关于解变分不等方程的投影方法

Projection Methods for Variational Inequalities

本文证明了对于算子T=A′TA,>0,使以初值x_0∈X和步尺λ∈(0,],变分不等方程(x—x~*)′A′T(Ax~*)≥0,xR^n的解x~*∈X可借助于投影方法x_(k+1)=P[x_k-λQ^(-1)A′T(Ax_k)]得到。这里A:R^n→R^m是一个线性映射,且设算子T_2:R^n→R^m是Lipschitz连续和严格单调的,而X是一多面凸集。

This paper proves if T is of the form T=A'TA then a solution x~* ∈XR^n of the variational inequality (x—x~*)'A'T(Ax~*)≥0, x∈X can be obtained iteratively by means of the projection method x_(k+1)=P[x_k-λQ^(-1)A'T(Ax_k)], x_0∈X, provided >0, the stepsize λ∈(0, ]. Here A: R^n→R^m is a Linear mapping, Provided T: R^n→R^m is Lipschitz Continuous and strongly monotone operator, and the set X is polyhedral.