摘要
近年来,关于多个凸函数和的优化问题受到广泛关注.本文研究三个凸函数和f(x)+g(x)+h(Bx)的一类凸优化问题,其中f (x)可微且具有Lipschitz连续梯度, g(x)和h(x)是正则下半连续简单凸函数, B是一个有界线性算子.此类优化问题在信号恢复和图像处理等实际问题中有着广泛的应用.为充分利用问题中的可微函数,本文基于向前向后分裂算法和三算子分裂算法框架,建立若干具有内外迭代形式的算法.在推导迭代算法的过程中,本文提出基于对偶和原始对偶方法求解函数g+h?B和h?B的邻近算子.在对参数一定假设条件下,本文证明所提出的迭代算法收敛性.通过与Condat和Vu算法、原始对偶不动点(primal-dual?xed point, PDFP)算法和原始对偶三算子(primal-dual three-operator, PD3O)算法比较,建立三种迭代算法与本文提出的迭代算法之间的联系.最后,通过对融合Lasso问题、约束全变分正则化问题和低秩全变分图像超分辨率重建问题实施一系列数值实验,验证所提出的迭代算法的有效性.
In recent years,the optimization problem of the sum of several convex functions has received much attention.In this paper,we consider solving a class of convex optimization problem which minimizes the sum of three convex functions f(x)+g(x)+h(Bx),where f(x)is differentiable with a Lipschitz continuous gradient,g(x)and h(x)have a closed-form expression of their proximity operators and B is a bounded linear operator.Such optimization problems have wide application in signal recovery and image processing.To make full use of the differentiable function in the problem,we propose several inner-outer iterative algorithms based on the forward-backward splitting algorithm and the three-operator splitting algorithm frameworks.In the process of deriving the iterative algorithms,we use dual and primal-dual methods to solve the proximity operator of the functions g+h?B and h?B.Under mild assumptions on the parameters,we prove the convergence of the proposed iterative algorithms.By comparing with the Condat and Vu algorithm,the primal-dual fixed point(PDFP)algorithm and the primal-dual three-operator(PD3 O)algorithm,we establish the connection between these algorithms with ours.Numerical experiments applied to the fused Lasso problem,the constrained total variation regularization problem and the low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms.
作者
唐玉超
吴国荣
朱传喜
Yuchao Tang;Guorong Wu;Chuanxi Zhu
出处
《中国科学：数学》
CSCD
北大核心
2019年第5期831-858,共28页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:11661056
11401293和11771198)资助项目
关键词
向前向后分裂算法
三算子分裂算法
对偶
原始对偶
全变分
forward-backward splitting algorithm
three operator splitting algorithm
dual
primal-dual
total variation
作者简介
唐玉超,E-mail:hhaaoo1331@163.com;吴国荣,E-mail:guorong wu@med.unc.edu;通信作者:朱传喜,E-mail:chuanxizhu@126.com
