一个充分下降的修正PRP型谱共轭梯度法  

A Modiffed PRP Type Spectral Conjugate Gradient Method with Sufficient Descent Property

作　　者：简金宝 宋丹 江羡珍 JIAN Jinbao;SONG Dan;JIANG Xianzhen(College of Mathematics and Physics,Guangxi University for Nationalities,Nanning 530006)

机构地区：[1]广西民族大学数学与物理学院,南宁530006

出　　处：《工程数学学报》2022年第2期265-276,共12页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11771383);广西自然科学基金(2018GXNSFFA281007);广西民族大学科研基金(2018KJQD02);广西民族大学研究生创新项目(gxun-chxzs2019034)。

摘　　要：谱共轭梯度法是共轭梯度法的一种重要延拓,可以通过共轭参数和谱参数二维度调整,使得所设计算法的搜索方向满足某一预设条件,比如充分下降条件或共轭条件等。谱参数和共轭参数的设计是谱共轭梯度法的两大核心工作,决定方法的收敛性和数值效果。基于PRP方法,构造了一个修正的PRP型共轭参数,该共轭参数不仅保持了PRP公式的结构和性能,而且具有FR方法的收敛性质。利用充分下降条件取定一个谱参数,与修正的PRP型共轭参数结合,建立一个新的谱共轭梯度算法。该算法不依赖于任何线搜索就可以满足充分下降条件。常规假设条件下,采用强Wolfe线搜索准则产生步长,证明了新算法的全局收敛性。通过100个算例对该算法进行数值测试并与其他五个算法进行比较,同时采用性能图对数值结果进行直观展示,结果表明该算法是有效的。A modified PRP type spectral conjugate gradient method with sufficient descent property spectral conjugate gradient method is an important extension of conjugate gradient method.By adjusting the conjugate parameters and spectral parameters,the search direction of the designed algorithm can meet a certain preset condition,such as sufficient descent condition or conjugate condition.The two core tasks of designing spectral parameters and conjugate parameters are spectral conjugate gradient method,which determine the convergence and numerical effect of the method.Based on the PRP method,a modified PRP type conjugate parameter is proposed and a spectral parameter is chosen by sufficient descent conditions,and then a new spectral conjugate gradient method is established.Sufficient descent condition of the new algorithms does not depend on any line search.Under the usual assumptions,using the strong Wolfe line search to generate the step-length,the global convergence of the presented method is proved.By testing 100 numerical experiments,the corresponding performance profiles for the proposed method and other five comparisons are reported,which indicate that the proposed method is effective.

关 键 词：无约束优化 谱共轭梯度法 强Wolfe线搜索 全局收敛性 

分 类 号：O221.2[理学—运筹学与控制论]