杨洋,唐玉超.求解两块可分离凸极小化问题的惯性交替方向乘子法[J].数学研究及应用,2021,41(2):204~220
求解两块可分离凸极小化问题的惯性交替方向乘子法
An Inertial Alternating Direction Method of Multipliers for Solving a Two-Block Separable Convex Minimization Problem
投稿时间:2020-03-12  修订日期:2020-09-27
DOI:10.3770/j.issn:2095-2651.2021.02.008
中文关键词:  交替方向乘子法  惯性方法  Douglas-Rachford分裂算法
英文关键词:alternating direction method of multipliers  inertial method  Douglas-Rachford splitting algorithm
基金项目:国家自然科学基金(Grant Nos.12061045; 12061046; 11661056; 11771198; 11771347; 91730306; 41390454; 11401293), 中国博士后科学基金(Grant No.2015M571989), 江西省博士后科学基金(Grant No.2015KY51).
作者单位
杨洋 南昌大学数学系, 江西 南昌 330031 
唐玉超 南昌大学数学系, 江西 南昌 330031 
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中文摘要:
      交替方向乘子法是解决信号和图像处理中出现的许多凸极小化模型的常用方法.本文提出一种惯性的交替方向乘子法求解具有线性等式约束的两块可分离凸极小化问题.该算法是利用原问题对应的对偶问题的惯性Douglas-Rachford分裂算法得到的.在无穷维Hilbert空间中,我们证明该算法的收敛性.进而,我们将所提出的算法应用于鲁棒主成分分析问题,并与其它现有算法进行比较.数值结果表明所提算法的优越性.
英文摘要:
      The alternating direction method of multipliers (ADMM) is a widely used method for solving many convex minimization models arising in signal and image processing. In this paper, we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints. This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem. We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces. Furthermore, we apply the proposed algorithm on the robust principal component analysis problem and also compare it with other state-of-the-art algorithms. Numerical results demonstrate the advantage of the proposed algorithm.
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