求解单调包含问题的惯性混合非精确邻近点算法  

Inertial Hybrid Inexact Proximal Point Method for Solving Monotone Inclusions Problems

作　　者：何明明 彭建文 HE MINGMING;PENG JIANWEN(School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《应用数学学报》2020年第4期700-727,共28页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金重大项目(11991024);国家自然科学基金面上项目(11171363);重庆市基础科学与前沿技术研究专项重点项目(cstc2015jcyjBX0029)资助.

摘　　要：本文提出了求解单调包含问题的一类新的惯性混合非精确邻近点算法(简记为iHIPPA).在适当的参数假设下,我们证明了求解单调包含问题的iHIPPA所产生点列的弱收敛性,获得了iHIPPA的非渐近收敛率为O(1/√k)及iHIPPA的遍历迭代复杂性为O(1/k).作为应用,我们还建立了求解单调变分包含问题的惯性邻近收缩算法,求解广义变分不等式问题的惯性投影邻近点算法,及求解原始—对偶问题的惯性非精确调比部分逆算法产生点列的收敛性及相应算法的非渐近收敛率及遍历迭代复杂性.本文结果推广和改进了文献中的相应结论.最后,本文应用新的惯性交替方向乘子法用以求解LASSO问题,而且一些初步的试验结果表明了新的算法的优越性.In this paper,we propose a new inertial hybrid inexact proximal point algorithm for solving monotone inclusion problems.Under certain conditions,we are able to establish the weak convergence of the point sequence generated by the algorithm,the O(1/√k)non-asymptotic global convergence rate and the O(1/k)ergodic iterative complexity result of the proposed inertial hybrid inexact proximal point method.As an application,we propose and analyze three special cases:the inertial proximal contraction algorithm for solving monotone inclusions problems with monotone and Lipschitz continuous operators,the inertial projection-proximal point algorithm for solving generalized variational inequalities with monotone operators,and the inertial inexact scaled partial inverse method for solving primal-dual problem with maximal monotone operators.And we obtain the convergence,non-asymptotic global convergence rate and ergodic iterative complexity of the corresponding algorithm under the farmework of the inertial hybrid inexact proximal point algorithm.The results of this paper generalize and improve the corresponding conclusions in the literature.Last,We apply our new inertial ADMM method to LASSO problem,and some preliminary numerical experiments are reported to illustrate the advantages of the new method.

关 键 词：单调包含问题 惯性混合非精确邻近点算法 弱收敛性 非渐近全局收敛率 遍历迭代复杂性 

分 类 号：O221.2[理学—运筹学与控制论]