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ON THE $O(1/K)$ CONVERGENCE RATE OF THE ALTERNATING DIRECTION METHOD OF MULTIPLIERS IN A COMPLEX DOMAIN

Part of: Control systems

Published online by Cambridge University Press:  30 August 2018

L. LI Open the ORCID record for L. LI [Opens in a new window] ,
G. Q. WANG  and
J. L. ZHANG
L. LI*
Affiliation:
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, China email [email protected], [email protected], [email protected]
G. Q. WANG
Affiliation:
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, China email [email protected], [email protected], [email protected]
J. L. ZHANG
Affiliation:
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, China email [email protected], [email protected], [email protected]
*
✉[email protected]
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Abstract

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We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the $O(1/K)$ convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the $O(1/K)$ convergence rate and that it has certain advantages compared with the ADMM in a real domain.

Keywords

the alternating direction method of multipliersconvergence rateWirtinger calculusleast absolute shrinkage and selectionator operator

MSC classification

Primary: 90C25: Convex programming
Secondary: 65K10: Optimization and variational techniques 93C83: Control problems involving computers (process control, etc.)
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 1 , July 2018 , pp. 95 - 117
Copyright
© 2018 Australian Mathematical Society 

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