一种惯性邻近的Peaceman-Rachford分裂方法  

An inertial proximal Peaceman-Rachford splitting method

作　　者：窦明圆 李慧云[2] 刘新为[1] DOU MingYuan LI HuiYun LIU XinWei

机构地区：[1]河北工业大学理学院,天津300401 [2]河北工业大学控制科学与工程学院,天津300401

出　　处：《中国科学：数学》2017年第2期333-348,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11271107;11671116和91630202);河北省自然科学基金(批准号:A2015202365)资助项目

摘　　要：严格压缩的Peaceman-Rachford(PR)分裂方法是一种收敛速度快于交替方向乘子法的求解线性约束可分离凸优化问题的有效方法.最近提出的半邻近PR分裂方法是严格压缩的PR分裂方法的一种改进方法.基于惯性邻近交替方向乘子法的思想,本文进一步改进了半邻近PR分裂方法,提出了一种惯性邻近PR分裂方法.该方法利用前两次产生的迭代点来产生新的迭代点,可以加速半邻近PR分裂方法的收敛.本文提出的方法具有一般性,它包含严格压缩的PR分裂方法和半邻近PR分裂方法作为特殊情形.在一定的假设下,本文证明了该算法产生的迭代序列的渐进可行性及函数值的收敛性,进而得到了迭代序列的全局收敛性.最后,本文通过数值试验说明了算法的有效性.The strictly contractive Peaceman-Rachford splitting method （SC-PRSM） is a very efficient firstorder approach for linearly constrained separable convex optimization problems, and its convergence rate is faster than that of ADMM. Recently, a semi-proximal Peaceman-Rachford splitting method was proposed to modify SC-PRSM by introducing semi-proximal terms to the subproblems. In this paper, motivated by the idea of the inertial proximal ADMM, we improve the semi-proximal Peaceman-Rachford splitting method by employing an inertial technique to accelerate its convergence, i.e., at each iteration the semi-proximal Peaceman-Rachtord splitting method is applied to a point extrapolated at the current iterate in the direction of last movement. The proposed algorithmic framework is very general so that SC-PRSM and semi-proximal Peaceman-Rachford splitting methods are covered as special cases. Based on the asymptotic feasibility of the iterative sequence and the convergence of the function values, we establish the convergence of the whole sequence generated by the proposed algorithm under very mild assumptions. We demonstrate the efficiency of the inertial extrapolation step via experimental results on least squares semi-definite programming problems and total variation based medical image reconstruction problems.

关 键 词：凸优化 半邻近分裂方法 Peaceman-Rachford分裂方法 惯性邻近点方法 

分 类 号：O221[理学—运筹学与控制论]