一个基于 BFGS 更新的惯性三项共轭梯度投影法

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1698-1710.

一个基于 BFGS 更新的惯性三项共轭梯度投影法

刘鹏杰(),邵虎^*(),李佳宜(),董端端()

中国矿业大学数学学院, 江苏省应用数学中心江苏徐州 221116

收稿日期:2024-04-12 修回日期:2024-12-20 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 邵虎, E-mail:shaohu@cumt.edu.cn
作者简介:刘鹏杰, E-mail:liupengjie2019@163.com|李佳宜, E-mail:10213705@cumt.edu.cn|董端端,E-mail:03201428@cumt.edu.cn
基金资助:
中央高校基本科研业务费(2025QN1147)

An Inertial Three-Term Conjugate Gradient Projection Method Based on BFGS Update

Pengjie Liu(),Hu Shao^*(),Jiayi Li(),Duanduan Dong()

School of Mathematics, Jiangsu Center for Applied Mathematics, China University of Mining and Technology, Jiangsu Xuzhou 221116

Received:2024-04-12 Revised:2024-12-20 Online:2025-10-26 Published:2025-10-14
Supported by:
Fundamental Research Funds for the Central Universities(2025QN1147)

摘要/Abstract

摘要：

该文首先基于 Broyden-Fletcher-Goldfarb-Shanno 更新, 给出一类改进的三项搜索方向. 结合惯性策略和超平面投影技术, 提出一个惯性的三项共轭梯度投影法用于求解无约束非线性单调方程组. 所提惯性算法具有如下特征: (i) 其搜索方向独立于任何线搜索, 有充分下降性和信赖域性质; (ii) 惯性技术采用三步迭代信息以产生惯性迭代点; (iii) 底层映射无需满足 Lipschitz 连续性, 亦可获得算法的全局收敛性结果. 通过初步数值试验结果验证了算法的有效性.

关键词: 非线性单调方程组, 三项共轭梯度法, 惯性策略, BFGS 更新, 全局收敛性

Abstract:

Based on the BFGS update, in this paper, we present a class of improved three-term search directions. Then, by combining the inertial strategy and hyperplane projection approach, an inertial three-term conjugate gradient projection method is proposed for solving unconstrained nonlinear monotone equations. The proposed inertial algorithm exhibits the following features: (i) its search direction always possesses sufficient descent and trust region properties, independent of any line search; (ii) the inertial strategy generates iterative points by utilizing information from three iterations; (iii) the method achieves global convergence results without assuming the underlying mapping to satisfy Lipschitz continuity. Preliminary numerical experiments verify the effectiveness of the proposed method.

Key words: nonlinear monotone equations, three-term conjugate gradient method, inertial strategy, BFGS Update, global convergence

中图分类号:

O221.2

刘鹏杰, 邵虎, 李佳宜, 董端端. 一个基于 BFGS 更新的惯性三项共轭梯度投影法[J]. 数学物理学报, 2025, 45(5): 1698-1710.

Pengjie Liu, Hu Shao, Jiayi Li, Duanduan Dong. An Inertial Three-Term Conjugate Gradient Projection Method Based on BFGS Update[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1698-1710.

