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非凸两分块优化超松弛步长邻近ADMM的收敛性分析

简金宝1,徐笑2,晁绵涛2   

  1. 1. 广西民族大学理学院, 南宁 530006; 2. 广西大学数学与信息科学学院, 南宁 530004
  • 出版日期:2021-11-25 发布日期:2021-12-25

简金宝, 徐笑, 晁绵涛. 非凸两分块优化超松弛步长邻近ADMM的收敛性分析[J]. 系统科学与数学, 2021, 41(11): 3139-3150.

JIAN Jinbao, XU Xiao, CHAO Miantao. Convergence of Proximal ADMM with an Over-Relaxation Stepsize for Nonconvex Two-Block Problem[J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(11): 3139-3150.

Convergence of Proximal ADMM with an Over-Relaxation Stepsize for Nonconvex Two-Block Problem

JIAN Jinbao1 ,XU Xiao2, CHAO Miantao2   

  1. 1. College of Science, Guangxi University for Nationalities, Nanning 530006; 2. College of Mathematics and Information Science, Guangxi University, Nanning 530004
  • Online:2021-11-25 Published:2021-12-25
讨论带线性约束的非凸两分块优化问题, 旨在分析带超松弛 步长参数的邻近乘子交替方向法(PADMM)的收敛性. 已有乘子交替方向法均要求对偶变量迭代步长参数$\theta\in(0,\frac{1+\sqrt{5}}{2}]$. 文章在$\theta\in(0,2)$的情形下分析PADMM的收敛性. 首先, 在适当的假设条件下, 证明了该算法的全局收敛性. 其次, 当效益函数满足Kurdyka-{\L}ojasiewicz性质时, 证明了该算法的强收敛性. 最后, 通过初步的数值实验验证了算法的有效性.
In this paper, we study the convergence of proximal alternating direction method of multipliers (PADMM) with over-relaxation stepsize parameter for nonconvex two-block optimization with linear constraints. The existing alternating direction method of multipliers all require the iteration step parameter of dual variable $\theta\in (0, \frac{1+\sqrt{5}}{2}] $. In this paper, we analyze the convergence of PADMM when $\theta\in(0,2)$. First, we prove that PADMM is globally convergent under suitable assumptions. Second, under the assumption that the merit function satisfied the Kurdyka-{\L}ojasiewicz property, we prove that the PADMM is strongly convergent. Finally, some preliminary numerical results are reported to support the efficiency of the proposed algorithm.
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