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

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.