两个充分下降的RMIL型共轭梯度法及图像去噪应用

doi:10.15960/j.cnki.issn.1007-6093.2026.02.006

摘要/Abstract

摘要： 共轭梯度法因其低存储、迭代简单等优点,广泛应用于求解大规模无约束优化问题。本文基于Rivaie-Mustafa-Ismail-Leong (RMIL)共轭参数,提出两个拓展的RMIL型共轭参数,并建立相应的共轭梯度算法。在强Wolfe非精确线搜索下,证明第一个算法产生的搜索方向满足充分下降性,并给出其全局收敛性证明。第二个算法不依赖任何线搜索,搜索方向有充分下降性;利用标准Wolfe线搜索产生步长,得到算法的全局收敛性。为测试两算法的数值效果,将其应用于求解无约束优化数值算例和图像去噪问题。与其他算法对比,本文结果表明两个新算法是有效的。

关键词: 无约束优化, 共轭梯度法, 全局收敛性, 图像去噪

Abstract: The conjugate gradient method possesses the advantages of lower storage requirement and simplicity to iterate, therefore it has been widely used for solving the large-scale optimization problems. Based on the Rivaie-Mustafa-Ismail-Leong (RMIL) conjugate coefficient, we propose two extended RMIL-type coefficients and establish the corresponding conjugate gradient algorithms for solving unconstrained optimization problems. Under the strong Wolfe line search, we prove that the search direction sequence generated by the first algorithm satisfies the descent property. The global convergence property is established under the normal assumptions. The descending property of the second algorithm is independent of any line search condition. By using the standard Wolfe line search, the global convergence of the second algorithm is obtained. To test the numerical effects of two proposed algorithms, we apply them to solve unconstrained optimization problems and restore the blurred images affected by impulse noise. Compared with some existing conjugate gradient methods, experimental results show that the two proposed algorithms are promising.

Key words: unconstrained optimization, conjugate gradient method, global convergence, image restoration

中图分类号:

O221.2

吴晓宇, 邵虎, 刘鹏杰, 周金诚. 两个充分下降的RMIL型共轭梯度法及图像去噪应用[J]. 运筹学学报（中英文）, 2026, 30(2): 79-92.

WU Xiaoyu, SHAO Hu, LIU Pengjie, ZHOU Jincheng. Two RMIL-type conjugate gradient methods with sufficient descent property and applications in image restoration[J]. Operations Research Transactions, 2026, 30(2): 79-92.

