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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (4): 591-604.doi: 10.3969/j.issn.1005-3085.2023.04.006

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一个修正的谱共轭梯度法

朱艺轩,  宋恩彬   

  1. 四川大学数学学院，成都 610044
  • 收稿日期:2022-08-19 接受日期:2023-03-04 出版日期:2023-08-15 发布日期:2024-02-28
  • 基金资助:
    国家自然科学基金(U2066203).

A Modified Spectral Conjugate Method

ZHU Yixuan,  SONG Enbin   

  1. School of Mathematics, Sichuan University, Chengdu 610044
  • Received:2022-08-19 Accepted:2023-03-04 Online:2023-08-15 Published:2024-02-28
  • Supported by:
    The National Natural Science Foundation of China (U2066203).

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摘要：

对无约束优化问题进行了研究，提出了一个修正的谱共轭梯度法。该算法的搜索方向是下降方向，在标准的Wolfe-Powell线搜索下具有全局收敛性，且在适当的条件下，证明了该算法具有线性收敛率。对一些标准的测试函数进行了数值实验，数值实验结果表明所提算法在算法迭代次数，函数调用次数以及程序运行时间等方面是有效的，且与相关算法相比有一定的优势。最后将该算法应用到图像去噪问题，对经典图像Lena与Camera施加了不同的噪声效果并用该算法进行图像去噪，与文献中相关算法进行了对比，通过信噪比这一指标说明该算法有良好的去噪效果。

关键词: 共轭梯度法, 谱共轭梯度法, 全局收敛, 线性收敛率, 图像去噪问题

Abstract:

A modified spectral conjugate gradient method is proposed to solve unconstrained optimization problems. The search direction of the algorithm is the descent direction, and it has global convergence under the Wolfe-Powell line search. It is proved that the algorithm has a linear convergence rate under some suitable conditions. Some numerical experiments on some standard functions show that the algorithm is effective on the facts including number of iteration, number of function calculations and program running time, and has certain advantages compared with related algorithms. Finally, the algorithm is applied to image denoising problems. Some different noise effects are applied for the classical images Lena and Camera. The comparison with related algorithms in references shows that the algorithm has good image denoising effect about the fact of the PSNR.

Key words: conjugate method, spectral conjugate method, global convergence, linear convergent rate, image deconvolution problem

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