一般单调变分不等式的近似邻近外梯度算法

An Approximate Proximal Extragradient Algorithm for General Monotone Variational Inequalities

近似邻近点算法是求解单调变分不等式的一个有效方法,该算法通过解决一系列强单调子问题,产生近似邻近点序列来逼近变分不等式的解,而外梯度算法则通过每次迭代中增加一个投影来克服一般投影算法限制太强的缺点,但它们均未能改变迭代步骤中不规则闭凸区域上投影难计算的问题.于是,本文结合外梯度算法的迭代格式,构造包含原投影区域的半空间,将投影建立在半空间上,简化了投影的求解过程,并对新的邻近点序列作相应限制,使得改进的算法具有较好的收敛性.

Inexact proximal point algorithm is an effective method for solving monotone variationM inequalities, which generates the proximal point sequence by solving a series of strong monotonous subproblems to approach to the solution of variational inequalities, and extragradient algorithm adds a projection at each iteration to avoid the strong restriction. But they still fail to change the phenomenon that it is hard to accurately calculate the projection on the irregular convex domain. The paper constructs a subgradient half space depended on the extragradient algorithm which contains the projection domain. It simplifies the process of projection, and the corresponding restrictions are added on the proximal point sequence to make the improved algorithm numerically perform well.