The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems

Zehui Jia; Xue Gao; Xingju Cai; Deren Han

doi:10.3934/jimo.2020053

Article Contents

2021, Volume 17, Issue 4: 1943-1971. Doi: 10.3934/jimo.2020053

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The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems

1.
Department of Information and Computing Science, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2.
School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, China
3.
LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China

^* Corresponding author: Xingju Cai
^* Corresponding author: Xingju Cai

Received: July 2019

Revised: October 2019

Early access: March 2020

Published: July 2021

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Abstract

The symmetric alternating direction method of multipliers is an efficient algorithm, which updates the Lagrange multiplier twice at each iteration and the variables are treated in a symmetric manner. Considering that the convergence range of the parameters plays an important role in the implementation of the algorithm. In this paper, we analyze the convergence rate of the symmetric ADMM with a more relaxed parameter range for solving the two block nonconvex separable optimization problem under the assumption that the generated sequence is bounded. Two cases are considered. In the first case, both components of the objective function are nonconvex, we prove the convergence of the augmented Lagrangian function sequence, and establish the $ O(1/\sqrt{k}) $ worst-case complexity measured by the difference of two consecutive iterations. In the second case, one component of the objective function is convex and the error bound condition is assumed, then we can prove that the iterative sequence converges locally to a KKT point in a R-linear rate; and an auxiliary sequence converges in a Q-linear rate. Furthermore, a practical inexact symmetric ADMM with relative error criteria is proposed, and the associated convergence analysis is established under the same conditions.

Keywords:

Symmetric alternating direction method of multipliers,
nonconvex minimization,
inexact,
relative error criteria,
sublinear convergence,
linear convergence.

Mathematics Subject Classification: 90C26, 49M27, 90C33.

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Figure 1. The relationship between $ \alpha $ and $ \theta $

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