交替最小化算法求解强凸函数与弱凸函数和的极小值问题  

Alternating Minimization Algorithm for Minimizing the Sum of Strongly Convex Function and Weakly Convex Function

作　　者：叶明露[1] 陈玉洁 YE Minglu;CHEN Yujie(College of Mathematics and Information,China West Normal University,Nanchong,Sichuan,637002,P.R.China)

机构地区：[1]西华师范大学数学与信息学院,四川南充637002

出　　处：《数学进展》2020年第2期225-233,共9页Advances in Mathematics（China）

基　　金：Supported by NSFC(Nos.11871059,11801455);Sichuan Science and Technology Program(No.2019YFG0299);the Talent Research Fund Project of China West Normal University(No.17YC394)。

摘　　要：交替最小化算法(简称AMA)最早由[SIAM J.Control Optim.,1991,29(1):119-138]提出,并能用于求解强凸函数与凸函数和的极小值问题.本文直接利用AMA算法来求解强凸函数与弱凸函数和的极小值问题.在强凸函数的模大于弱凸函数的模的假设下,我们证明了AMA生成的点列全局收敛到优化问题的解,并且若该优化问题中的某个函数是光滑函数时,AMA所生成的点列的收敛率是线性的.Alternating Minimization Algorithm(AMA for short) is proposed by [Tseng P,Applications of a splitting algorithm to decomposition in convex programming and variational inequalities,SIAM J.Control Optim.,1991,29(1):119-138] and can be used to minimize the sum of a strongly convex function and convex function.In this paper,we extend directly AMA for minimizing the sum of a strongly convex function and a weakly convex function.Under the assumption that the strong convex modulus is larger than the weakly convex modulus,we show that the sequence generated by AMA is globally convergent to the solution of optimization problem.Moreover,if in addition one of underlying functions is smooth,the convergent rate of AMA is linear.

关 键 词：交替最小化算法 弱凸 强凸 凸规划 

分 类 号：O221[理学—运筹学与控制论]