凸约束非光滑方程组一个新的谱梯度投影算法

doi:10.12286/jssx.2020.4.457

计算数学 ›› 2020, Vol. 42 ›› Issue (4): 457-471.DOI: 10.12286/jssx.2020.4.457

凸约束非光滑方程组一个新的谱梯度投影算法

尹江华^1,2,3,4, 简金宝², 江羡珍²

1 内蒙古大学数学科学学院, 呼和浩特 010021;
2 广西民族大学数学物理学院, 南宁 530006;
3 玉林师范学院, 复杂系统优化与大数据处理广西高校重点实验室, 玉林 537000;
4 广西科技师范学院数学与计算机科学学院, 来宾 546199

收稿日期:2018-12-09 出版日期:2020-11-15 发布日期:2020-11-15
通讯作者: 简金宝,Email:jianjb@gxu.edu.cn.
基金资助:
国家自然科学基金（11771383），广西自然科学基金（2016GXNSFDA380019，2016GXNSFAA380028），广西高校中青年教师基础能力提升项目（2017KY0537，2018KY0700），复杂系统优化与大数据处理广西高校重点实验室开放课题（2017CSOBDP0105），广西科技厅项目（AD16450003）资助.

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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.

Yin Jianghua, Jian Jinbao, Jiang Xianzhen. A SPECTRAL GRADIENT PROJECTION ALGORITHM FOR CONVEX CONSTRAINED NONSMOOTH EQUATIONS BASED ON AN ADAPTIVE LINE SEARCH[J]. Mathematica Numerica Sinica, 2020, 42(4): 457-471.

A SPECTRAL GRADIENT PROJECTION ALGORITHM FOR CONVEX CONSTRAINED NONSMOOTH EQUATIONS BASED ON AN ADAPTIVE LINE SEARCH

Yin Jianghua^1,2,3,4, Jian Jinbao², Jiang Xianzhen²

1 School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China;
2 College of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, China;
3 Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China;
4 College of Mathematics and Computer Science, Guangxi Science&Technology Normal University, Laibin 546199, China

Received:2018-12-09 Online:2020-11-15 Published:2020-11-15

基于寻找分离超平面的三种经典线搜索技术，本文提出了一种自适应线搜索技术.结合谱梯度投影法，提出了凸约束非光滑单调方程组的一个谱梯度投影算法.该算法不需要计算和存储任何矩阵，因而适合求解大规模非光滑的非线性单调方程组.在较弱的条件下，证明了方法的全局收敛性，并分析了算法的收敛率.数值试验结果表明算法是有效的和鲁棒的.

关键词： 非光滑方程组, 自适应线搜索, 谱梯度投影算法, 全局收敛性, 收敛率

Based on three classic line search techniques for finding separating hyperplane, this paper proposes an adaptive line search method. Combining this with the spectral gradient projection method, a spectral gradient projection algorithm for nonsmooth monotone equations with convex constraints is proposed. The proposed method does not calculate and store any matrix, so it is suitable for solving large-scale nonsmooth monotone nonlinear equations. Under mild conditions, the global convergence of the proposed method is proved, and its rate of convergence is analyzed. Numerical experiments show that the proposed algorithm is efficient and robust.

Key words: nonsmooth equations, adaptive line search, spectral gradient projection algorithm, global convergence, convergence rate

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摘要

凸约束非光滑方程组一个新的谱梯度投影算法

A SPECTRAL GRADIENT PROJECTION ALGORITHM FOR CONVEX CONSTRAINED NONSMOOTH EQUATIONS BASED ON AN ADAPTIVE LINE SEARCH

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