一个充分下降的修正 PRP 型谱共轭梯度法

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

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.