基于拟牛顿方程一个改进的非线性共轭梯度算法
A Modified Nonlinear Conjugate Gradient Algorithm Using Secant Conditions
DOI: 10.12677/AAM.2019.84076, PDF,    科研立项经费支持
作者: 梁 静, 马国栋, 时瑜婷, 林志梅, 邹文婷, 李柳娜：玉林师范学院，数学与统计学院，广西 玉林
关键词: 共轭梯度法；充分下降性；全局收敛性；Conjugate Gradient Method； Sufficient Descent Property； Global Convergence
摘要: 基于拟牛顿方程，借鉴文[1]的思想，本文给出一个改进的具有全局收敛非线性共轭梯度算法。该文的算法对文[1]中算法进行了改进，所产生的搜索方向具有充分下降性。在温和的假设下，新算法具有全局收敛性。最后，数值结果检验其有效性。
Abstract: In this paper, based on the idea of Ref. [1], we propose a modified conjugate gradient method using secant conditions for unconstrained optimization problems. The proposed algorithm improves the method in Ref. [1], which possesses the following properties: the search direction has the sufficient descent property; the global convergence of the given algorithm will be established under suitable assumptions; numerical results are reported to test its efficiency.
文章引用：梁静, 马国栋, 时瑜婷, 林志梅, 邹文婷, 李柳娜. 基于拟牛顿方程一个改进的非线性共轭梯度算法[J]. 应用数学进展, 2019, 8(4): 676-683. https://doi.org/10.12677/AAM.2019.84076

参考文献

[1] 陈海. 一个修改的LS非线性共轭梯度算法[J]. 应用数学进展, 2013(2): 48-54.
[2] Polak, E. and Ribiere, G. (1969) Note sur la xonvergence de directions conjugees. Rev. Francaise informat Recherche Operatinelle, 3e Annee, 16, 35-43.
[3] Polyak, B.T. (1969) The Conjugate Gradient Method in Extreme Problems. USSR Computational Mathematics and Mathematical Physics, 9, 94-112
[4] Hestenes, M.R. and Stiefel, E. (1952) Method of Conjugate Gradient for Solving Linear Equations. Journal of Research of the National Bureau of Standards, 49, 409-436. [Google Scholar] [CrossRef] 
[5] Fletcher, R. and Reeves, C. (1964) Function Minimization bu Conjugate Gradients. Compute Journal, 7, 149-154.
[6] Dai, Y. and Yuan, Y. (1999) A Nonlinear Conjugate Gradient with a Strong Global Convergence Properties. SIAM Journal on Optimization, 10, 177-182. [Google Scholar] [CrossRef] 
[7] Fletcher, R. (1980) Practical Method of Optimization, Vol. I: Unconstrained Optimization. 2nd Edition, Wiley, Hoboken.
[8] Liu, Y. and Storey, C. (1992) Efficient Generalized Conjugate Gradient Algorithms, Part 1: Theory. Journal of Optimization Theory and Application, 69, 17-41.
[9] Yuan, G. (2009) Modified Nonlinear Conjugate Gradient Methods with Sufficient Descent Property for Large-Scale Optimization Problems. Optimization Letters, 3, 11-21. [Google Scholar] [CrossRef] 
[10] Yuan, G. and Wei, Z. (2010) Convergence Analysis of a Modified BFGS Method on Convex Minimizations. Computational Optimization and Applications, 47, 237-255. [Google Scholar] [CrossRef] 
[11] Zhang, J., Deng, N. and Chen, L. (1999) New Quasi-Newton Equation and Related Methods for Unconstrained Optimization. Journal of Optimization Theory and Application, 102, 147-167. [Google Scholar] [CrossRef] 
[12] Yuan, G., Wei, Z. and Zhao, Q. (2014) A Modified Polak-Ribière-Polyak Conjugate Gradientalgorithm for Large-Scale Optimization Problems. IIE Transactions, 46, 397-413. [Google Scholar] [CrossRef] 
[13] Yuan, G. and Zhang, M. (2013) A Modified Hestenes-Stiefel Conjugate Gradient Algorithm for Large-Scale Optimization. Numerical Functional Analysis and Optimization, 34, 914-937. [Google Scholar] [CrossRef] 
[14] Gilbert, J.C. and Nocedal, J. (1992) Global Convergence Properties of Conjugate Gradient Methods for Optimization. SIAM Journal on Optimization, 2, 21-42. [Google Scholar] [CrossRef] 
[15] Hager, W.W. and Zhang, H. (2005) A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search. SIAM Journal on Optimization, 16, 170-192. [Google Scholar] [CrossRef] 
[16] Yao, S., He, D. and Shi, L. (2018) An Improved Perry Conjugate Gradient Method with Adaptive Parameter Choice. Numerical Algorithms, 78, 1255-1269. [Google Scholar] [CrossRef] 
[17] Yuan, G. and Lu, X. (2009) A Modified PRP Conjugate Gradient Method. Annals of Operations Research, 66, 73-90. [Google Scholar] [CrossRef] 
[18] Dai, Y. and Liao, L. (2001) New Conjugacy Conditions and Re-lated Nonlinear Conjugate Methods. Applied Mathematics and Optimization, 43, 87-101. [Google Scholar] [CrossRef] 
[19] Hager, W.W. and Zhang, H. (2006) Algorithm 851: CGDESENT, A Conjugate Gradient Method with Guaranteed Descent. ACM Transactions on Mathematical Software, 32, 113-137. [Google Scholar] [CrossRef] 
[20] Li, G., Tang, C. and Wei, Z. (2007) New Conjugacy Condition and Related New Conjugate Gradient Methods for Unconstrained Optimization Problems. Journal of Computational and Applied Mathemathics, 202, 532-539.
[21] Wei, Z., Li, G. and Qi, L. (2006) New Nonlinear Conjugate Gradient Formulas for Large-Scale Unconstrained Optimization Problems. Applied Mathematics and Computation, 179, 407-430. [Google Scholar] [CrossRef] 
[22] Wei, Z., Li, G. and Qi, L. (2008) Global Convergence of the PRP Conjugate Gradient Methods with Inexact Line Search for Nonconvex Unconstrained Optimization Problems. Mathematics of Computation, 77, 2173-2193. [Google Scholar] [CrossRef] 
[23] Wei, Z., Yao, S. and Lin, L. (2006) The Convergence Properties of Some New Conjugate Gradient Methods. Applied Mathematics and Computation, 183, 1341-1350. [Google Scholar] [CrossRef] 
[24] Yuan, G., Lu, X. and Wei, Z. (2009) A Conjugate Gradient Method with Descent Direction for Unconstrained Optimization. Journal of Computational and Applied Mathematics, 102, 519-530. [Google Scholar] [CrossRef] 
[25] 李向荣. 修改的DY和HS共轭梯度算法及其全局收敛性[J]. 理论数学, 2011(1): 1-7.

为你推荐