An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation. (English) Zbl 1181.65089

Zhang, Li

Appl. Math. Comput. 215, No. 6, 2269-2274 (2009).

The author considers the unconstrained minimization problem consisting in a minimizing continuously differentiable function \(f\) defined on \(\mathbb{R}^n\). Conjugate gradient methods knows from the literature are compared and two slight modifications of Z. Wei, S. Yao and L. Liu’s nonlinear conjugate method [Appl. Math. Comput. 183, No. 2, 1341–1350 (2006; Zbl 1116.65073)] are proposed. The modified methods posses better convergence properties and converge globally if the strong Wolfe line search with a restriction on one of its parameters is used.
The second of the two methods is proved to be globally convergent even if the standard Wolfe line search is used. Numerical results reported in the concluding part of the paper show that the methods are efficient for problems from the CUTE library [see I. Bongartz, A. R. Conn, N. Gould and Ph. L. Toint, CUTE: constrained and unconstrained testing environments, ACM Trans. Math. Softw 21, No. 1, 123–160 (1995; Zbl 0886.65058)]. The efficiency of the proposed methods is compared with the efficiency of some other conjugate gradient methods.

Reviewer: Karel Zimmermann (Praha)

MSC:

65K05	Numerical mathematical programming methods
90C30	Nonlinear programming
65Y20	Complexity and performance of numerical algorithms

Keywords:

conjugate gradient method; descent direction; global convergence; Wolfe line; numerical results; efficiency

Citations:

Zbl 1116.65073; Zbl 0886.65058

Software:

CUTE; CUTEr

References:

[1]	Al-Baali, M., Descent property and global convergence of the fletcher – reeves method with inexact line search, IMA J. numer. anal., 5, 121-124, (1985) · Zbl 0578.65063
[2]	Bongartz, K.E.; Conn, A.R.; Gould, N.I.M.; Toint, P.L., CUTE: constrained and unconstrained testing environments, ACM trans. math. softw., 21, 123-160, (1995) · Zbl 0886.65058
[3]	Dai, Y.H.; Yuan, Y., A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. optim., 10, 177-182, (1999) · Zbl 0957.65061
[4]	Dai, Y.H.; Yuan, Y., Nonlinear conjugate gradient methods, (2000), Shanghai Science and Technology Publisher Shanghai · Zbl 1030.90141
[5]	Fletcher, R.; Reeves, C., Function minimization by conjugate gradients, Comput. J., 7, 149-154, (1964) · Zbl 0132.11701
[6]	Gilbert, J.C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. optim., 2, 21-42, (1992) · Zbl 0767.90082
[7]	Hager, W.W.; Zhang, H., A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. optim., 16, 170-192, (2005) · Zbl 1093.90085
[8]	Hager, W.W.; Zhang, H., A survey of nonlinear conjugate gradient methods, Pacific J. optim., 2, 35-58, (2006) · Zbl 1117.90048
[9]	Hestenes, M.R.; Stiefel, E.L., Methods of conjugate gradients for solving linear systems, J. res. nat. bur. stand. sec. B, 49, 409-432, (1952) · Zbl 0048.09901
[10]	Huang, H.; Wei, Z.; Yao, S., The proof of the sufficient descent condition of the wei – yao – liu conjugate gradient method under the strong wolfe – powell line search, Appl. math. comput., 189, 1241-1245, (2007) · Zbl 1131.65049
[11]	Liu, Y.L.; Storey, C.S., Efficient generalized conjugate gradient algorithms. part 1: theory, J. optim. theor. appl., 69, 129-137, (1991) · Zbl 0702.90077
[12]	Moré, J.J.; Thuente, D.J., Line search algorithms with guaranteed sufficient decrease, ACM trans. math. softw., 20, 286-307, (1994) · Zbl 0888.65072
[13]	Polak, B.; Ribiere, G., Note surla convergence des méthodes de directions conjuguées, Rev. francaise imformat recherche opertionelle., 16, 35-43, (1969) · Zbl 0174.48001
[14]	Polyak, B.T., The conjugate gradient method in extreme problems, USSR comput. math. math. phys., 9, 94-112, (1969) · Zbl 0229.49023
[15]	Wei, Z.; Li, G.; Qi, L., Global convergence of the polak – ribière – polyak conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems, Math. comput., 77, 2173-2193, (2008) · Zbl 1198.65091
[16]	Wei, Z.; Yao, S.; Liu, L., The convergence properties of some new conjugate gradient methods, Appl. math. comput., 183, 1341-1350, (2006) · Zbl 1116.65073
[17]	Yao, S.; Wei, Z.; Huang, H., A notes about wyl’s conjugate gradient method and its applications, Appl. math. comput., 191, 381-388, (2007) · Zbl 1193.90213
[18]	Yuan, G.; Lu, X., A modified PRP conjugate gradient method, Ann. oper. res., 166, 73-90, (2009) · Zbl 1163.90798
[19]	Yuan, G., Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems, Optim. lett., 3, 11-21, (2009) · Zbl 1154.90623
[20]	Zhang, L.; Zhou, W.; Li, D., A descent modified polak – ribière – polyak conjugate gradient method and its global convergence, IMA J. numer. anal., 26, 629-640, (2006) · Zbl 1106.65056
[21]	Zhang, L.; Zhou, W.; Li, D., Global convergence of a modified fletcher – reeves conjugate gradient method with Armijo-type line search, Numer. math., 104, 561-572, (2006) · Zbl 1103.65074
[22]	Zhang, L.; Zhou, W.; Li, D., Some descent three-term conjugate gradient methods and their global convergence, Optim. method softw., 22, 697-711, (2007) · Zbl 1220.90094