一类带参数自适应步长的快速迭代收缩阈值算法及其应用  

An Improved Parameterized Fast Iterative Shrinkage-thresholding Algorithm with Adaptive Step Size and Its Applications

作　　者：唐艳[1,2] 张叶雨 吉智慧 邹雨阳 Yan TANG;Ye Yu ZHANG;Zhi Hui JI;Yu Yang ZOU(School of Mathematics and Statistics,Chongqing Technology and Business University,Chongqing 400067,P.R.China;Chongqing Key Laboratory of Social Economy and Applied Statistics,Chongqing 400067,P.R.China;Chongqing Yubei Experimental Middle School,Chongqing 401120,P.R.China)

机构地区：[1]重庆工商大学数学与统计学院,重庆400067 [2]经济社会应用统计重庆市重点实验室,重庆400067 [3]重庆市渝北区实验中学校,重庆401120

出　　处：《数学学报（中文版）》2024年第6期1163-1178,共16页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(12071316);重庆市自然科学基金(CSTC2021jcyj-msxmX0177);重庆市研究生导师团队建设项目(yds223010);重庆工商大学研究生科研创新项目(yjscxx2023-211-71)资助

摘　　要：本文针对非光滑凸优化问题提出了一种带参数的自适应步长的快速迭代收缩阈值算法.利用参数化策略带来的自由度在实Hilbert空间上分别研究了目标函数O(1/k^(2))和迭代算法o(1/k^(2))的收敛速率,并在目标函数F一致凸的条件下获得了算法的强收敛性.此外,在该算法的基础上建立了连续动力系统模型,并获得了连续动力系统解的逼近性质.最后通过图像去噪实例验证了算法的优越性.In this paper,a parameterized fast iterative shrinkage-thresholding algorithm with adaptive step size is proposed for nonsmooth optimization problems.The convergence rates of the objective function and the iterative algorithm are studied separately in the real Hilbert space using the degrees of freedom brought by the parameterization strategy,and the strong convergence of the sequence generated by the algorithm is obtained under the condition that the objective function is uniformly convex.In addition,the connection between the algorithm and inertial dynamical system is established and the related inference of the dynamical system solution trajectory is obtained.Meanwhile,the specific applications and comparisons of the algorithms listed in this paper to the image denoising problem demonstrate their superiority.

关 键 词：快速迭代收缩阈值算法 自适应步长 连续动力系统 图像去噪 

分 类 号：O177.91[理学—数学]