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Andrei, Neculai

Another hybrid conjugate gradient algorithm for unconstrained optimization. (English) Zbl 1141.65041

Numer. Algorithms 47, No. 2, 143-156 (2008).
The author studies a conjugate gradient algorithm for unconstrained optimization. The article begins with an introduction to the general nonlinear unconstrained optimization problem and a short review of the existing literature. The second section outlines the hybrid conjugate gradient algorithm as a convex combination of two existing algorithms. This is followed by the third and fourth sections which present the details of the algorithm itself and study its convergence properties, where several theorems are presented with proof. The paper concludes with an extensive section containing results of the numerical experimentation and a list of relevant references.
Reviewer: Efstratios Rappos (Athens)

Cited in 4 Reviews
Cited in 37 Documents

MSC:

65K05 Numerical mathematical programming methods
90C06 Large-scale problems in mathematical programming
90C30 Nonlinear programming

Keywords:

unconstrained optimization; hybrid conjugate gradient method; Newton direction; numerical comparisons; convergence

Software:

CUTEr; Algorithm 500; SCALCG; CUTE
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\textit{N. Andrei}, Numer. Algorithms 47, No. 2, 143--156 (2008; Zbl 1141.65041)
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References:

[1] Andrei, N.: Test functions for unconstrained optimization. http://www.ici.ro/camo/neculai/SCALCG/evalfg.for · Zbl 1161.90486
[2] Andrei, N.: Scaled conjugate gradient algorithms for unconstrained optimization. Comput. Optim. Appl. 38, 401–416 (2007) · Zbl 1168.90608
[3] Andrei, N.: Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization. Optim. Methods Softw. 22, 561–571 (2007) · Zbl 1270.90068
[4] Andrei, N.: A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization. Appl. Math. Lett. 20, 645–650 (2007) · Zbl 1116.90114
[5] Andrei, N.: Numerical comparison of conjugate gradient algorithms for unconstrained optimization. Stud. Inform. Control 16, 333–352 (2007)
[6] Birgin, E., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43, 117–128 (2001) · Zbl 0990.90134
[7] Bongartz, I., Conn, A.R., Gould, N.I.M., Toint, P.L.: CUTE: constrained and unconstrained testing environments. ACM Trans. Math. Soft. 21, 123–160 (1995) · Zbl 0886.65058
[8] Dai, Y.H.: New properties of a nonlinear conjugate gradient method. Numer. Math. 89, 83–98 (2001) · Zbl 1006.65063
[9] Dai, Y.H., Liao, L.Z.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43, 87–101 (2001) · Zbl 0973.65050
[10] Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F., Yin, X., Yuan, Y.: Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10, 348–358 (1999) · Zbl 0957.65061
[11] 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
[12] Dai, Y.H., Yuan, Y.: An efficient hybrid conjugate gradient method for unconstrained optimization. Ann. Oper. Res. 103, 33–47 (2001) · Zbl 1007.90065
[13] Dolan, E.D. Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002) · Zbl 1049.90004
[14] Fletcher, R.: Practical Methods of Optimization, vol. 1: Unconstrained Optimization. John Wiley & Sons, New York (1987) · Zbl 0905.65002
[15] Fletcher, R., Reeves, C.: Function minimization by conjugate gradients. Comput. J. 7, 149–154 (1964) · Zbl 0132.11701
[16] Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992) · Zbl 0767.90082
[17] 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
[18] Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35–58 (2006) · Zbl 1117.90048
[19] Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952) · Zbl 0048.09901
[20] Hu, Y.F., Storey, C.: Global convergence result for conjugate gradient methods. J. Optim. Theory Appl. 71, 399–405 (1991) · Zbl 0794.90063
[21] Liu, Y., Storey, C.: Efficient generalized conjugate gradient algorithms, Part 1: Theory. J. Optim. Theory Appl. 69, 129–137 (1991) · Zbl 0724.90067
[22] Polak, E., Ribière, G.: Note sur la convergence de directions conjuguée. Rev. Francaise Informat Recherche Operationelle, 3e Année 16, 35–43 (1969) · Zbl 0174.48001
[23] Polyak, B.T.: The conjugate gradient method in extreme problems. USSR Comp. Math. Math. Phys. 9, 94–112 (1969) · Zbl 0229.49023
[24] Powell, M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. In: Numerical Analysis (Dundee, 1983), Lecture Notes in Mathematics, vol. 1066, pp. 122–141, Springer-Verlag, Berlin (1984)
[25] Shanno, D.F., Phua, K.H.: Algorithm 500, Minimization of unconstrained multivariate functions. ACM Trans. Math. Soft. 2, 87–94 (1976) · Zbl 0319.65042
[26] Touati-Ahmed, D., Storey, C.: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64, 379–397 (1990) · Zbl 0687.90081
[27] Zhang, J.Z., Deng, N.Y., Chen, L.H.: New quasi-Newton equation and related methods for unconstrained optimization. J. Optim. Theory Appl. 102, 147–167 (1999) · Zbl 0991.90135
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