参考文献

[1]	Awwal A M, Kumam P, Sitthithakerngkiet K, et al. Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application. AIMS Math, 2021, 6(8): 8792-8814
[2]	Bing Y, Lin G. An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations. SIAM J Optimiz, 1991, 1(2): 206-221
[3]	Cheng W Y. A PRP type method for systems of monotone equations. Math Comput Modell, 2009, 50: 15-20
[4]	Cruz W L. A spectral algorithm for large-scale systems of nonlinear monotone equations. Numer Algorithms, 2017, 76: 1109-1130
[5]	Dolan E D, Moré J J. Benchmarking optimization software with performance profiles. Math Program, 2002, 91(2): 201-213
[6]	Dai Y H, Yuan Y X. A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim, 1999, 10(1): 177-182
[7]	Gao P T, He C J. An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo, 2018, 55(4): Article 53
[8]	Hu Y P, Wei Z X. A modified Liu-Storey conjugate gradient projection algorithm for nonlinear monotone equations. Int J Math, 2014, 9: 1767-1777
[9]	Ibrahim A H, Kumama P, Rapaji? S, et al. Approximation methods with inertial term for large-scale nonlinear monotone equations. Appl Numer Math, 2022, 181: 417-435
[10]	Ibrahim A H, Kumam P, Sun M, et al. Projection method with inertial step for nonlinear equations: Application to signal recovery. J Ind Manag Optim, 2023, 19(1): 30-55
[11]	Jian J B, Yin J H, Tang C M, et al. A family of inertial derivative-free projection methods for constrained nonlinear pseudo-monotone equations with applications. Comput Appl Math, 2022, 41(7): Article 309
[12]	Kou C X, Dai Y H. A modified self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno method for unconstrained optimization. J Optim Theory Appl, 2015, 165: 209-224
[13]	La Cruz W, Martínez J M, Raydan M. Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math Comput, 2006, 75(255): 1429-1448
[14]	Li M. A modified Hestense-Stiefel conjugate gradient method close to the memoryless BFGS quasi-Newton method. Optim Methods Softw, 2018, 33(2): 336-353
[15]	刘金魁, 孙悦, 赵永祥. 凸约束伪单调方程组的无导数投影算法. 计算数学, 2021, 43(3): 388-400
[15]	Liu J K, Sun Y, Zhao Y X. A derivative-free projection algorithm for solving pseudo-monotone equations with convex constraints. Math Numer Sin, 2021, 43(3): 388-400
[16]	Liu P J, Shao H, Yuan Z H, et al. A family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial extensions for the constrained nonlinear pseudo-monotone equations with applications. Numer Algorithms, 2023, 94: 1055-1083
[17]	Liu P J, Wu X Y, Shao H, et al. Three adaptive hybrid derivative-free projection methods for constrained monotone nonlinear equations and their applications. Numer Linear Algebra Appl, 2023, 30(2): e2471
[18]	Ma G D, Jin J C, Jian J B, et al. A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing. Numer Algorithms, 2023, 92(3): 1621-1653
[19]	Moré J J, Garbow B S, Hillstrom K E. Testing unconstrained optimization software. ACM Trans Math Softw, 1981, 7(1): 17-41
[20]	Narushima Y, Yabe H, Ford J A. A three-term conjugate gradient method with sufficient descent property for unconstrained optimization. SIAM J Optim, 2011, 21(1): 212-230
[21]	Nocedal J. Updating quasi-Newton matrices with limited storage. Math Comput, 1980, 35(151): 773-782
[22]	Polyak B T. Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys, 1964, 4(5): 1-17
[23]	Polyak B T. Introduction to Optimization. New York: Inc Publications Division, 1987
[24]	Rao J Y, Huang N. A derivative-free scaling memoryless DFP method for solving large scale nonlinear monotone equations. J Glob Optim, 2023, 87(2): 641-677
[25]	Shanno D F. Conjugate gradient methods with inexact searches, Math Oper Res, 1978, 3(3): 244-256
[26]	Solodov M V, Svaiter B F. A globally convergent inexact Newton method for systems of monotone equations//Fukushima M, Qi L. Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Dordrecht: Kluwer, 1998. 355-369
[27]	Song T Y, Liu Z X. An efficient inertial subspace minimization CG algorithm with convergence rate analysis for constrained nonlinear monotone equations. J Comput Appl Math, 2024, 446: 115873
[28]	Stanimirovi? P S, Ivanov B, Djordjevi? S, et al. New hybrid conjugate gradient and Broyden-Fletcher-Goldfarb-Shanno conjugate gradient methods. J Optim Theory Appl, 2018, 178: 860-884
[29]	Wu X Y, Shao H, Liu P J, et al. An inertial spectral CG projection method based on the memoryless BFGS update. J Optim Theory Appl, 2023, 198: 1130-1155
[30]	尹江华, 简金宝, 江羡珍. 凸约束非光滑方程组基于自适应线搜索的谱梯度投影算法. 计算数学, 2020, 42(4): 457-471
[30]	Yin J H, Jian J B, Jiang X J. A spectral gradient projection algorithm for convex constrained nonsmooth equations based on an adaptive line search. Math Numer Sin, 2020, 42(4): 457-471
[31]	Yin J H, Jian J B, Jiang X Z, et al. A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications. Numer Algorithms, 2021, 88: 389-418
[32]	Yin J H, Jian J B, Jiang X Z, et al. A family of inertial-relaxed DFPM-based algorithms for solving large-scale monotone nonlinear equations with application to sparse signal restoration. J Comput Appl Math, 2023, 419: Art 114674
[33]	Yu G H, Niu S Z, Ma J H, et al. An adaptive prediction-correction method for solving large-scale nonlinear systems of monotone equations with applications. Abstr Appl Anal, 2013, 2013: 1-13
[34]	Zhang L, Zhou W J. Spectral gradient projection method for solving nonlinear monotone equations. J Comput Appl Math, 2006, 196: 478-484
[35]	张宁, 刘金魁. 非线性伪单调方程组的谱 LS 型投影算法. 数学物理学报, 2022, 42A(6): 1886-1897
[35]	Zhang N, Liu J K. Spectral LS-type projection algorithm for solving nonlinear pseudo-monotone equations. Acta Math Sci, 2022, 42A(6): 1886-1897
[36]	Zhang N, Liu J K, Zhang L Q, et al. A fast inertial self-adaptive projection based algorithm for solving large-scale nonlinear monotone equations. J Comput Appl Math, 2023, 426: Art 115087
[37]	Zhou W J, Shen D M. Convergence properties of an iterative method for solving symmetric non-linear equations. J Optim Theory Appl, 2015, 164: 277-289