参考文献

[1] Fletcher R, Reeves C. Function minimization by conjugate gradients [J]. Computer Journal, 1964, 7(2): 149-154.
[2] Dai Y H, Yuan Y X. A nonlinear conjugate gradient method with a strong global convergence property [J]. SIAM Journal on Optimization, 1999, 10(1): 177-182.
[3] Polyak B T. The conjugate gradient method in extreme problems [J]. USSR Computational Mathematics and Mathematical Physics, 1969, 9: 94-112.
[4] Hestenes M R, Stiefel E. Method of conjugate gradient for solving linear equations [J]. Journal of Research of National Bureau of Standards, 1952, 49: 409-436.
[5] Wei Z X, Yao S W, Liu L Y. The convergence properties of some new conjugate gradient methods [J]. Applied Mathematics and Computation, 2006, 183: 1341-1350.
[6] Huang H, Wei Z X, Yao S W. The proof of the sufficient descent condition of the Wei-Yao-Liu conjugate gradient method under the strong Wolfe-Powell line search [J]. Applied Mathematics and Computation, 2007, 189: 1241-1245.
[7] 江羡珍, 简金宝, 马国栋. 具有充分下降性的两个共轭梯度法 [J]. 数学学报, 2014, 57(2): 365-372.
[8] Jiang X Z, Jian J B. A sufficient descent Dai-Yuan type nonlinear conjugate gradient method for unconstrained optimization problems [J]. Nonlinear Dynamics, 2013, 72: 101-112.
[9] Jiang X Z, Jian J B. Two modified nonlinear conjugate gradient methods with disturbance factors for unconstrained optimization [J]. Nonlinear Dynamics, 2014, 77: 387-397.
[10] 江羡珍, 马国栋, 简金宝. Wolfe 线搜索下一个新的全局收敛共轭梯度法 [J]. 工程数学学报, 2011, 28(6): 779-786.
[11] Liu J K, Feng Y M, Zou L M. A spectral conjugate gradient method for solving large-scale unconstrained optimization [J]. Computers and Mathematics with Applications, 2019, 77(3): 731-739.
[12] Jian J B, Lin H, Jiang X Z. A hybrid conjugate gradient method with descent property for unconstrained optimization [J]. Applied Mathematical Modelling, 2015, 39: 1281-1290.
[13] Jiang X Z, Liao W, Yin J H. A new family of hybrid three-term conjugate gradient methods with applications in image restoration [J]. Numerical Algorithms, 2022, 91: 161-191.
[14] 刘金魁. 广义Wolfe 线搜索下一类修正的Fletcher-Reeves方法的收敛性 [J]. 应用数学学报, 2013, 36(6): 1109-1117.
[15] 刘鹏杰, 江羡珍, 宋丹. 一类具有充分下降性的谱共轭梯度法 [J]. 运筹学学报, 2022, 26(4): 87-97.
[16] 江羡珍, 廖伟, 简金宝, 等. 一个带重启步的改进PRP 型谱共轭梯度法 [J]. 数学物理学报, 2022, 42(1): 216-227.
[17] 简金宝, 尹江华, 江羡珍. 一个充分下降的有效共轭梯度法 [J]. 计算数学, 2015, 37(4): 415-424.
[18] 朱志斌, 耿远航. 一个改进的WYL 型三项共轭梯度法 [J]. 数学物理学报, 2021, 41(6): 1871-1879.
[19] Tsegay G W, 张海斌, 张鑫, 等. 一种求解非线性无约束优化问题的充分下降的共轭梯度法 [J]. 运筹学学报, 2018, 22(3): 59-68.
[20] 夏丽娜, 朱志斌. Wolfe 线搜索下的两类修正FR 谱共轭梯度法 [J]. 应用数学, 2021, 34(3): 647- 656.
[21] 张慧玲, 赛·闹尔再, 吴晓云. 修正PRP共轭梯度法求解无约束优化问题 [J]. 运筹学学报, 2022, 26(2): 64-72.
[22] Wu X Y, Shao H, Liu P J, et al. An efficient conjugate gradient-based algorithm for unconstrained optimization and its projection extension to large-scale constrained nonlinear equations with applications in signal recovery and image denoising problems [J]. Journal of Computational and Applied Mathematics, 2023, 422: 114879.
[23] Shao H, Guo H, Wu X Y, et al. Two families of self-adjusting spectral hybrid DL conjugate gradient methods and applications in image denoising [J]. Applied Mathematical Modelling, 2023, 118: 393-411.
[24] Jiang X Z, Zhu Y H, Jian J B. Two efficient nonlinear conjugate gradient methods with restart procedures and their applications in image restoration [J]. Nonlinear Dynamics, 2023, 111: 5469-5498.
[25] Jiang X Z, Yang H H, Yin J H, et al. A three-term conjugate gradient algorithm with restart procedure to solve image restoration problems [J]. Journal of Computational and Applied Mathematics, 2023, 424: 115020.
[26] 简金宝, 刘鹏杰, 江羡珍. 一个充分下降的谱三项共轭梯度法 [J]. 应用数学学报, 2020, 43(6): 1000-1012.
[27] 王云, 黄敬频, 邵虎, 等. 一个全局收敛的改进PRP-HS混合共轭梯度法 (英文) [J]. 数学理论与应用, 2022, 42(4): 58-70.
[28] Rivaie M, Mustafa M, June L W, et al. A new class of nonlinear conjugate gradient coefficient with global convergence properties [J]. Applied Mathematics and Computation, 2012, 218: 11323-11332.
[29] Dai Z F. Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties [J]. Applied Mathematics and Computation, 2016, 276: 297-300.
[30] Yousef M B, Mamat M, Rivaie M. A new modified RMIL CG method with global convergence properties [C]//AIP Conference Proceedings, 2019, 2184(1): 060050.
[31] Zoutendijk G. Nonlinear programming computational methods [M]//Abadie J. (ed.) Integer and Nonlinear Programming. Amsterdam: North-Holland, 1987.
[32] Bongartz K E, Conn A R, Gould N, et al. CUTE: Constrained and unconstrained testing environments [J]. ACM Transactions on Mathematical Software, 1995, 21(1): 123-160.
[33] Andrei N. An unconstrained optimization test functions collection [J]. Advanced Modeling and Optimization, 2008, 10(1): 147-161.
[34] Moré J J, Garbow B S, Hillstrome K E. Testing unconstrained optimization software [J]. ACM Transactions on Mathematical Software, 1981, 7: 17-41.
[35] 马国栋, 简金宝, 江羡珍. 一个具有下降性的改进Fletcher-Reeves 共轭梯度法 [J]. 应用数学学报, 2015, 38(1): 89-97.
[36] Zhu Z B, Zhang D D, Wang S. Two modified DY conjugate gradient methods for unconstrained optimization problems [J]. Applied Mathematics and Computation, 2020, 373: 125004.
[37] Dolan E D, Moré J J. Benchmarking optimization software with performance profiles [J]. Mathematical Programming, 2002, 91(2): 201-213.
[38] Chan R H, Ho C W, Nikolova M. Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization [J]. IEEE Transactions on Image Processing, 2005, 14(10): 1479-1485.
[39] Cai J F, Chan R, Di Fiore C. Minimization of a detail-preserving regularization functional for impulse noise removal [J]. Journal of Mathematical Imaging and Vision, 2007, 29(1): 79-91.
[40] Cai J F, Chan R, Morini B. Minimization of an edge-preserving regularization functional by conjugate gradient type methods [M]//Tai X C, Lie K-A, Chan T, et al. (eds.) Image Processing Based on Partial Differential Equations, Berlin: Springer, 2007: 109-122.
[41] Hwang H, Haddad R A. Adaptive median filters: New algorithms and results [J]. IEEE Transactions on Image Processing, 1995, 4(4): 499-502.
[42] Bovik A. Handbook of Image and Video Processing [M]. San Diego: Academic Press, 2000.