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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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
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Abstract	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.
AbstractList	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.
Author	Sun, Tao Yin, Penghang Jiang, Hao Cheng, Lizhi
Author_xml	– sequence: 1 givenname: Tao surname: Sun fullname: Sun, Tao email: nudtsuntao@163.com, nudttaosun@gmail.com organization: College of Science, National University of Defense Technology – sequence: 2 givenname: Penghang surname: Yin fullname: Yin, Penghang organization: Department of Mathematics, University of California – sequence: 3 givenname: Lizhi surname: Cheng fullname: Cheng, Lizhi organization: College of Science & The State Key Laboratory for High Performance Computation, National University of Defense Technology – sequence: 4 givenname: Hao surname: Jiang fullname: Jiang, Hao organization: College of Computer, National University of Defense Technology
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SubjectTerms	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
Title	Alternating direction method of multipliers with difference of convex functions
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