Three modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. (English) Zbl 1310.90108
Summary: In this paper, three modified Polak-Ribière-Polyak (PRP) conjugate gradient methods for unconstrained optimization are proposed. They are based on the two-term PRP method proposed by W.-Y. Cheng [Numer. Funct. Anal. Optim. 28, No. 11–12, 1217–1230 (2007; Zbl 1138.90028)], the three-term PRP method proposed by L. Zhang et al. [IMA J. Numer. Anal. 26, No. 4, 629–640 (2006; Zbl 1106.65056)], and the descent PRP method proposed by G.-H. Yu et al. [Optim. Methods Softw. 23, No. 2, 275–293 (2008; Zbl 1279.90166)]. These modified methods possess the sufficient descent property without any line searches. Moreover, if the exact line search is used, they reduce to the classical PRP method. Under standard assumptions, we show that these three methods converge globally with a Wolfe line search. We also report some numerical results to show the efficiency of the proposed methods.
References:
| [1] | Fletcher, R, Reeves, C: Function minimization by conjugate gradients. J. Comput. 7, 149-154 (1964) · Zbl 0132.11701 · doi:10.1093/comjnl/7.2.149 |
| [2] | Polak, B, Ribière, G: Note sur la convergence des méthodes de directions conjuguées. Rev. Fr. Inform. Rech. Oper. 16, 35-43 (1969) · Zbl 0174.48001 |
| [3] | Gilbert, JC, 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 |
| [4] | Liu, YL, Storey, CS: Efficient generalized conjugate gradient algorithms, part 1: theory. J. Optim. Theory Appl. 69, 129-137 (1991) · Zbl 0702.90077 · doi:10.1007/BF00940464 |
| [5] | Dai, YH, Yuan, YX: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10, 177-182 (2000) · Zbl 0957.65061 · doi:10.1137/S1052623497318992 |
| [6] | Hestenes, MR, Stiefel, EL: Method of conjugate gradient for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409-432 (1952) · Zbl 0048.09901 · doi:10.6028/jres.049.044 |
| [7] | Fletcher, R: Practical Methods of Optimization. Volume 1: Unconstrained Optimization. Wiley, New York (1987) · Zbl 0905.65002 |
| [8] | Grippo, L, Luidi, S: A globally convergent version of the Polak-Ribière gradient method. Math. Program. 78, 375-391 (1997) · Zbl 0887.90157 |
| [9] | Cheng, WY: A two-term PRP-based descent method. Numer. Funct. Anal. Optim. 28, 1217-1230 (2007) · Zbl 1138.90028 · doi:10.1080/01630560701749524 |
| [10] | Zhang, L, Zhou, WJ, Li, DH: 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 |
| [11] | Yu, GH, Guan, LT, Chen, WF: Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optim. Methods Softw. 23, 275-293 (2008) · Zbl 1279.90166 · doi:10.1080/10556780701661344 |
| [12] | Dolan, ED, Moré, JJ: Benchmarking optimization software with performance profiles. Math. Program. 91, 201-213 (2002) · Zbl 1049.90004 · doi:10.1007/s101070100263 |
| [13] | Wei, ZX, Li, G, Qi, LQ: Global convergence of the Polak-Ribière-Polyak conjugate gradient method with inexact line searches for non-convex unconstrained optimization problems. Math. Comput. 77, 2173-2193 (2008) · Zbl 1198.65091 · doi:10.1090/S0025-5718-08-02031-0 |
| [14] | Li, G, Tang, CM, Wei, ZX: New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. J. Comput. Appl. Math. 202, 523-539 (2007) · Zbl 1116.65069 · doi:10.1016/j.cam.2006.03.005 |
| [15] | Yu, G, Guan, L, Li, G: Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. J. Ind. Manag. Optim. 3, 565-579 (2008) · Zbl 1168.65030 |
| [16] | Dai, YH, Kou, CX: 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 |
| [17] | Neculai, A: Unconstrained optimization by direct searching (2007). http://camo.ici.ro/neculai/UNO/UNO.FOR · Zbl 1116.90114 |
| [18] | Wei, ZX, Yao, SW, Liu, LY: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341-1350 (2006) · Zbl 1116.65073 · doi:10.1016/j.amc.2006.05.150 |
| [19] | Dai, ZF, Wen, FH: Another improved Wei-Yao-Liu nonlinear conjugate gradient method with sufficient descent property. Appl. Math. Comput. 218, 7421-7430 (2012) · Zbl 1254.65074 · doi:10.1016/j.amc.2011.12.091 |
| [20] | Yu, GH, Zhao, YL, Wei, ZX: A descent nonlinear conjugate gradient method for large-scale unconstrained optimization. Appl. Math. Comput. 187, 636-643 (2007) · Zbl 1117.65097 · doi:10.1016/j.amc.2006.08.087 |
| [21] | Zoutendijk, G.; Abadie, J. (ed.), Nonlinear programming, computational methods, 37-86 (1970), Amsterdam · Zbl 0336.90057 |
| [22] | Dai, ZF, Tian, BS: Global convergence of some modified PRP nonlinear conjugate gradient methods. Optim. Lett. 5(4), 615-630 (2011) · Zbl 1228.90153 · doi:10.1007/s11590-010-0224-8 |
| [23] | Dai, YH, Liao, LZ: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43, 97-101 (2001) · Zbl 0973.65050 · doi:10.1007/s002450010019 |
| [24] | Hager, WW, Zhang, HC: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35-58 (2006) · Zbl 1117.90048 |
| [25] | Wu, QZ, Zheng, ZY, Deng, W: Operations Research and Optimization, MATLAB Programming, pp. 66-69. China Machine Press, Beijing (2010) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
