Journal of Optimization Theory and Applications

, Volume 166, Issue 3, pp 906–929

# Further Study on the Convergence Rate of Alternating Direction Method of Multipliers with Logarithmic-quadratic Proximal Regularization

Article

## Abstract

In the literature, the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization has been proved to be convergent, and its worst-case convergence rate in the ergodic sense has been established. In this paper, we focus on a convex minimization model and consider an inexact version of the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization. Our primary purpose is to further study its convergence rate and to establish its worst-case convergence rates measured by the iteration complexity in both the ergodic and non-ergodic senses. In particular, existing convergence rate results for this combination are subsumed by the new results.

## Keywords

Convex programming Alternating direction method of multipliers Logarithmic-quadratic proximal Convergence rate Iteration complexity

## Mathematics Subject Classification

90C25 90C33 65K05

## Notes

### Acknowledgments

The first author was supported in part by the Natural Science Foundation of Jiangsu Province Grant BK20130550 and the NSFC Grant 11401300. The second author was supported by the Program for New Century Excellent Talents in University Grant NCET-12-0111, Qing Lan Project and the Natural Science Foundation of Jiangsu Province Grant BK2012662. The third author was partially supported by the Grant from Hong Kong Baptist University: FRG2/13-14/061 and the General Research Fund from Hong Kong Research Grants Council: 203613.

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