Alternating direction method of multipliers with difference of convex functions

In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM...

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Bibliographic Details
Published in:Advances in computational mathematics Vol. 44; no. 3; pp. 723 - 744
Main Authors: Sun, Tao, Yin, Penghang, Cheng, Lizhi, Jiang, Hao
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2018
Springer Nature B.V
Subjects:
Algorithms
Composite functions
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Convex analysis
Critical point
Image processing
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Multipliers
Noise reduction
Signal processing
Visualization
Alternating direction method of multipliers
Kurdyka-Łojasiewicz property
90C30
Difference of convex functions
90C26
47N10
Nonconvex
ISSN:1019-7168, 1572-9044
Online Access:Get full text
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Summary:In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM often leads to difficult nonconvex subproblems. To address this issue, we propose to convexify the subproblems through a linearization technique as done in the difference of convex functions algorithm (DCA). By assuming the Kurdyka-Łojasiewicz property, we prove that the resulting algorithm sequentially converges to a critical point. It turns out that in the applications of signal and image processing such as compressed sensing and image denoising, the proposed algorithm usually enjoys closed-form solutions of the subproblems and thus can be very efficient. We provide numerical experiments to demonstrate the effectiveness of our algorithm.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-017-9559-3