• 论文 • 上一篇    下一篇

建立在修正BFGS公式基础上的新的共轭梯度法

王开荣, 刘奔   

  1. 重庆大学数学与统计学院, 重庆 401331
  • 收稿日期:2011-06-15 出版日期:2012-02-15 发布日期:2012-02-21
  • 基金资助:

    重庆市2010年高等教育教学改革研究重点项目(项目编号102104).

王开荣, 刘奔. 建立在修正BFGS公式基础上的新的共轭梯度法[J]. 计算数学, 2012, 34(1): 81-92.

Wang Kairong, Liu Ben. NEW CONJUGATE GRADIENT METHOD BASED ON THE MODIFIED BFGS FORMULA[J]. Mathematica Numerica Sinica, 2012, 34(1): 81-92.

NEW CONJUGATE GRADIENT METHOD BASED ON THE MODIFIED BFGS FORMULA

Wang Kairong, Liu Ben   

  1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
  • Received:2011-06-15 Online:2012-02-15 Published:2012-02-21
共轭梯度法是一类非常重要的用于解决大规模无约束优化问题的方法. 本文通过修正的BFGS公式提出了一个新的共轭梯度方法. 该方法具有不依赖于线搜索的充分下降性. 对于一般的非线性函数, 证明了该方法的全局收敛性. 数值结果表明该方法是有效的.
Conjugate gradient methods are a class of important methods for large scale unconstrained optimization. In this paper, motivated by the modified BFGS formula, we proposed a new conjugate gradient method with the sufficient descent direction independent of the line search and obtained the global convergence of the method for the general functions. The numerical results show that the proposed method is efficient.

MR(2010)主题分类: 

()
[1] Hestenes M R, Stiefel E L. Methods of conjugate gradients for solving linear systems[J]. Journal of Research of the National Bureau of Standards, 1952, 49(6): 409-32.

[2] Polak B, Ribière G. Note sur la convergence des méthodes de directions conjuguées[J]. Rev. Fr. Inform. Rech. Oper., 1969, 16: 35-43.

[3] Polyak B T. The conjugate gradient method in extreme problems[J]. USSR Comp. Math. Math. Phys., 1969, 9: 94-112.

[4] Liu Y L, Storey C S. Efficient generalized conjugate gradient algorithms, Part1: Theory[J]. Journal of Optimization Theory and Applications, 1991, 69(1): 129-137.

[5] Dai Y H, Yuan Y X. A nonlinear conjugate gradient method with a strong global convergence property[J]. SIAM J. Optim., 2000, 10: 177-182.

[6] Fletcher R, Reeves C M. Function minimization by conjugate gradients[J]. Comput. J., 1964, 7: 149-154.

[7] Fletcher R. Practical Methods of Optimization, Vol I: Unconstrained Optimization[M]. New York, Wiley and Sons, 1987.

[8] Zoutendijk G. Nonlinear Programming, Computational Methods[M]. North-Holland, Amsterdam, J. Abadie, 1970: 37-86.

[9] Al-Baali M. Descent property and global convergence of the Fletcher-Reeves method with inexact line search[J]. IMA Journal of Numerical Analysis, 1985, 5(1): 121-124.

[10] Gilbert J C, Nocedal J. Global convergence properties of conjugate gradient methods for optimization[ J]. SIAM J. Optim., 1992, 2(1): 21-42.

[11] Powell MJ D. Nonconvex minimization calculations and the conjugate gradient method[M]. Berlin, Springer, 1984.

[12] 戴志锋, 陈兰平. 一种混合的HS-DY共轭梯度法[J]. 计算数学, 2005, 27(4): 429-436.

[13] 焦宝聪, 陈兰平, 潘翠英. Goldstein线搜索下混合共轭梯度法的全局收敛性[J].计算数学, 2007, 29(2): 137-146.

[14] Dai Y H, Liao L Z. New conjugacy conditions and related nonlinear conjugate gradient methods[J]. Appl. Math. Optim., 2001, 43: 87-101.

[15] Yabe H, Takano M. Global convergence properties of nonlinear conjugate gradient methods with modified secant conditions[J]. Comput. Optim. Appl., 2004, 28: 203-225.

[16] Li G Y, Tang C M,Wei Z X. New conjugacy condition and related new conjugate gradient methods for unconstrained optimization[J]. Journal of Computational and Applied Mathematics, 2007, 202: 523-539.

[17] Cheng W Y, Liu Q F. Sufficient descent nonlinear conjugate gradient methods with conjugacy condition[J]. Numer. Algor., 2010, 53: 113-131.

[18] 戴彧虹,袁亚湘. 非线性共轭梯度法[M]. 上海: 上海科学技术出版社, 2000.

