Convergence revisit on generalized symmetric ADMM

Abstract

In this note, we show a sublinear nonergodic convergence rate for the algorithm developed in Bai et al. [Generalized symmetric ADMM for separable convex optimization. Comput Optim Appl. 2018;70:129–170], as well as its linear convergence under assumptions that the sub-differential of each component objective function is piecewise linear and all the constraint sets are polyhedra. These remaining convergence results are established for the stepsize parameters of dual variables belonging to a special isosceles triangle region, which aims to strengthen our understanding for convergence of the generalized symmetric ADMM.

Keywords:

• Convex optimization
• alternating direction method of multipliers
• symmetric parameter domain
• convergence rate

2010 Mathematics Subject Classifications:

• 65K10
• 68W40
• 90C25

Acknowledgments

The authors wish to thank the Editor-in-Chief Prof. Christiane Tammer and the anonymous referees for providing their valuable suggestions, which have significantly improved the quality of our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the Fundamental Research Funds for the Central Universities [grant number G2019KY05106], the National Natural Science Foundation of China [grant numbers 11671318; 11571271; 11631013; 11971372] and the Natural Science Foundation of Fujian Province [grant number 2016J01028]. The second author Xiaokai Chang was supported by the Hongliu Foundation of First-class Disciplines of Lanzhou University of Technology.