一个应用于图像恢复问题的修正共轭梯度算法

doi:10.15960/j.cnki.issn.1007-6093.2026.01.015

运筹学学报（中英文） ›› 2026, Vol. 30 ›› Issue (1): 207-216.doi: 10.15960/j.cnki.issn.1007-6093.2026.01.015

• • 上一篇

一个应用于图像恢复问题的修正共轭梯度算法

刘聪^1,2, 简艾伦^1,2, 袁功林^1,2,†

1. 广西大学数学与信息科学学院, 广西南宁 530004;
2. 广西壮族自治区应用数学中心 (广西大学), 广西南宁 530004

收稿日期:2022-08-17 发布日期:2026-03-16
通讯作者: 袁功林 E-mail:glyuan@gxu.edu.cn
基金资助:
国家自然科学基金 (No. 11661009), 赋能行动计划 (广西重点研发计划)项目 (No. FN2504240023), 广西科技基地和人才专项基金资助项目 (No. 桂科 AD22080047)

A modified conjugate gradient algorithm with its applications in image recovery problems

LIU Cong^1,2, JIAN Ailun^1,2, YUAN Gonglin^1,2,†

1. School of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, China;
2. Center for Applied Mathematics of Guangxi (Guangxi University), Nanning 530004, Guangxi, China

Received:2022-08-17 Published:2026-03-16

摘要/Abstract

摘要： 在弱Wolfe-Powell线搜索技术下,通过PRP共轭梯度法如何获得非凸函数的全局收敛仍是一个公开问题。本文针对大规模无约束优化问题,提出了一种混合的共轭梯度方法(MPRP)。该方法是将修正的BFGS方法与修正的PRP共轭梯度法混合,采用了弱Wolfe-Powell线搜索技术来寻找步长,其搜索方向具有充分下降的性质。在理论上,通过对条件合理的假设, 确保了非凸函数的全局收敛性。在数值实验上,通过对Muskingum模型的参数估计,减少了计算量和存储量,说明了MPRP 的有效性; 在不同噪声情况下,通过对比多种图像的恢复情况,证明了MPRP 有较强的竞争力; 并且在低脉冲噪声图像下,图像的恢复情况较为显著。

关键词: 非凸函数, PRP算法, 弱Wolfe-Powell线搜索, 全局收敛, 图像恢复

Abstract: It is well-known that, under the WWP (weak Wolfe-Powell) line search technique, the global convergence of the PRP conjugate gradient method for non-convex functions is still open. In this paper, a hybrid conjugate gradient method is proposed for large-scale unconstrained optimization problems. In this method, the modified BFGS method is mixed with the modified PRP conjugate gradient method, and the weak Wolfe-Powell line search technique is used to find the step size, and the search direction has the property of sufficient descent. Theoretically, the global convergence of nonconvex functions is ensured by assuming reasonable conditions. In numerical experiments, the parameter estimation of the Muskingum model reduces the amount of computation and storage, which illustrates the effectiveness of MPRP. In different noise situations, the MPRP is proved to be highly competitive by comparing the recovery of multiple images. In addition, the restoration of the image is more significant under the image of low impulse noise.

Key words: non-convex functions, PRP algorithm, weak Wolfe-Powell line search, global convergence, image recovery

中图分类号:

O221

刘聪, 简艾伦, 袁功林. 一个应用于图像恢复问题的修正共轭梯度算法[J]. 运筹学学报（中英文）, 2026, 30(1): 207-216.

LIU Cong, JIAN Ailun, YUAN Gonglin. A modified conjugate gradient algorithm with its applications in image recovery problems[J]. Operations Research Transactions, 2026, 30(1): 207-216.

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一个应用于图像恢复问题的修正共轭梯度算法

A modified conjugate gradient algorithm with its applications in image recovery problems

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