A Note on the Alternating Direction Method of Multipliers

We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m =2, while it remains o...

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Published in:	Journal of optimization theory and applications Vol. 155; no. 1; pp. 227 - 238
Main Authors:	Han, Deren, Yuan, Xiaoming
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.10.2012 Springer Nature B.V
Subjects:	Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Constraints Convergence Convex analysis Engineering Lagrange multiplier Mathematical functions Mathematics Mathematics and Statistics Multipliers Operations Research/Decision Theory Optimization Programming Studies Theory of Computation Alternating direction method of multipliers Strongly convex functions Global convergence
ISSN:	0022-3239, 1573-2878
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Abstract	We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m =2, while it remains open whether its convergence can be extended to the general case m ≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.
AbstractList	We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m =2, while it remains open whether its convergence can be extended to the general case m ≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex. We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m greater than or equal to 3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex. We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.[PUBLICATION ABSTRACT]
Author	Yuan, Xiaoming Han, Deren
Author_xml	– sequence: 1 givenname: Deren surname: Han fullname: Han, Deren organization: School of Mathematical Science and Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University – sequence: 2 givenname: Xiaoming surname: Yuan fullname: Yuan, Xiaoming email: xmyuan@hkbu.edu.hk organization: Department of Mathematics, Hong Kong Baptist University
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Snippet	We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled... We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled...
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SubjectTerms	Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Constraints Convergence Convex analysis Engineering Lagrange multiplier Mathematical functions Mathematics Mathematics and Statistics Multipliers Operations Research/Decision Theory Optimization Programming Studies Theory of Computation
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Title	A Note on the Alternating Direction Method of Multipliers
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