AAM  >> Vol. 2 No. 1 (February 2013)

    一个修改的LS非线性共轭梯度算法
    A Modified LS Nonlinear Conjugate Gradient Algorithms

  • 全文下载: PDF(409KB)    PP.48-54   DOI: 10.12677/AAM.2013.21007  
  • 下载量: 1,613  浏览量: 6,269   国家自然科学基金支持

作者:  

陈 海:广西大学数学与信息科学学院,南宁

关键词:
共轭梯度法下降性全局收敛性 Conjugate Gradient Method; Descent Property; Global Convergence

摘要:

我们给出一个修改的LS共轭梯度公式,此公式能保证参数βk非负且搜索方向在不需要任何线搜索下具有充分下降性。在适当条件下,证明该方法对一般函数具有全局收敛性,同时给出数值检验结果。

In this paper, a modified LS conjugate gradient formula is proposed. This formula can ensure that the scalar holds and the search direction possesses the sufficiently descent property without any line search. The global convergence will be established for general functions under suitable conditions and numerical results are reported.

文章引用:
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(1): 48-54. http://dx.doi.org/10.12677/AAM.2013.21007

参考文献

[1] E. Polak, G. Ribiere. Note sur la convergence de méthodes de directions conjugées. Revue Française d’Informatique et Recherche Opératinelle, Série Rouge Tome 3, 1969, 16(1): 35-43.
[2] B. T. Polyak. The conjugate gradient method in extreme problems. USSR Computational Mathematics and Mathematical Physics, 1969, 9: 94-112.
[3] R. Fletcher, C. M. Reeves. Function minimization by conjugate gradients. Computer Journal, 1964, 7(2): 149-154.
[4] R. Fletcher. Practical method of optimization, Volume I: Unconstrained optimization (2nd Edition). Hoboken: John Wiley & Sons, 2007.
[5] Y. Liu, C. Storey. Efficient generalized conjugate gradient algorithms, part 1: Theory. Journal of Optimization Theory and Application, 1992, 69(1): 17-41.
[6] M. R. Hestenes, E. Stiefel. Method of conjugate gradient for solving linear equations. Journal of Research of the National Bureau Standards, 1952, 49(6): 409-436.
[7] Y. Dai, Y. Yuan. A nonlinear conjugate gradient with a strong global convergence property. SIAM Journal on Optimization, 2000, 10(1): 177-182.
[8] Y. Dai, L. Z. Liao. New conjugacy conditions and related nonlinear conjugate methods. Applied Mathematics and Optimization, 2001, 43(1): 87-101.
[9] W.W. Hager, H. Zhang. A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization, 2005, 16(1): 170-192.
[10] W.W. Hager, H. Zhang. Algorithm 851: CGDESENT, a conjugate gradient method with guaranteed descent. ACM Transactions on Mathematical Software, 2006, 32: 113-137.
[11] G. Li, C. Tang and Z. Wei. New conjugacy condition and related new conjugate gradient methods for unconstrained optimization problems. Journal of Computational and Applied Mathematics, 2007, 202(2): 532-539.
[12] Z. Wei, G. Li and L. Qi. New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Applied Mathematics and Computation, 2006, 179(2): 407-430.
[13] Z. Wei, G. Li and L. Qi. Global convergence of the PRP conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems. Mathematics of Computation, 2008, 77: 2173-2193.
[14] Z. Wei , S. Yao and L. Lin. The convergence properties of some new conjugate gradient methods. Applied Mathematics and Computation, 2006, 183(2): 1341-1350.
[15] G. L. Yuan. Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optimization Letters, 2009, 3(1): 11-21
[16] G. L. Yuan, X. W. Lu. A modified PRP conjugate gradient method. Annals of Operations Research, 2009, 166(1): 73-90.
[17] G. L. Yuan, X. W. Lu and Z. X. Wei. A conjugate gradient method with descent direction for unconstrained optimization. Journal of Computational and Applied Mathematics, 2009, 233(2): 519-530.
[18] J.C. Gilbert, J. Nocedal. Global Convergence properties of conjugate gradient methods for optimization. SIAM Journal on Optimization, 1992, 2(1): 21-42.
[19] http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume24/ortizboyer05a-html/node6.html