非光滑凸优化问题的一个非精确梯度镜面下降算法  

An Inexact Gradient Mirror Descent Algorithm for Non-smooth Convex Optimization

作　　者：徐宇淼 董榕恩 胡清洁 XU YUMIAO;DONG RONGEN;HU QINGJIE(Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,Guilin 541004,China;School of Mathematics and Computational Science,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]广西高校数据分析与计算重点实验室,桂林541004 [2]桂林电子科技大学数学与计算科学学院,桂林541004

出　　处：《应用数学学报》2022年第4期560-577,共18页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(No.11761014,11961011);广西科技基地和人才专项(2021AC06001)资助项目;海南省普通高等学校研究生创新科研课题(Qhyb2021-19).

摘　　要：本文提出一个求解非光滑凸优化问题非精确梯度镜面下降算法.该算法是Allen Zhu 2016年提出求解光滑凸优化问题梯度镜面下降算法的推广,而且该算法允许目标函数中光滑部分梯度计算和非光滑部分邻近算子计算都存在误差,并且在适当条件下分析了该算法函数值序列的0(1/k2)收敛速度,这里k表示迭代数最后关于Lasso问题和Logistic问题的数值结果表明该算法是有效的.In this paper,we present an inexact gradient mirror descent algorithm for nonsmooth convex optimization problem.It is a generalization of gradient mirror descent algorithm for smooth convex optimization problem which is proposed by Allen-Zhu in 2016.At each iteration,it allows the errors of gradient calculations of smooth terms and proximity operator calculations of non-smooth terms in the objective function.Moreover,the convergence Tate of the proposed algorithm is analyzed under mild conditions.Finally,the algorithm is used to solve the lasso and logistic problems,numerical results illustrate the efficiency of the proposed algorithm.

关 键 词：非光滑凸优化 非精确梯度镜面下降算法 收敛速度 

分 类 号：O224[理学—运筹学与控制论]