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A linearly convergent majorized ADMM with indefinite proximal terms for convex composite programming and its applications


Authors: Ning Zhang, Jia Wu and Liwei Zhang
Journal: Math. Comp. 89 (2020), 1867-1894
MSC (2010): Primary 90C25, 90C30, 65K05, 62J12
DOI: https://doi.org/10.1090/mcom/3506
Published electronically: January 30, 2020
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Abstract: This paper aims to study a majorized alternating direction method of multipliers with indefinite proximal terms (iPADMM) for convex composite optimization problems. We show that the majorized iPADMM for 2-block convex optimization problems converges globally under weaker conditions than those used in the literature and exhibits a linear convergence rate under a local error bound condition. Based on these, we establish the linear rate convergence results for a symmetric Gauss-Seidel based majorized iPADMM, which is designed for multiblock composite convex optimization problems. Moreover, we apply the majorized iPADMM to solve different types of regularized logistic regression problems. The numerical results on both synthetic and real datasets demonstrate the efficiency of the majorized iPADMM and also illustrate the effectiveness of the introduced indefinite proximal terms.


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Additional Information

Ning Zhang
Affiliation: School of Computer Science and Technology, Dongguan University of Technology, Dongguan 523808, People’s Republic of China
Email: ningzhang_2008@yeah.net

Jia Wu
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: wujia@dlut.edu.cn

Liwei Zhang
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: lwzhang@dlut.edu.cn

DOI: https://doi.org/10.1090/mcom/3506
Keywords: Alternating direction method of multiplier, linear rate convergence, indefinite proximal term, logistic regression, symmetric Gauss-Seidel decomposition
Received by editor(s): February 6, 2018
Received by editor(s) in revised form: February 22, 2019, and October 28, 2019
Published electronically: January 30, 2020
Additional Notes: The first author’s research was supported by the National Natural Science Foundation of China (11901083).
The second author is the corresponding author.
The third author’s research was supported by the National Natural Science Foundation of China (11971089, 11571059 and 11731013).
Article copyright: © Copyright 2020 American Mathematical Society