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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2017, Vol. 34 ›› Issue (6): 609-621.doi: 10.3969/j.issn.1005-3085.2017.06.004

• • 上一篇    下一篇

基于原始对偶分裂方法求解一类约束可分离凸优化问题及其应用

唐玉超1,2,   陈   宝1,   朱传喜1,2,   余   晖3   

  1. 1- 南昌大学数学系，南昌  330031
    2- 南昌大学管理学院，南昌  330031
    3- 南昌大学信息工程学院，南昌  330031
  • 收稿日期:2015-10-21 接受日期:2016-10-18 出版日期:2017-12-15 发布日期:2018-02-15
  • 基金资助:
    国家自然科学基金(11401293; 11461046; 11661056)；中国博士后科学基金(2015M571989)；中国科学院数学与系统科学研究院访问基金(AM201622C04)；江西省自然科学基金(20151BAB211010; 20142BAB211016)；江西省博士后科学基金(2015KY51).

Algorithm Based on Primal-dual Splitting Method for Solving a Class of Constrained Separable Convex Optimization Problem and its Application

TANG Yu-chao1,2,   CHEN Bao1,   ZHU Chuan-xi1,2,   YU Hui3   

  1. 1- Department of Mathematics, Nanchang University, Nanchang 330031
    2- School of Management, Nanchang University, Nanchang 330031
    3- School of Information Engineering, Nanchang University, Nanchang 330031
  • Received:2015-10-21 Accepted:2016-10-18 Online:2017-12-15 Published:2018-02-15
  • Supported by:
    The National Natural Science Foundations of China (11401293; 11461046; 11661056); the China Postdoctoral Science Foundation (2015M571989); the Visiting Scholarship of Institute of Mathematics and Systems Science, Chinese Academy of Sciences (AM201622C04); the Natural Science Foundations of Jiangxi Province (20151BAB211010; 20142BAB211016); the Jiangxi Province Postdoctoral Science Foundation (2015KY51).

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摘要： 本文研究一类具有代表性的约束可分离凸优化模型，其目标函数中的数据误差项满足可微性条件，许多图像恢复和图像重建等问题都可以归结为该模型的求解．为克服现有求解该模型方法的不足，文中首先借助指示函数，将原模型转化为无约束的凸优化模型；然后基于原始对偶分裂方法思想，提出一种新的迭代算法，该算法具有结构简单和参数选取容易的特点，同时证明所提算法的收敛性．最后，为验证算法的有效性，我们将其应用于CT 图像重建问题，数值实验结果表明所提出的算法在重建时间和重建图像质量上优于现有的其他算法．

关键词: 可分凸规划, 原始对偶分裂方法, 图像重建, 邻近算子

Abstract: In this paper, we study a constrained separable convex optimization model, in which the data error term in the objective function is differentiable. Many problems arising in image restoration and image reconstruction are the special cases of this convex optimization problem. In order to overcome the shortcomings of existing methods for solving this model, we transform the original problem into an unconstrained convex optimization problem by using an indicator function. Then, we propose a new iterative algorithm, which is based on the idea of the primal-dual splitting method. The proposed algorithm has a simple structure and is easy for parameter selection. At the same time, we prove the convergence of the new iterative algorithm. Finally, to verify its effectiveness, we apply the algorithm to CT image reconstruction. Numerical experiments show that the proposed algorithm outperforms existing iterative algorithms in terms of reconstruction time and reconstruction image quality.

Key words: separable convex programs, primal-dual splitting method, image reconstruction, proximity operator

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