一种求解低秩矩阵补全的惯性加速交替方向法  

An inertial alternating direction method of multiplier for low rank matrix completion

作　　者：闫喜红[1] 唐晓妮 YAN Xihong;TANG Xiaoni(School of Mathematics and Statistics,Taiyuan Normal University,Jinzhong 030619,Shanxi,China)

机构地区：[1]太原师范学院数学与统计学院,山西晋中030619

出　　处：《运筹学学报(中英文)》2025年第1期172-184,共13页Operations Research Transactions

基　　金：国家自然科学基金(No.11901424);山西省回国留学人员科研教研项目(No.2022-170);山西省研究生教育创新项目(No.2022Y758);山西省科技创新人才团队专项基金。

摘　　要：交替方向法作为求解矩阵补全问题的经典方法之一,具有能够将一个极小化问题分解成多个规模更小、更容易求解的子问题的优势,近年来在图像处理和数据分析等领域备受青睐。本文采用交替方向法的框架,结合惯性策略,提出了一种求解矩阵补全问题的惯性加速交替方向法。新算法在每步迭代中,对于其中一部分变量利用交替方向法的前两次迭代点外推得到新一步的迭代点,从而提高计算效率。本文在合理的假设条件下,证明了新算法的收敛性。最后,通过随机矩阵补全的数值实验及图像修复的实例验证了新算法的有效性和可行性。The alternating direction method of multiplier is attractive for solving matrix completion problems,which has the advantage of being able to decompose a minimization problem into many smaller and easier subproblems.Therefore,it is popular in the field of image processing and data analysis in recent years.Based on the framework of the alternating direction method of multiplier,we combine the inertial strategy and propose an inertial accelerated alternating direction method of multiplier to solve the matrix completion problems in this paper.In each iteration of the new method,for some of variables,the iterative points of the previous two iterations of the alternating direction method of multiplier are extrapolated to obtain the new iteration points,which improve the computational efficiency of the new method.Under the reasonable assumptions,we prove convergence of the new method.Finally,we also verify the effectiveness and feasibility of the new method by numerical experiments of random matrix completion and an example of image restoration.

关 键 词：矩阵补全问题 交替方向法 惯性加速 

分 类 号：O221.2[理学—运筹学与控制论]