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Journal of Applied Mathematics and Computation

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Frequency: quarterly ISSN Print: 2576-0645 CODEN: JAMCEZ
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Article Open Access http://dx.doi.org/10.26855/jamc.2018.09.002

New Hybrid Conjugate Gradient Method as A Convex Combination of HS and FR Methods

Snezana S. Djordjevic

Faculty of Technology, University of Nis, 16000 Leskovac, Serbia.

*Corresponding author: Snezana S. Djordjevic

Published: September 27,2018

Abstract

In this paper we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Hestenes-Stiefel conjugate gradient method and Fletcher-Reeves conjugate gradient method. The parameter  is chosen in such a way that the search direction satisfies the condition of the Newton direction. The strong Wolfe line search conditions are used. The global convergence of new method is proved. Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.

References

[1] M. Al-Baali, Descent property and global convergence of the Fletcher-Reeves method with inexact line search, IMA J. Numer. Anal., 5 (1985), 121-124. 

[2] N. Andrei, 40 Conjugate Gradient Algorithms for unconstrained optimization, A survey on their definition, ICI Technical Report, 13/08, 2008. 

[3] N. Andrei, New hybrid conjugate gradient algorithms for unconstrained optimization, Encyclopedia of Optimization, 2560-2571, 2009. 

[4] N. Andrei, A Hybrid Conjugate Gradient Algorithm with Modified Secant Condition for Unconstrained Optimization as a Convex Combination of Hestenes-Stiefel and Dai-Yuan Algorithms, STUDIES IN INFORMATICS AND CONTROL, 17, 4 (2008), 373-392. 

[5] N. Andrei, A hybrid conjugate gradient algorithm for unconstrained optimization as a convex combination of Hestenes-Stiefel and Dai-Yuan, Studies in Informatics and Control, 17, 1 (2008), 55-70. 

[6] N. Andrei, Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47, 2 (2008), 143-156. 

[7] N. Andrei, An unconstrained optimization test functions, Advanced Modeling and Optimization. An Electronic International Journal, 10 (2008), 147-161. 

[8] N. Andrei, Accelerated hybrid conjugate gradient algorithm with modified secant condition for unconstrained optimization, Numer. Algorithms 54 (2010) 23-46. 

[9] Y.H. Dai, Y.Yuan, Convergence properties of the Fletcher-Reeves method, IMA J. Numer. Anal., 16 (1996), 155-164. 

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

[11] Y. H. Dai, Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182. 

[12] Y.H.Dai, Han J.Y., Liu G.H., Sun D.F., Yin X., Yuan Y., Convergence properties of nonlinear conjugate gradient methods, SIAM Journal on Optimization, 10 (1999), 348-358. 

[13] S. S. Đorđević, New hybrid conjugate gradient method as a convex combination of FR and PRP methods, Filomat, 30:11 (2016), 3083-3100. 

[14] E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Programming, 91 (2002), 201-213. 

[15] R. Fletcher, Practical methods of optimization vol. 1: Unconstrained Optimization, John Wiley and Sons, New York, 1987. 

[16] R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154. 

[17] J. C. Gilbert, J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal of Optimization, 2 (1992), 21-42. 

[18] W. W. Hager, H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16, 1 (2003), 170-192. 

[19] W. W. Hager, H. Zhang, CG-DESCENT, a conjugate gradient method with guaranteed descent, ACM Transactions on Mathematical Software, 32, 1 (2006), 113-137. 

[20] W.W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific journal of Optimization, 2 (2006), 35-58. 

[21] M. R. Hestenes, E. L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436. 

[22] Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods, J. Optim. Theory Appl., 71 (1991), 399-405. 

[23] J.K. Liu, S.J. Li, New hybrid conjugate gradient method for unconstrained optimization, Applied Mathematics and Computation, 245 (2014), 36-43. 

[24] G. H. Liu, J. Y. Han and H. X. Yin, Global convergence of the Fletcher-Reeves algorithm with an inexact line search, Appl. Math. J. Chinese Univ. Ser. B, 10 (1995), 75-82. 

[25] Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 1: Theory, JOTA, 69 (1991), 129-137. 

[26] J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 1999. 

[27] E. Polak, G. Ribiére, Note sur la convergence de méthodes de directions conjugués, Revue Française d'Informatique et de Recherche Opérationnelle, 16 (1969), 35-43. 

[28] B. T. Polyak, The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9 (1969), 94-112.

[29] M. J. D. Powell, Nonconvex minimization calculations and the conjugate gradient method, in D.F. Griffiths, ed., Numerical Analysis Lecture Notes in Mathematics 1066 (Springer-Verlag, Berlin, 1984), 122-141. 

[30] M. J. D. Powell, Restart procedures of the conjugate gradient method, Mathematical Programming, 2 (1977), 241-254. 

[31] D. Touati-Ahmed, C. Storey, Efficient hybrid conjugate gradient techniques, J. Optim. Theory Appl., 64 (1990), 379-397. 

[32] X. Yang, Z. Luo, X. Dai, A Global Convergence of LS-CD Hybrid Conjugate Gradient Method, Advances in Numerical Analysis, 2013 (2013), Article ID 517452, 5 pages. 

[33] Y. Yuan, J. Stoer, A subspace study on conjugate gradient algorithm, Z. Angew. Math. Mech. 75 (1995), 69-77. 

[34] P. Wolfe, Convergence conditions for ascent methods, SIAM Review, 11 (1969), 226-235. 

[35] P. Wolfe, Convergence conditions for ascent methods. II: Some corrections, SIAM Review, 11 (1969), 226-235. 

[36] G. Zoutendijk, Nonlinear programming, computational methods, in Integer and Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam, (1970), 37-86.

How to cite this paper

New Hybrid Conjugate Gradient Method as A Convex Combination of HS and FR Methods

How to cite this paper: Snezana S. Djordjevic. (2018) New Hybrid Conjugate Gradient Method as A Convex Combination of HS and FR MethodsJournal of Applied Mathematics and Computation, 2(9), 366-378.

DOI: http://doi.org/10.26855/jamc.2018.09.002