求解广义互补问题

1 引言 设为一闭凸锥,f是R~n到自身的一映射.广义互补问题,记作GCP(K,f),即找一向量x满足 GCP(K,f) x∈K,f(x)∈且x~Tf(x)=0,(1) 其中,是K的对偶锥(即对任一K中向量x,满足x~Ty≤0的所有y的集合).该问题首先 由Habetler和Price提出.当K=R_+~n(R~n空间的正卦限),此问题就是一般的互补问题.许多作者已经提出了很多求解线性或非线性互补问题的方法.例如:Dafermos,Fukushima,Harker和Price以及其它如参考文献所列.近年来,何针对单调线性变分不等式提出了一些投影收缩算法. Fang在函数是Lipschitz连续及强单调的条件下,在[3]给出一简单的迭代投影法,在[4]中给出一线性化方法去求解广义互补问题(1).在[3]中,他的迭代模式是

SOLVING GENERALIZED COMPLEMENTARITY PROBLEMS

Given a generalized complementarity problem, Fang introduces an itera-tive projection method to solve it under the conditions that the function is Lipschitz continuous and strongly monotone. Fang's method has to make a projection on the cone in each iteration. In this note, we propose to solve this problem by solving an e-quivalent large variational inequality. The projection and constraction methods, re-placing the projection on a cone by the projection on an orthant, solve the variational inequality provided that the function is continuous and monotone (not necessarily strongly monotone).