求解单调变分包含问题的自适应-正则化方法及应用

Adaptive Regularization Method for Solving Monotonic Variational Inclusion Problems and Applications

摘要

在Hilbert空间中给出一种求解单调变分包含问题的自适应-正则化方法.首先,利用自适应规则去除现有算法收敛性分析中迭代步长对算子的依赖;其次,结合正则化技巧建立关于求解变分包含问题的强收敛性定理;最后,通过图像重建实验说明该算法的收敛性和优越性.

Abstract

We gave an adaptive regularization method for solving monotonic variational inclusion problems in Hilbert space. Firstly, we used adaptive rules to eliminate the dependence of iterative step-sizes on operators in the convergence analysis of existing algorithms. Secondly, we combined regularization technique to establish strong convergence theorems for solving variational inclusion problems. Finally, the convergence and superiority of the proposed algorithm were demonstrated through image reconstruction experiment.

关键词

变分包含问题 / 自适应算法 / 正则化 / 强收敛 / 图像重建

Key words

variational inclusion problem / adaptive algorithm / regularization / strong convergence / image reconstruction

参考文献

[1] BYRNE C. A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image Reconstruction[J]. Inverse Problems, 2004, 20(1): 103-120.

[2] CENSOR Y, ELFVING T, KOPF N, et al. The Multiple-Sets Split Feasibility Problem and Its Applications for Inverse Problems[J]. Inverse Problems, 2005, 21(6): 2071-2084.

[3] CENSOR Y, BORTFELD T, MARTIN B, et al. A Unified Approach for Inversion Problems in Intensity-Modulated Radiation Therapy[J]. Phys Med Biol, 2006, 51(10): 2353-2365.

[4] 闻道君, 张蓉, 何光. 单调变分不等式问题外梯度方法的推广[J]. 吉林大学学报(理学版), 2020, 58(4):847-852.

[5] (WEN D J, ZHANG R, HE G. Extension of Extragradient Method for Monotone Variational Inequality Problems[J]. Journal of Jilin University(Science Edition), 2020, 58(4):847-852.)

[6] COMBETTES P L, WAJS V R. Singnal Recovery by Proximal Forward-Backward Splitting[J]. Multiscale Model Simul, 2005, 4(4): 1168-1200.

[7] TSENG P. A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings[J]. SIAM J Control Optim, 2000, 38(2): 431-446.

[8] WEN D J. Modified Krasnoselski-Mann Type Iterative Algorithm with Strong Convergence for Hierarchical Fixed Point Problem and Split Monotone Variational Inclusions[J]. J Comput Appl Math, 2021, 393: 113501-1-113501-13.

[9] SUANTAI S, INKRONG P, CHOLAMJIAK P. Forward-Backward-Forward Algorithms Involving Two Inertial Terms for Monotone Inclusions[J]. Comput Appl Math, 2023, 42(6): 255-1-255-17.

[10] DEY S, IZUCHUKWU C, TAIWO A, et al. New Iterative Algorithms for Solving Split Variational Inclusions[J]. J Glob Optim, 2025, 91(3): 587-609.

[11] VAN HIEU D, ANH P K, MUU L D, et al. Iterative Regularization Methods with New Stepsize Rules for Solving Variational Inclusions[J]. J Appl Math Comput, 2022, 68(1): 571-599.

[12] BRÉZIS H. Opérateurs Maximaux Monotones et Semi-groupes Contractions Dans Les Espaces de Hilbert[M].North-Holland Math Stud, No.5. Amsterdam: North-Holland Publishing Co., 1973: 1-183.

[13] XU H K. Iterative Algorithms for Nonlinear Operators[J]. J London Math Soc, 2002, 66(1): 240-256.

[14] COTTLER W, YAO J C. Pseudo-monotone Complementarity Problems in Hilbert Space[J]. J Optim Theory Appl, 1992, 75(2): 281-295.

[15] ANH P K, THONG D V, DUNG V T. A Strongly Convergent Mann-Type Inertial Algorithm for Solving Split Variational Inclusion Problems[J]. Optim Eng, 2021, 22(1): 159-185.

基金资助

国家自然科学基金(11471059)

重庆市自然科学基金(cstc2020jcyj-msxmX0316)

重庆工商大学研究生科研创新项目(yjsc-xx2025-269-245)