强伪单调变分不等式的线性收敛

The Linear Convergence for Strongly Pseudomonotone Variational Inequalities

在强伪单调假设下,外梯度投影算法具有线性收敛性.鉴于外梯度投影算法需要向非空闭凸集做两次投影,且当闭凸集结构复杂时,投影计算困难.故为了克服这一困难,先对外梯度投影算法进行改进,用次梯度外梯度投影算法求解强伪单调变分不等式,每次迭代过程中只需向闭凸集做一次投影.研究表明,次梯度外梯度投影算法在求解强伪单调变分不等式所产生的迭代序列具有线性收敛性,用次梯度外梯度投影算法求解强伪单调变分不等式一定程度上加速了迭代序列的收敛速度.

Under the assumptions of the strong pseudomonotonicity,the extragradient method for solving the variational inequality is linearly convergent.Since the extragradient method needs to project twice onto the closed convex set,the projection is usually difficult to calculate especially when closed convex set is complex.In order to overcome this shortcoming,one needs modifiy the extragradient method,thus the subgradient extragradient method for solving strongly pseudomonotone variational inequality was proposed.At each iteration,only one projection needs to be executed onto the closed convex set.The research has shown that iterative sequences generated by the subgradient extragradient method for solving the strongly pseudomonotone variational inequality is linearly convergent.By using of the subgradient extragradient method for solving the strongly pseudomonotone variational inequality,the convergence of iterative sequences is accelerated to some extent.