算子分裂法求解一类变分不等式问题的收敛率分析  

Convergence Rate Analysis of An Operator Splitting Method for Solving a Class of Variational Inequality Problems

作　　者：葛志利 蔡邢菊[2] 张欣 Ge Zhili;Cai Xingju;Zhang Xin(Basic Sciences Department,Nanjing Polytechnic Institute,Nanjing 210048,China;School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023,China;School of Arts and Science,Suqian College,Suqian 223800,China)

机构地区：[1]南京科技职业学院基础科学部,江苏南京210048 [2]南京师范大学数学科学学院,江苏南京210023 [3]宿迁学院文理学院,江苏宿迁223800

出　　处：《南京师大学报（自然科学版）》2020年第1期5-12,共8页Journal of Nanjing Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11401315、11871279);江苏省高校自然科学研究面上项目(17KJD110003);江苏省高职院校教师专业带头人高端研修项目;江苏省青蓝工程资助项目.

摘　　要：考虑一类变分不等式问题:寻找x^*∈Ω,满足F(x^*)T(x-x^*)≥0,x∈Ω,其中Ω是R n上的闭凸子集,F=f+g是R n到R n的连续算子,f和g单调但f的表达式未知.针对此类应用较广的问题,本文研究了一种新的算子分裂法.根据已有的收敛性结果,进一步分析了该方法在非遍历意义下O(1/k)和o(1/k)的次线性收敛率,其中k表示迭代步数.最后,通过数值实验展示了算法的有效性.Consider a class of variational inequality problems:finding x^*∈Ω,such that F(x^*)T(x-x^*)≥0,x∈Ω,whereΩR n is nonempty,closed and convex,F=f+g is a continuous mapping from R n to R n,f and g are monotone but f is unknown.We study an operator splitting method for this class of problems with a variety of applications.Based on the previous convergence results,we further analyze the O(1/k)and o(1/k)sublinear convergence rate in non-ergodic sense for this operator splitting method,where k counts the iteration number.Finally,numerical results demonstrate the efficiency of the algorithm.

关 键 词：部分算子未知 单调变分不等式 算子分裂法 次线性收敛率 

分 类 号：O221.4[理学—运筹学与控制论]