一类非单调保守BFGS算法研究

doi:10.12286/jssx.2011.4.387

计算数学 ›› 2011, Vol. 33 ›› Issue (4): 387-396.DOI: 10.12286/jssx.2011.4.387

一类非单调保守BFGS算法研究

万中, 冯冬冬

中南大学数学科学与计算技术学院, 长沙 410083

收稿日期:2011-05-11 出版日期:2011-11-15 发布日期:2011-11-15
基金资助:
国家自然科学基金(71071162, 70921001).

PDF ( 1225 ) 引用被引次数 ( 2 )

万中, 冯冬冬. 一类非单调保守BFGS算法研究[J]. 计算数学, 2011, 33(4): 387-396.

Wan Zhong, Feng Dongdong. INVESTIGATION ON A CLASS OF NONMONOTONE CAUTIOUS BFGS ALGORITHMS[J]. Mathematica Numerica Sinica, 2011, 33(4): 387-396.

INVESTIGATION ON A CLASS OF NONMONOTONE CAUTIOUS BFGS ALGORITHMS

Wan Zhong, Feng Dongdong

School of Math. Sci. & Comput. Tech., Central South Univer., Changsha 410083, China

Received:2011-05-11 Online:2011-11-15 Published:2011-11-15

基于非单调线搜索在寻求优化问题最优解中的优越性, 提出了一类新的非单调保守BFGS算法. 同已有方法不同, 该算法中用来控制非单调性程度的算法参数不是取固定值, 而是利用已有目标函数和梯度函数的信息自动调整其取值, 以改善算法的数值表现. 在合适的假设条件下, 建立了新的非单调保守 BFGS算法的全局收敛性. 用基准测试优化问题测试了算法, 其数值结果表明该算法比以往同类算法具有更高的计算效率.

关键词： 无约束最优化, 保守BFGS方法, 非单调线搜索, 全局收敛性

With the superiority of nonmonotone line search in finding a solution of optimization problem, a class of nonmonotone cautious BFGS algorithms are developed. Different from the existing techniques of nonmonotone line search, the parameter, which is employed to control the magnitude of nonmonotonicity, is modified (not a fixed value) by the known information of the objective function and the gradient function such that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence is proved for this algorithm. Implementing the algorithm to solve some benchmark test problems, the results demonstrate that it is more effective than the similar algorithms.

Key words: unconstrained optimization, cautious BFGS method, nonmonotone line search, global convergence

MR(2010)主题分类:

90C26
90B50

分享此文:

[1] Li Donghui, Masao Fukushima. On the global convergence of the BFGS method for nonconvex unconstrained optimization problems[J]. Society for Industrial and Applied Mathematics, 2001, 11(4): 1054-1064.

[2] Rauf A I, FukushimaM. Globally convergent BFGS method for nonsmooth convex optimization[J]. Journal of Optimization Theory and Applications, 2000, 104(3): 539-558.

[3] Liu JiangGuo, Guo Qiang. Global convergence properties of the modified BFGS method associating with general line search model[J]. J. Appl. Math. Computing, 2004, 18(1-2): 195-205.

[4] Guo Qiang, Liu JianGuo. Global convergence of a modified BFGS-type method for unconstrained monconvex minimization[J]. J. Appl. Math. Computing, 2007, 24(1-2): 325-331.

[5] Guo Qiang, Liu Jian-Guo, Wang Dan-Hong. A modified BFGS method and its superlinear convergence in nonconvex minimization with general line search rule[J]. J. Appl. Math. Computing, 2008, 28: 435-446.

[6] 李董辉, 童小娇, 万中. 数值最优化算法与理论[M]. 北京: 科学出版社, 2010.

[7] Grippo L, Lampariello F, Lucidi S. Newton-type algorithms with nonmonotone line search for large-scale unconstrained optimization[J]. System Modelling and Optimization, 1988, 113: 187- 196.

[8] Grippo L, Lampariello F, Lucidi S. A class of nonmonotone stabilization methods in unconstrained optimization[J]. Numer. Math., 199l, 59: 779-805.

[9] Dai Y H, Di Pillo G. On the nonmonotone line search[J]. Journal of Optimization Theory and Applications, 2002, 112(2): 315-330.

[10] Zhou Weijun, Zhang Li, Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization[J]. Journal of Computational and Applied Mathematics, 2009, 223: 40-47.

[11] Yuan Gonglin, Wei Zengxin. New line search methods for unconstrained optimization[J]. Journal of the Korean Statistical Society, 2009, 38: 29-39.

[12] Yuan Gong Lin, Wei Zeng Xin. The superlinear convergence analysis of a nonmonotone BFGS algoriyhm on convex objective functions[J]. Acta Mathematica Sinica, English Series, 2008, 24(1): 35-42.

[13] Xiao Yunhai, Sun Huijuan, Wang Zhiguo. A globally convergent BFGS method with nonmonotone line search for non-convex minimization[J]. Journal of Computational and Applied Mathematics, 2009, 230: 95-106.

[14] Zhang Hongchao, W Hager William. A nonmonotone line search technique and its application to unconstrained optimization[J]. Society for Industrial and Applied Mathematics, 2004, 14(4): 1043-1056.

[1]	孙青青, 王川龙. 低秩稀疏矩阵恢复的快速非单调交替极小化方法[J]. 计算数学, 2021, 43(4): 516-528.
[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]	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.
[4]	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.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	毕亚倩, 刘新为. 求解界约束优化的一种新的非单调谱投影梯度法[J]. 计算数学, 2013, 35(4): 419-430.
[11]	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.
[12]	孙清滢, 段立宁, 陈颖梅, 王宣战, 宫恩龙, 徐胜来. 基于修正拟牛顿方程的两阶段步长非单调稀疏对角变尺度梯度投影算法[J]. 计算数学, 2013, 35(2): 113-124.
[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]	刘景辉, 马昌凤, 陈争. 解无约束优化问题的一个新的带线搜索的信赖域算法[J]. 计算数学, 2012, 34(3): 275-284.

阅读次数

全文

摘要

一类非单调保守BFGS算法研究

INVESTIGATION ON A CLASS OF NONMONOTONE CAUTIOUS BFGS ALGORITHMS

扫码分享