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

Authors: Ahmad Alhawarat, Mustafa Mamat, Mohd Rivaie, Ismail Mohd

Abstract:

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.

Keywords: Conjugate gradient method, conjugate gradient coefficient, global convergence.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1336925

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.