广义鞍点问题的改进的类SOR算法

doi:10.12286/jssx.2020.1.39

计算数学 ›› 2020, Vol. 42 ›› Issue (1): 39-50.DOI: 10.12286/jssx.2020.1.39

广义鞍点问题的改进的类SOR算法

张纯^1,2, 贾泽慧³, 蔡邢菊¹

1. 南京师范大学数学科学学院, 南京 210023;
2. 中国人民解放军陆军工程大学基础部, 南京 211101;
3. 南京信息工程大学数学与统计学院, 南京 210044;
4. 北京航空航天大学数学科学学院, 北京 100191

收稿日期:2018-03-07 出版日期:2020-02-15 发布日期:2020-02-15
基金资助:
国家自然科学基金（11625105，11926358，11871279，11571178，11801279），江苏省自然科学基金（BK2018078），南京信息工程大学科研启动基金（2017r059）.

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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.

Zhang Chun, Jia Zehui, Cai Xingju, Han Deren. AN IMPROVED SOR-TYPE ALGORITHM FOR SLOVING GENERALIZED SADDLE-POINT PROBLEMS[J]. Mathematica Numerica Sinica, 2020, 42(1): 39-50.

AN IMPROVED SOR-TYPE ALGORITHM FOR SLOVING GENERALIZED SADDLE-POINT PROBLEMS

Zhang Chun^1,2, Jia Zehui³, Cai Xingju¹, Han Deren⁴

1. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China;
2. Department of Basic Courses, The PLA Army Engineering University, Nanjing 211101, China;
3. School of Mathematics and Statistics, Nanjing University of Information Science&Technology, Nanjing 210044, China;
4. School of Mathematical Sciences, Beihang University, Beijing 100191, China

Received:2018-03-07 Online:2020-02-15 Published:2020-02-15

针对广义鞍点问题，本文提出了一个改进的类逐次超松弛迭代算法，在较弱的条件下，分析了算法的收敛性及线性收敛率.新算法的每步计算量与已有的算法类似，都是需要（近似）求解线性方程组，但新算法有更好的灵活度通过合适地选取参数矩阵，每一步子问题可以容易地求解，甚至可以有闭式解（closed-form solution）.数值实验结果显示了新算法的有效性.

关键词： 鞍点问题, 类SOR算法, 全局收敛性, 收敛率

For the generalized saddle point problem, we develop an improved class of successive over relaxation algorithms. Under mild conditions, we prove its convergence and establish its linear rate of convergence. While, as the classical methods, it needs to solve some linear system of equations approximately to get the next iterate, the flexibility in choosing the involved matrices makes the subproblems easy or even to have closed form solutions, which leads the algorithm to be an efficient one. Preliminary numerical results show the effectiveness of the new method.

Key words: saddle point problem, SOR algorithm, global convergence, convergence rate

MR(2010)主题分类:

65F10
65H10

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摘要

广义鞍点问题的改进的类SOR算法

AN IMPROVED SOR-TYPE ALGORITHM FOR SLOVING GENERALIZED SADDLE-POINT PROBLEMS

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