A New Modification of Nonlinear Conjugate Gradient Coefficients with Global Convergence Properties

Conjugate gradient method has been enormously used to solve large scale unconstrained optimization problems due to the number of iteration, memory, CPU time, and convergence property, in this paper we find a new class of nonlinear conjugate gradient coefficient with global convergence properties proved by exact line search. The numerical results for our new βK give a good result when it compared with well known formulas.

Authors:

• Ahmad Alhawarat
• Mustafa Mamat
• Mohd Rivaie
• Ismail Mohd

Keywords:

• Conjugate gradient method
• conjugate gradient coefficient
• global convergence.

References:

[1] M.R. Hestenes, E.L. Stiefel, Methods of conjugate gradients for solving
linear systems, J. Res. Na- tl. Bur. Stand. Sec. B 49 (1952), 409–432.
[2] R. Fletcher, C. Reeves, Function minimization by conjugate gradients,
comput. J. 7 (1964), 149–1 -54. [3] B. Polak, G. Ribiére, Note surla convergence des méthodes de directions
conjuguées, Rev. Fr. Aut- om. Inform. Rech. Oper.,3e Année. 16 (1969),
35–43.
[4] R. Fletcher, Practical method of optimization, vol 1, unconstrained
optimization, John Wiley & Sons, New York, 1987.
[5] Y.L. Liu, C.S. Storey, Efficient generalized conjugate gradient
algorithms, part 1: theory, J. Optim. Theory Appl. 69 (1991), 129–137.
[6] R. Fletcher, and C. Reeves, Function minimization by conjugate
gradients, Comput. J. 7(1964), 149-154.
[7] Z. Wei, S. Yao, L. Liu, The convergence properties of some new
conjugate gradient methods, App-l. Math. Comput. 183 (2006), 1341–
1350.
[8] G. Zoutendijk, Nonlinear progrAMRing computational methods, in: J.
Abadie (Ed.), Integer and N- onlinear ProgrAMRing, North-Holland,
Amsterdam (1970), 37–86.
[9] J.C. Gilbert, J. Nocedal, Global convergence properties of conjugate
gradient methods for optimi- zation, SIAM J. Optimizat. 2 (1) (1992),
21–42.
[10] Cheng, W.Y.: A two-term PRP-based descent method. Numer. Funct.
Anal. Optim. 28(2007)1217–1230
[11] Dai, Z.F., Tian, B.S.: Global convergence of some modified PRP
nonlinear conjugate gradient methods. Opt. Lett. (2010),
doi:10.1007/s11590-010-0224-8
[12] Yu, G.H., Zhao, Y.L., Wei, Z.X.: A descent nonlinear conjugate gradient
method for largescale unconstrained optimization. Appl. Math. Comput.
187 (2007), 636–643
[13] Zhang, L., Zhou, W., Li, D.: A descent modified Polak-Ribi-re-Polyak
conjugate gradient method and its global convergence. IMA J. Numer.
Anal. 26(2006), 629–640
[14] Zhang, L., Zhou, W., Li, D.: Some descent three-term conjugate gradient
methods and their global convergence. Optim. Methods Softw. 22
(2007), 697–711
[15] Z. Wei, S. Yao, L. Liu, The convergence properties of some new
conjugate gradient methods, Appl. Math. Comput. 183 (2006), 1341–
1350.
[16] L. Zhang, An improved Wei–Yao–Liu nonlinear conjugate gradient
method for optimization computation, Appl. Math. Comput. 215 (2009),
2269–2274.
[17] Mohd Rivaie, Mustafa Mamat, Ismail Mohd, Leong Wah June, A new
class of nonlinear conjugate gradient coefficients with global
convergence properties, Appl.Math. Comput218 (2012), 11323-11332.
[18] Z.F. Dai, Two modified HS type conjugate gradient methods for
unconstrained optimization problems, Nonliner Anal. 74 (2011), 927–
936.
[19] G. Yuan, X. Lu, and Z. Wei, A conjugate gradient method with descent
direction for unconstrained optimization, J. Comp. App. Maths.
233(2009), 519-530.
[20] E. Dolan, J.J. More, Benchmarking optimization software with
performance profile, Math. Prog. 91 (2002), 201–213.