非线性共轭梯度法研究进展  

Advances in nonlinear conjugate gradient methods

作　　者：戴彧虹[1,2] 刘泽显 Yu-Hong Dai;Zexian Liu

机构地区：[1]中国科学院数学与系统科学研究院计算数学与科学工程计算研究所,北京100190 [2]中国科学院大学数学科学学院,北京100049 [3]贵州大学数学与统计学院,贵阳550025

出　　处：《中国科学:数学》2025年第2期427-450,共24页Scientia Sinica:Mathematica

基　　金：中国科学院战略性先导科技专项(A类)(批准号:XDA27010101);国家自然科学基金(批准号:12021001,11991021和12261019);贵州省自然科学基金(批准号:黔科合基础-ZK[2022]一般084)资助项目。

摘　　要：共轭梯度法具有迭代简单、所需存储量少和数值效果好等优点,是求解大规模无约束优化问题极具竞争力的方法.本文首先介绍非线性共轭梯度法的收敛理论及其进展;然后将非线性共轭梯度法分为早期的共轭梯度法、下降的共轭梯度法和充分下降的共轭梯度法,逐一介绍其理论与算法的最新进展,并讨论共轭梯度法的杂交策略、重启策略及其研究前沿;最后介绍Dai-Liao共轭梯度法和其他共轭梯度法的最新进展,并指出未来可能的研究方向.Due to the simple iterative form,low storage requirement,and good numerical performance,nonlinear conjugate gradient methods have become a class of highly competitive iterative methods for solving large-scale unconstrained optimization problems.Firstly,we review the convergence theories and their advance for nonlinear conjugate gradient methods.Then,we divide nonlinear conjugate gradient methods into early conjugate gradient methods,descent conjugate gradient methods,and sufficiently descent conjugate gradient methods,and summarize the advances in their theories and algorithms respectively;we also discuss the hybrid and restart strategies of nonlinear conjugate gradient methods and their progress.Finally,we focus on the development of Dai-Liao conjugate gradient methods and other conjugate gradient methods and point out some future research topics.

关 键 词：共轭条件 共轭梯度法 无约束优化 全局收敛性 线搜索 

分 类 号：O224[理学—运筹学与控制论]