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Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search. (English) Zbl 1185.65100

It is well known that global convergence has not been established for the Polak-Ribiere-Polyak (PRP) conjugate gradient method using the standard Wolfe conditions. In this paper some global convergence results for the PRP-type conjugate gradient method are established, where the step-length is computed by a modified Wolfe line search. Some efficient choices for \(\beta_k\) which can ensure the descent property of the search direction are also discussed. Preliminary numerical experiments on a set of a large-scale problems show that the computational efficiency of the PRP method is not deteriorated.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C06 Large-scale problems in mathematical programming
65Y20 Complexity and performance of numerical algorithms

Software:

minpack
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References:

[1] Ahmed, T.; Storey, D., Efficient hybrid conjugate gradient techniques, Journal of optimization theory and application, 64, 379-394, (1990) · Zbl 0666.90063
[2] Al-Baali, A., Descent property and global convergence of the fletcher – reeves method with inexact line search, IMA journal of numerical analysis, 5, 121-124, (1985) · Zbl 0578.65063
[3] Dai, Y., Conjugate gradient methods with Armijo-type line searches, Acta mathematicae applicatae sinica, English series, 18, 1, 123-130, (2002) · Zbl 1114.90479
[4] Dai, Y.; Han, J.; Liu, G.; Sun, D.; Yin, H.; Yan, Y., Convergence properties of nonlinear conjugate methods, SIAM journal on optimization, 2, 345-358, (1999) · Zbl 0957.65062
[5] Dolan, E.; Moré, J., Benchmarking optimization software with performance profiles, Mathematical programming series A, 91, 201-213, (2002) · Zbl 1049.90004
[6] Fletcher, R.; Reeves, C., Function minimization by conjugate gradients, Computer journal, 7, 149-154, (1964) · Zbl 0132.11701
[7] Gibert, J.C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM journal on optimization, 2, 21-42, (1992) · Zbl 0767.90082
[8] Grippo, L.; Lucidi, S., A globally convergent version of the polak – ribière gradient method, Mathematics programming, 78, 375-391, (1997) · Zbl 0887.90157
[9] Grippo, L.; Lucidi, S., Convergence conditions, line search algorithms and trust region implementations for the polak – ribière conjugate gradient method, Optimization methods and software, 20, 1, 71-98, (2005) · Zbl 1087.90086
[10] Hager, W.W.; Zhang, H., A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM journal on optimization, 16, 170-192, (2005) · Zbl 1093.90085
[11] Hestenes, M.R.; Stiefel, E., Method of conjugate gradient for solving linear equations, Journal of research of the national bureau of standards, 49, 409-436, (1952) · Zbl 0048.09901
[12] Hu, Y.F.; Storey, C., Global convergence result for conjugate gradient method, Journal of optimization theory and application, 71, 399-405, (1991) · Zbl 0794.90063
[13] Morè, J.J.; Garbow, B.S.; Hillstrome, K.E., Testing unconstrained optimization software, Transactions of the ACM on mathematical software, 7, 17-41, (1981) · Zbl 0454.65049
[14] Nocedal, J.; Wright, S.J., Numerical optimization, () · Zbl 1104.65059
[15] E. Polak, G. Ribière, Note Sur la convergence de directions conjugèes, Rev. Francaise Informat Recherche Operationelle, 3e Annèe 16 (1969) 35-43.
[16] Polyak, B.T., The conjugate gradient method in extreme problems, USSR computational mathematics and mathematical physics, 9, 94-112, (1969) · Zbl 0229.49023
[17] Powell, M.J.D., Nonconvex minimization calculations and the conjugate gradient method, (), 122-141 · Zbl 0531.65035
[18] Wei, Z.; Li, G.; Qi, L., Global convergence of the polak – ribière – polyak conjugate gradient method with inexact line searches for nonconvex unconstrained optimization problems, Mathematics of computation, 77, 2173-2193, (2008) · Zbl 1198.65091
[19] Yu, G.; Guan, L.; Wei, Z., A globally convergent polak – ribière – polyak conjugate gradient method with Armijo-type line search, Numerical mathematics: A journal of Chinese universities English series, 15, 357-367, (2006) · Zbl 1132.65059
[20] Yu, G.; Zhao, Y.; Wei, Z., A descent nonlinear conjugate gradient method for large-scale unconstrained optimization, Applied mathematics and computations, 187, 636-643, (2007) · Zbl 1117.65097
[21] Yuan, G., Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems, Optimization letters, 3, 11-21, (2009) · Zbl 1154.90623
[22] Zoutendijk, G., Nonlinear programming computational methods, (), 37-86 · Zbl 0336.90057
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