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A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization. (English) Zbl 1315.90051

Summary: Recently, conjugate gradient methods, which usually generate descent search directions, are useful for large-scale optimization. Y. Narushima et al. [SIAM J. Optim. 21, No. 1, 212–230 (2011; Zbl 1250.90087)] have proposed a three-term conjugate gradient method which satisfies a sufficient descent condition. We extend this method to two parameters family of three-term conjugate gradient methods which can be used to control the magnitude of the directional derivative. We show that these methods converge globally and work well for suitable choices of the parameters. Numerical results are also presented.

MSC:

90C30 Nonlinear programming
90C52 Methods of reduced gradient type

Citations:

Zbl 1250.90087
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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 · doi:10.1093/imanum/5.1.121
[2] Barzilai, J., Borwein, J.M.: Two-point stepsize gradient methods. IMA J. Numer. Anal. 8, 141-148 (1988) · Zbl 0638.65055 · doi:10.1093/imanum/8.1.141
[3] Bongartz, I., 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 · doi:10.1145/200979.201043
[4] Cheng, W.: A two-term PRP-based descent method. Numer. Funct. Anal. Optim. 28, 1217-1230 (2007) · Zbl 1138.90028 · doi:10.1080/01630560701749524
[5] Dai, Y.-H., Kou, C.-X.: A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search. SIAM J. Optim. 23, 296-320 (2013) · Zbl 1266.49065 · doi:10.1137/100813026
[6] 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 · doi:10.1007/s002450010019
[7] 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 · doi:10.1137/S1052623497318992
[8] Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201-213 (2002) · Zbl 1049.90004 · doi:10.1007/s101070100263
[9] Fletcher, R.: Practical Methods of Optimization (Second Edition). Wiley, New York (1987) · Zbl 0905.65002
[10] Fletcher, R.: On the Barzilai-Borwein Method, Optimization and Control with Applications, Springer series in Applied Optimization, 96, 235-256, Springer, New York (2005) · Zbl 1118.90318
[11] Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7, 149-154 (1964) · Zbl 0132.11701 · doi:10.1093/comjnl/7.2.149
[12] Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21-42 (1992) · Zbl 0767.90082 · doi:10.1137/0802003
[13] Gould, N.I.M., Orban, D., Toint, P.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29, 373-394 (2003) · Zbl 1068.90526 · doi:10.1145/962437.962439
[14] Hager, W.W. http://people.clas.ufl.edu/hager/. Accessed 10 Mar 2014 · Zbl 1093.90085
[15] 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 · doi:10.1137/030601880
[16] Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35-58 (2006) · Zbl 1117.90048
[17] Hager W.W., Zhang, H.: CG_DESCENT Version 1.4 User’s Guide, University of Florida (2005). http://people.clas.ufl.edu/hager/. Accessed 10 Mar 2014 · Zbl 1068.90526
[18] Hager, W.W., Zhang, H.: Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32, 113-137 (2006) · Zbl 1346.90816 · doi:10.1145/1132973.1132979
[19] Hager, W.W., Zhang, H.: The limited memory conjugate gradient method. SIAM J. Optim. 23, 2150-2168 (2013) · Zbl 1298.90129
[20] Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409-436 (1952) · Zbl 0048.09901 · doi:10.6028/jres.049.044
[21] Liu, Y., Storey, C.: Efficient generalized conjugate gradient algorithms, Part1: Theory. J. Optim. Theory. Appl. 69, 129-137 (1991) · Zbl 0702.90077 · doi:10.1007/BF00940464
[22] Narushima, Y., Yabe, H., Ford, J.A.: A three-term conjugate gradient method with sufficient descent property for unconstrained optimization. SIAM J. Optim. 21, 212-230 (2011) · Zbl 1250.90087 · doi:10.1137/080743573
[23] Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research, Springer, New York (2006) · Zbl 1104.65059
[24] Sugiki, K., Narushima, Y., Yabe, H.: Globally convergent three-term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization. J Optim Theory. Appl. 153, 733-757 (2012) · Zbl 1262.90170 · doi:10.1007/s10957-011-9960-x
[25] Sorenson, H.W.: Comparison of some conjugate direction procedures for function minimization. J. Frankl. Inst. 288, 421-441 (1969) · Zbl 0228.90040 · doi:10.1016/0016-0032(69)90253-1
[26] Yu, G., Guan, L., Li, G.: Global convergence of modified Polak-Ribiére-Polyak conjugate gradient methods with sufficient descent property. J. Ind. Manag. Optim. 4, 565-579 (2008) · Zbl 1168.65030 · doi:10.3934/jimo.2008.4.565
[27] Zhang, L.: A new Liu-Storey type nonlinear conjugate gradient method for unconstrained optimization problems. J. Comput. Appl. Math. 225, 146-157 (2009) · Zbl 1185.65101 · doi:10.1016/j.cam.2008.07.016
[28] Zhang, L., Zhou, W., Li, D.H.: Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search. Numer. Math. 104, 561-572 (2006) · Zbl 1103.65074 · doi:10.1007/s00211-006-0028-z
[29] Zhang, L., Zhou, W., Li, D.H.: A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629-640 (2006) · Zbl 1106.65056 · doi:10.1093/imanum/drl016
[30] Zhang, L., Zhou, W., Li, D.H.: Some descent three-term conjugate gradient methods and their global convergence. Optim. Methods Softw. 22, 697-711 (2007) · Zbl 1220.90094 · doi:10.1080/10556780701223293
[31] Zoutendijk, G.: Nonlinear Programming, Computational Methods, in Integer and Nonlinear Programming. Abadie, J. (ed.) North-Holland, Amsterdam, 37-86, 1970 · Zbl 0336.90057
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