Volume 34, Issue 6
A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Peijing Chen, Jianguo Huang & Xiaoqun Zhang

DOI: 10.4208/jcm.1612-m2016-0536

J. Comp. Math., 34 (2016), pp. 723-738.

Published online: 2016-12

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  • Abstract

We have proposed a primal-dual fixed point algorithm (PDFP) for solving minimization of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.

  • Keywords

Primal-dual fixed point algorithm Multi-block optimization problems

  • AMS Subject Headings

65K05 46N10 90C06 90C25.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chenpeijun@sjtu.edu.cn (Peijing Chen)

jghuang@sjtu.edu.cn (Jianguo Huang)

xqzhang@sjtu.edu.cn (Xiaoqun Zhang)

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@Article{JCM-34-723, author = {Chen , Peijing and Huang , Jianguo and Zhang , Xiaoqun }, title = {A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {6}, pages = {723--738}, abstract = { We have proposed a primal-dual fixed point algorithm (PDFP) for solving minimization of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1612-m2016-0536}, url = {http://global-sci.org/intro/article_detail/jcm/9822.html} }
TY - JOUR T1 - A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization AU - Chen , Peijing AU - Huang , Jianguo AU - Zhang , Xiaoqun JO - Journal of Computational Mathematics VL - 6 SP - 723 EP - 738 PY - 2016 DA - 2016/12 SN - 34 DO - http://doi.org/10.4208/jcm.1612-m2016-0536 UR - https://global-sci.org/intro/article_detail/jcm/9822.html KW - Primal-dual fixed point algorithm KW - Multi-block optimization problems AB - We have proposed a primal-dual fixed point algorithm (PDFP) for solving minimization of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.
Peijing Chen , Jianguo Huang & Xiaoqun Zhang . (2020). A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization. Journal of Computational Mathematics. 34 (6). 723-738. doi:10.4208/jcm.1612-m2016-0536
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