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  

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] 

机构地区：[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

出　　处：《南京师大学报（自然科学版）》2013年第3期1-5,共5页Journal of Nanjing Normal University(Natural Science Edition)

基　　金：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)

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.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 PDFP2 O. 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 PDFP2 O. 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 PDFP20 for unconstrained problem, the convergence and convergence rate analysis can be obtained directly. Finally, we illustrate the efficiency of PDFP2Oc through computerized tomographic reconstruction.

关 键 词：凸约束 可分离凸优化 邻近算子 不动点算法 

分 类 号：O224[理学—运筹学与控制论] O29[理学—数学]