两分块非凸优化Peaceman-Rachford分裂序列二次规划双步长算法  

A Peaceman-Rachford splitting sequential quadratic programming method with double step-lengths for two-block nonconvex optimization

作　　者：简金宝 张晨[4] 尹江华 Jinbao Jian;Chen Zhang;Jianghua Yin

机构地区：[1]广西民族大学数学与物理学院,南宁530006 [2]广西应用数学中心,南宁530006 [3]广西混杂计算与集成电路分析重点实验室,南宁530006 [4]上海理工大学机械工程学院,上海200093

出　　处：《中国科学：数学》2022年第12期1449-1476,共28页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12171106);广西省自然科学基金(批准号:2020GXNSFDA238017)资助项目。

摘　　要：本文研究大规模两分块非凸约束优化的分解降维算法,提出Peaceman-Rachford(PR)分裂序列二次规划双步长求解方法.本文主要工作和贡献如下:(1)借助PR分裂算法思想将传统二次规划(quadratic programming,QP)子问题的增广Lagrange问题分解为两个小规模QP子问题;(2)通过求解小规模QP产生搜索方向;(3)以增广Lagrange函数为效益函数,沿搜索方向先后进行Armijo线搜索产生双迭代步长,在较弱的条件下保证了算法的全局收敛性、强收敛性和合理的迭代复杂性,克服了Maratos效应;(4)提出乘子新的对称型修正技术;(5)基于一类数学模型和电力系统经济调度模型以及?2正则二分类问题,对算法进行大量中等规模的比较数值实验,验证了算法的有效性.In this paper,the decomposing and dimension-decreasing algorithms for large-scale two-block nonconvex optimization problems are discussed,and a Peaceman-Rachford(PR)splitting sequential quadratic programming(SQP)method with double step-lengths for the discussed problems is proposed.The main work and contributions are as follows.(1)Based on the idea of the PR splitting algorithm,the augmented Lagrangian quadratic programming(QP)in the classical SQP method is decomposed into two small-scale QPs.(2)The improved search directions are obtained by solving the two small-scale QPs.(3)By taking the augmented Lagrangian function as the merit function,Armijo line searches are performed along the two improved directions such that two step-lengths are yielded.Such a line search technique not only guarantees the global and strong convergence as well as reasonable iteration complexity of the method but also overcomes the Maratos effect under weak conditions.(4)A new symmetric update technique for the multiplier associated with the equality constraints is put forward.(5)Via a class of mathematical model and a kind of economic dispatch model of the power system as well as?2 regularized binary classification problems,a large number of medium-scale numerical experiments of the proposed method,compared with three associated algorithms,are carried out.The numerical results show that the proposed method is effective and promising.

关 键 词：两分块非凸优化 Peaceman-Rachford分裂算法 序列二次规划 双步长算法 收敛性 

分 类 号：O224[理学—运筹学与控制论]