• 论文 •

### 两种有效的非线性共轭梯度算法

1. 重庆三峡学院 数学与统计学院, 重庆万州 404100
• 收稿日期:2013-01-17 出版日期:2013-08-15 发布日期:2013-09-07
• 基金资助:

重庆市教委项目(KJ121112)

Liu Jinkui. TWO EFFICIENT NONLINEAR CONJUGATE GRADIENT METHODS[J]. Mathematica Numerica Sinica, 2013, 35(3): 286-296.

### TWO EFFICIENT NONLINEAR CONJUGATE GRADIENT METHODS

Liu Jinkui

1. School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, Chongqing, China
• Received:2013-01-17 Online:2013-08-15 Published:2013-09-07

By the structure of CG-DESCENT method[1] and Powell's suggestion in[11], two efficient nonlinear conjugate gradient methods are given. The given methods can be guaranteed the sufficient descent property without out any line search, and be proved the global convergence property for the general functions under the standard Wolfe line search. In particular, by the famous evaluation method of Dolan & Moré[2], the numerical results also show that the proposed methods are more efficient by comparing with the famous CG-DESCENT method using a classical set of problems from CUTEr library.

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 [1] Hager W W and Zhang H. A new conjugate gradient method with guaranteed descent and an efficient line search[J]. SIAM Journal on Optimization, 2005, 16: 170-192.[2] Dolan E D and Mor| J J. Benchmarking optimization software with performance profiles[J]. Mathematical Programming, 2002, 91: 201-213.[3] Hestenes M R. Iterative method for sovling linear equations, NANL Report No 53-9, National Bureau of Standards, Washington, D.C. 1951(later published in JOTA, 1973, 1:322-334). [4] Stiefel E L. Über einige Methodern der Relationsrechnung, Zeitschrifit für Angewandte Mathematik under Physik 1952, 3.[5] Fletcher R, Reeves C. Function minimization by conjugate gradients[J]. Computer Journal, 1964, 7: 149-154.[6] Hestenes M R, Stiefel E L. Methods of conjugate gradients for solving linear systems[J]. J Res Nat Bur Standards Sect. 1952, 5: 409-436.[7] Liu Y and Story C. E fficient generalized conjugate gradient algorithms. Part 1: Theory, J. Optimize. Theory Appl. 1992, 69: 129-137.[8] Polak E, Ribire G. Note sur la xonvergence de directions conjugees[J]. Rev Francaise informat Recherche Operatinelle 3e Annee 1969, 16: 35-43.[9] Polak B T, The conjugate gradient method in extreme problems[J]. USSR Comput. Math. Math. Phys., 1969, 9: 94-112.[10] Dai Y H, Yuan Y X. Nonlinear Conjugate Gradient with a Strong Global Convergence Property[J]. SIAM Journal of Optimization, 2000, 10: 177-182.[11] Powell M J D. Convergence properties of algorithms for nonlinear optimization[J]. SIAM Review, 1986, 28: 487-500.[12] Grippo L, Lucidi S. A globally convergent version of the Polak-Rebiére conjugate gradient method[J]. Math Prog. 1997, 56: 375-391.[13] Zoutendijk G. Nonlinear Programming, Computational Methods, In: J.Abadie(eds0, Integer and Nonlinear Programming, North-Holland, 1970, 37-86.
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