非凸不可分优化线性近似Bregman型Peaceman-Rachford分裂算法  

A Linear Approximate Bregman-type Peaceman-Rachford Splitting Method for Nonconvex Nonseparable Optimization

作　　者：刘鹏杰 简金宝 马国栋 许佳伟 Peng Jie LIU;Jin Bao JIAN;Guo Dong MA;Jia Wei XU(College of Mathematics and Physics,Guangxi University for Nationalities,Nanning 530006,P.R.China;School of Mathematics,China University of Mining and Technology,Xuzhou 221l16.P.R.China;College of Mathematics and Information Science,Guangxi University,Nanning 530004,P.R.China)

机构地区：[1]广西民族大学数学与物理学院,南宁530006 [2]中国矿业大学数学学院,徐州221116 [3]广西大学数学与信息科学学院,南宁530006

出　　处：《数学学报（中文版）》2023年第1期75-94,共20页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(12171106);广西自然科学基金资助项目(2020GXNSFDA238017,2018GXNSFFA281007,2018GXNSFAA281099)。

摘　　要：基于Peaceman-Rachford分裂算法,结合线性近似技术和Bregman距离,本文提出一种线性近似Bregman型Peaceman-Rachford分裂算法,用于求解目标函数带不可分结构的线性约束非凸优化问题.在常规假设下,得到算法的全局收敛性.在效益函数满足Kurdyka-Lojasiewicz性质前提下,论证算法的强收敛性.当KurdykaLojasiewicz性质关联函数为特殊结构时,分析并获得算法的收敛率结果.最后,初步数值试验说明算法有数值有效性.Based on the Peaceman-Rachford splitting method,combined with the linear approximate technique and Bregman distance,in this paper,we present a linear approximation Bregman-type Peaceman-Rachford splitting method for solving the nonconvex nonseparable optimization problem with linear constraints.Under the conventional assumptions,we get the global convergence of the proposed algorithm.On the premise that the merit function satisfies the Kurdyka-Lojasiewicz property,the strong convergence of the proposed algorithm is proved.When the associated KurdykaLojasiewicz property function has a special structure,the convergence rate results of the proposed algorithm are analyzed and obtained.Finally,some preliminary numerical results show that the proposed algorithm has numerical validity.

关 键 词：非凸不可分优化 线性近似技术 Peaceman-Rachford分裂算法 Kurdyka-Lojasiewicz性质 收敛率 

分 类 号：O221[理学—运筹学与控制论]