[19] Dennis J E, Moré J J. A characterization of superlinear convergence and its application to qusi- Newton methods[J]. Mathematics of Computation, 1974, 28(2): 549-560.

[20] Dennis J E, Moré J J. Quasi-Newton methods, motivation and theory[J]. SIMA Review, 1977, 19(1): 46-89.

[21] Nocedal J. Updaing quai-Newton matrix with limited storage[J]. Mathematics of Computation, 1980, 35(2): 773-782.

[22] Shanno D F. Conjugate gradient methods with inexact searches[J]. Mathematics of Operations Research, 1978, 3(1): 244-256.

[23] Zhang J Z, Deng N Y, Chen L H. New quasi-Newton equation and related methods for unconstrained optimization[J]. Journal of Optimization Theory and Applications, 1999, 102: 147-167.

[24] Zhang J Z, Xu C X. Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations[J]. Journal of Computational and Applied Mathematics, 2001, 137: 169- 278.

[25] Li D H, Fukushima M. A modified BFGS method and its global convergence in nonconvex minimization[ J]. Journal of Computational and Applied Mathematics, 2001, 129: 15-35.

[26] Yuan Y X. A Modified BFGS Algorithm for Unconstrained Optimization[J]. IMA Journal of Numerical Analysis, 1991, 11: 325-332.

[27] Wei Z X, Li G Y, Qi L Q. New quasi-Newton methods for unconstrained optimization problems[J]. Applied Mathematics and Computation, 2006, 175: 1156-1188.

[28] Wolfe P. Convergence conditions for ascent methods[J]. SIMA Review, 1969, 11: 226-235.

[29] Bongartz I, Conn A R, Gould N. CUTE: Constrained and Unconstrained Testing Environments[J]. ACM Trans. Math. Softw., 1995, 21: 123-160.

[30] GOULD N, TOINT P L. CUTEr and SifDec: A Constrained and Unconstrained Testing Environment, revisited[J]. ACM Trans. Math. Softw., 2003, 29(4): 373-394.

[31] Hager W W, Zhang H C. A new conjugate gradient method with guaranteed descent and an efficient line search[J]. SIAM J. Optim., 2005, 16: 170-192.
[1] 刘金魁, 孙悦, 赵永祥. 凸约束伪单调方程组的无导数投影算法[J]. 计算数学, 2021, 43(3): 388-400.
[2] 尹江华, 简金宝, 江羡珍. 凸约束非光滑方程组一个新的谱梯度投影算法[J]. 计算数学, 2020, 42(4): 457-471.
[3] 吴敏华, 李郴良. 求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法[J]. 计算数学, 2020, 42(2): 223-236.
[4] 张纯, 贾泽慧, 蔡邢菊. 广义鞍点问题的改进的类SOR算法[J]. 计算数学, 2020, 42(1): 39-50.
[5] 王福胜, 张瑞. 不等式约束极大极小问题的一个新型模松弛强次可行SQCQP算法[J]. 计算数学, 2018, 40(1): 49-62.
[6] 刘金魁. 解凸约束非线性单调方程组的无导数谱PRP投影算法[J]. 计算数学, 2016, 38(2): 113-124.
[7] 刘亚君, 刘新为. 无约束最优化的信赖域BB法[J]. 计算数学, 2016, 38(1): 96-112.
[8] 简金宝, 尹江华, 江羡珍. 一个充分下降的有效共轭梯度法[J]. 计算数学, 2015, 37(4): 415-424.
[9] 张凯院, 牛婷婷, 聂玉峰. 一类非线性矩阵方程对称解的双迭代算法[J]. 计算数学, 2014, 36(1): 75-84.
[10] 袁敏, 万中. 求解非线性P0互补问题的非单调磨光算法[J]. 计算数学, 2014, 36(1): 35-50.
[11] 简金宝, 唐菲, 黎健玲, 唐春明. 无约束极大极小问题的广义梯度投影算法[J]. 计算数学, 2013, 35(4): 385-392.
[12] 刘金魁. 两种有效的非线性共轭梯度算法[J]. 计算数学, 2013, 35(3): 286-296.
[13] 范斌, 马昌凤, 谢亚君. 求解非线性互补问题的一类光滑Broyden-like方法[J]. 计算数学, 2013, 35(2): 181-194.
[14] 简金宝, 马鹏飞, 徐庆娟. 不等式约束优化一个基于滤子思想的广义梯度投影算法[J]. 计算数学, 2013, 35(2): 205-214.
[15] 潘克家, 胡宏伶, 陈传淼, 汤井田. 外推瀑布式多网格法的OpenMP并行化[J]. 计算数学, 2012, 34(4): 425-436.
阅读次数
全文


摘要