b98874 DOI: https://doi.org/10.1007/b98874

Polak, E. and Ribiere, G. (1969) ' Note for Convergence Direction Conjugate, Revue Francaise Informant, Research. Operational, 16, 35-43. DOI: https://doi.org/10.1051/m2an/196903R100351

Zoutendijk G., 1970. Nonlinear Porgramming, computational methods, in: Integr and Nonlinear Programming, North-Holland, Amsterdam, pp. 37-86.

Yasushi N. and Hideaki I., (2011). Conjugate gradient methods using value of objective function for unconstrained optimization Optimization Letters, V.6, Issue 5, 941-955. DOI: https://doi.org/10.1007/s11590-011-0324-0




How to Cite

B. A. Hassan and H. A. Alashoor, “A New Nonlinear Conjugate Gradient Method Based on the Scaled Matrix”, Al-Mustansiriyah Journal of Science, vol. 27, no. 5, pp. 68–73, Jul. 2017.