Volume 37, Issue 5
A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

Yuchao Tang, Guorong Wu & Chuanxi Zhu

J. Comp. Math., 37 (2019), pp. 666-688.

Published online: 2019-03

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  • Abstract

In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum composed of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. Based on the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing model that is derived from computed tomography image reconstruction. Numerical results show that the proposed iterative algorithms outperform the compared algorithms including the alternating direction method of multipliers, the splitting primal-dual proximity algorithm, and the preconditioned splitting primal-dual proximity algorithm.

  • Keywords

Forward-backward splitting method, Primal-dual, Dual, Proximity operator.

  • AMS Subject Headings

90C25, 65K10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hhaaoo1331@163.com (Yuchao Tang)

guorong_wu@med.unc.edu (Guorong Wu)

chuanxizhu@126.com (Chuanxi Zhu)

  • BibTex
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@Article{JCM-37-666, author = {Tang , Yuchao and Wu , Guorong and Zhu , Chuanxi }, title = {A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {5}, pages = {666--688}, abstract = {

In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum composed of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. Based on the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing model that is derived from computed tomography image reconstruction. Numerical results show that the proposed iterative algorithms outperform the compared algorithms including the alternating direction method of multipliers, the splitting primal-dual proximity algorithm, and the preconditioned splitting primal-dual proximity algorithm.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1808-m2018-0027}, url = {http://global-sci.org/intro/article_detail/jcm/13040.html} }
TY - JOUR T1 - A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem AU - Tang , Yuchao AU - Wu , Guorong AU - Zhu , Chuanxi JO - Journal of Computational Mathematics VL - 5 SP - 666 EP - 688 PY - 2019 DA - 2019/03 SN - 37 DO - http://dor.org/10.4208/jcm.1808-m2018-0027 UR - https://global-sci.org/intro/jcm/13040.html KW - Forward-backward splitting method, Primal-dual, Dual, Proximity operator. AB -

In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum composed of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. Based on the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing model that is derived from computed tomography image reconstruction. Numerical results show that the proposed iterative algorithms outperform the compared algorithms including the alternating direction method of multipliers, the splitting primal-dual proximity algorithm, and the preconditioned splitting primal-dual proximity algorithm.

Yuchao Tang, Guorong Wu & Chuanxi Zhu. (2019). A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem. Journal of Computational Mathematics. 37 (5). 666-688. doi:10.4208/jcm.1808-m2018-0027
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