非凸非光滑不可分优化的两个线性邻近Peaceman-Rachford分裂算法  

TWO LINEAR PROXIMAL PEACEMAN-RACHFORD SPLITTING ALGORITHMS FOR NONCONVEX AND NONSMOOTH NONSEPARABLE OPTIMIZATION

作　　者：简金宝 蔡靖民 尹江华 Jian Jinbao;Cai Jingmin;Yin Jianghua(School of Mathematics and Physics,Center for Applied Mathematics of Guangxi,Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis,Guangxi Minzu University,Nanning 530006,China)

机构地区：[1]广西民族大学数学与物理学院,广西应用数学中心,广西混杂计算与集成电路分析重点实验室,南宁530006

出　　处：《计算数学》2023年第4期426-446,共21页Mathematica Numerica Sinica

基　　金：国家自然科学基金(项目任务书编号:12171106);广西自然科学基金项目(项目任务书编号:2023GXNSFBA026029,2020GXNSFDA238017)资助.

摘　　要：本文研究一类非凸非光滑不可分优化.基于Peaceman-Rachford(PR)分裂算法,并结合Armijo线搜索技术及线性正则化技术,提出了两个线性邻近PR分裂算法.利用PR分裂算法思想,将增广拉格朗日法涉及的子问题分解成两个小规模子问题.为便于子问题的求解和使其具有良好的理论性质,对子问题的目标函数中的光滑项作线性化处理,并分别添加必要的正则项.在常规假设下,论证了算法的全局收敛性及迭代复杂性.最后,数值实验结果表明算法是有效的.This work studies a class of nonconvex and nonsmooth nonseparable optimization.Based on the Peaceman-Rachford(PR)splitting algorithm,two linear proximal PR splitting algorithms are proposed by combining the Armijo line search technique and the linear regularization technique.Using the idea of the PR splitting algorithm decomposes the subproblem associated with the augmented Lagrangian method into two small-scale subproblems.In order to facilitate the solution of the just-mentioned subproblems and make them have good theoretical properties,the smooth terms in the objective functions are linearized,and then the regularization terms are added to each subproblem respectively.Under usual conditions,the global convergence and the iteration complexity of the two proposed methods are proved.Finally,numerical experiments show that the two proposed algorithms are effective.

关 键 词：非凸非光滑不可分优化 Peaceman-Rachford分裂算法 线性正则化技术 ARMIJO线搜索 收敛性 

分 类 号：O224[理学—运筹学与控制论]