[1]陈培军,黄建国,张小群.基于邻近算子求解带凸集约束可分离凸优化问题的原始对偶不动点算法[J].南京师大学报(自然科学版),2013,36(03):1-5.
　Chen Peijun,Huang Jianguo,Zhang Xiaoqun.A Primal-Dual Fixed Point Algorithm Based on Proximity Operator for Convex Set Constrained Separable Problem[J].Journal of Nanjing Normal University(Natural Science Edition),2013,36(03):1-5.
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基于邻近算子求解带凸集约束可分离凸优化问题的原始对偶不动点算法()
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《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

卷:
第36卷
期数:
2013年03期
页码:
1-5
栏目:
数学
出版日期:
2013-09-30

文章信息/Info

Title:
A Primal-Dual Fixed Point Algorithm Based on Proximity Operator for Convex Set Constrained Separable Problem
作者:
陈培军1; 2; 黄建国1; 张小群1; 3
(1.上海交通大学数学系,科学工程计算教育部重点实验室,上海 200240) (2.太原科技大学数学系,山西 太原 030024) (3.上海交通大学自然科学研究院,上海 200240)
Author(s):
Chen Peijun1; 2; Huang Jianguo1; Zhang Xiaoqun1; 3
(1.Department of Mathematics,MOE-LSC,Shanghai Jiao Tong University,Shanghai 200240,China) (2.Department of Mathematics,Taiyuan University of Science and Technology,Taiyuan 030024,China) (3.Institute of Natural Sciences,Shanghai Jiao Tong University,Shanghai 200240,China)
关键词:
凸约束; 可分离凸优化; 邻近算子; 不动点算法
Keywords:
convex constraint; convex separable minimization; proximity operator; fixed point algorithm
分类号:
O224; O29
摘要:
很多实际问题根据不同的物理背景,解的取值是有一定限制的.本文拟推广PDFP2O算法以求解带闭凸集约束的可分离凸优化问题.通过将闭凸集约束表示成示性函数而加入目标函数中的技巧,适当重组函数,可直接利用PDFP2O算法求解,再利用函数的可分离性,即可得到闭凸集上的基于邻近算子的原始对偶不动点算法(PDFP2OC).因为PDFP2OC本质上就是利用PDFP2O求解与原问题等价的无约束问题,根据PDFP2O的理论结果,可以方便地得到PDFP2OC的收敛性以及收敛速度.最后通过CT重构说明了算法的有效性.
Abstract:
In many real problems,the solutions may have constraints arising from their physical requirements.In this paper,we design an efficient algorithm for solving the separable convex minimization on closed convex set based on the algorithm PDFP2O.Precisely speaking,the constraint can be enforced by adding an indicator function to the objective function,and the function are reformulated and can be solved with PDFP2O.Using the separability of the function with respect to its variables,we thus get a primal-dual fixed point algorithm based on proximity operator on closed convex set(PDFP2OC).Since the algorithm PDFP2OC can be recast as the original PDFP2O for unconstrained problem,the convergence and convergence rate analysis can be obtained directly.Finally,we illustrate the efficiency of PDFP2OC through computerized tomographic reconstruction.

参考文献/References:

[1] Combettes P L,Wajs V R.Signal recovery by proximal forward-backward splitting[J].Multiscale Modeling and Simulation,2005,4(4):1 168-1 200.
[2]Rudin L I,Osher S,Fatemi E.Nonlinear total variation based noise removal algorithms[J].Physica D:Nonlinear Phenomena,1992,60(1/4):259-268.
[3]Chen P,Huang J,Zhang X.A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration[J].Inverse Problems,2013,29(2):025011(1-33).
[4]Micchelli C A,Shen L,Xu Y.Proximity algorithms for image models:denoising[J].Inverse Problems,2011,27(4):045009(1-30).
[5]Goldstein T,Osher S.The split Bregman method for b>1-regularized problems[J].SIAM Journal on Imaging Sciences,2009,2(2):323-343.
[6]Chambolle A,Pock T.A first-order primal-dual algorithm for convex problems with applications to imaging[J].Journal of Mathematical Imaging and Vision,2011,40(1):120-145.
[7]Zhang X,Burger M,Osher S.A unified primal-dual algorithm framework based on Bregman iteration[J].Journal of Scientific Computing,2011,46(1):20-46.
[8]Esser E,Zhang X,Chan T F.A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[J].SIAM Journal on Imaging Sciences,2010,3(4):1 015-1 046.
[9]Avinash C,Malcolm S.Principles of Computerized Tomographic Imaging[M].Philadelphia:SIAM,2001.

备注/Memo

备注/Memo:
收稿日期:2013-03-26.
基金项目:国家自然科学基金(11101277、11171219、11161130004)、上海市教育委员会E-研究院建设计划项目(E03004)、上海浦江人才计划基金(11PJ1405900).
通讯联系人:陈培军,博士,副教授,研究方向:科学计算.E-mail:chenpeijun@sjtu.edu.cn
更新日期/Last Update: 2013-09-30